REPRESENTATION TYPE OF COMMUTATIVE NOETHERIAN RINGS II: TAMENESS

LEE KLINGLER AND LAWRENCE S. LEVY

Abstract. This is the second of a series of three papers showing that every commutative noetherian ring has either wild or tame representation type (with a possible exception involving characteristic 2), and describing the finitely generated modules when the type is tame. The first paper established the wildness of a broad class of complete local rings. This second paper establishes the tameness of the remaining complete local rings (with the exception noted above), and of some noncommutative local rings, giving an explicit description of their finitely generated modules. The third paper will deal with the non-complete and non-local situations, with emphasis on local-versus-global behavior and direct-sum behavior of modules over tame rings.

1. Introduction

Let (Λ, 𝔪, k) be a complete local (commutative, noetherian) ring. In [KL2] we showed that Λ has wild representation type unless Λ is a Klein ring or a homomorphic image of a split or unsplit Dedekind-like ring. The bulk of the present paper is devoted to showing that Dedekind-like rings are tame — with a possible exception involving charactersitic 2. We accomplish this by explicitly describing their finitely generated modules. Then we describe briefly how to use this to give a description of the finitely generated modules over Klein rings and proper homomorphic images of Dedekind-like rings.

It turns out that — with essentially no additional effort — our methods yield all finitely gererated modules over some noncommutative rings too, most notably the (naive) integer quaternions ℤ[i, j, k]. In this paper we handle the 2-adic completion of this, leaving the full non-local case — as for commutative rings — to the third paper of this series.

The critical fact that allows some noncommutative content in our results is that our methods require commutativity only of the residue fields of Λ, rather than of Λ itself. However, in order to preserve readability of the commutative case, in this series of papers, we resist the temptation to engage in more substantial noncommutative generalization here. We begin by defining the rings with which the bulk of this paper deals.

Notation 1.1. By a residue-commutative local ring (Λ, 𝔪, k) we mean a left and right noetherian ring Λ with unique maximal left ideal 𝔪 such that 𝔪 is also a maximal right ideal and the residue ring k = Λ/𝔪 is commutative (and hence a field). We usually refer to such a ring merely as a local ring, occasionally inserting “residue-commutative” in parentheses, as a reminder.

Date: May 2, 1999.

1991 Mathematics Subject Classification. 13E05, 16G60.

Key words and phrases. Tame representation type.

Levy’s research was partially supported by an NSA grant.

We define a (residue-commutative) DVR (discrete valuation ring) V to be an integral domain containing a non-unit π ≠ 0 such that every nonzero one-sided ideal has the form V π^{e} = π^{e}V for some unique positive integer e, and such that the ring V/V π is commutative (hence a field). Note that V is not a division ring, and (V, V π, V/V π) is a local ring.

Now consider a commutative diagram of module-finite algebras over some commutative local (noetherian!) ring R of Krull dimension 1:

Λ ⊆ Γ
↓ ↓ρ  ^{↓ ↓}ρ (ker = 𝔪 ⊆ rad Γ)
k ⊆ Γ̄ (k = field)

(1.1.1)

where the right-hand map ρ is a surjective ring homomorphism, the left-hand map ρ is the restriction of this map to Λ, ρ(Λ) = k is a field, and Λ is the pullback of this diagram, that is,

Λ = {x ∈ Γ | ρ(x) ∈ k} (1.1.2)

In this situation, 𝔪 = ker(ρ) ⊂ Λ; and Λ and Γ are (left and right) noetherian rings. Moreover, 𝔪 = rad(Λ), and therefore (Λ, 𝔪, k) is a local ring.

The following two specializations of this situation are the principal rings considered in this paper.

We call (Λ, 𝔪, k) a (residue-commutative) unsplit Dedekind-like ring if (Γ, 𝔪, Γ̄) is a DVR and Γ̄ = F is a field such that dim_{k}(F) = 2. For the rest of this paper — when Λ is unsplit Dedekind-like — we fix an element π ∈ Γ such that Γπ = 𝔪 = πΓ, and call π the distinguished generator of 𝔪.

Keeping the notation of diagram (1.1.1), we call (Λ, 𝔪, k) a (residue-commutative) strictly split Dedekind-like ring if Γ = Γ_{1} ⊕ Γ_{2}, where (Γ_{1}, 𝔪_{1}, k) and (Γ_{2}, 𝔪_{2}, k) are DVRs with the common residue field k, and 𝔪 = 𝔪_{1} ⊕ 𝔪_{2}. In this situation, Γ̄ = k ⊕ k, and we arrange the notation so that we can identify k with the set of elements (x, x) ∈ k ⊕ k. Thus we again have dim_{k}(Γ̄) = 2. For the rest of this paper, we fix elements π_{i} ∈ Γ_{i} (i = 1, 2) such that Γ_{i}π_{i} = 𝔪_{i} = π_{i}Γ_{i}, and we call π_{i} the distinguished generator of 𝔪_{i}.

In both the unsplit and strictly split case we call Γ the normalization of Λ. The next result justifies this definition in our slightly noncommutative setting.

Theorem 1.2. The normalization Γ of any unsplit or strictly split Dedekind-like ring Λ is the unique ring, strictly between Λ and its Goldie quotient ring Δ, that is finitely generated as a (left) Λ-module.

Proof. We first claim that

Γ = Λ + Λγ (∀γ ∈ Γ − Λ) (1.2.1)

It suffices to verify (1.2.1) modulo the common ideal 𝔪 of Λ and Γ, whence it follows immediately from the fact that Γ/𝔪 is a 2-dimensional vector space over the field k = Λ/𝔪. For the rest of the proof we consider the unsplit and strictly split cases separately.

First let Λ be unsplit Dedekind-like. Since Λ is a noetherian domain, its Goldie quotient ring Δ is a division ring.

Claim 1. Γ is a maximal subring of Δ. Each nonzero element of the (residuecommutative) DVR Γ has the form uπ^{n} = π^{n}v for some u, v in Γ^{×}, the group of units of Γ, and n ≥ 0. Therefore every nonzero element of Δ has this same form, except that n can be negative as well as ≥ 0. Any ring Υ such that Γ ⊂ Υ ⊆ Δ therefore contains an element of the form uπ^{−n} with n > 0. Since Υ contains Γ it also contains π^{−1} = u^{−1}(uπ^{−n})π^{n−1}, and therefore contains every element of the form vπ^{−m} with v ∈ Γ^{×} and m any integer. Thus Υ = Δ, completing the proof of the claim.

We complete the unsplit case by proving the following assertion that — in view of (1.2.1) — is stronger than needed.

Claim 2. Γ is the unique ring strictly between Λ and Δ.

Let Υ be such a ring. Since Δ is the quotient division ring of Γ, any element of Υ has the form uπ^{n} with u ∈ Γ^{×}. Choose uπ^{n} ∈ Υ − Λ.

Note that n ≤ 0. For if not, we have uπ^{n} ∈ Γπ = 𝔪 ⊆ Λ. We now consider two cases.

Case 1: u ∉ Λ. Then since π ∈ Λ ⊂ Υ we have u = uπ^{n}π^{|n|} ∈ Γ ∩ Υ, and hence also u ∈ Γ − Λ. Therefore Υ ⊇ Λ + Λu, which equals Γ by (1.2.1). Claim 1 then shows that Υ = Γ, as desired.

Case 2: u ∈ Λ. Since u ∈ Γ^{×}, u is not in the maximal ideal 𝔪 of Γ. Since 𝔪 is also the maximal ideal of the local ring Λ, we therefore have u ∈ Λ^{×}. But then π^{n} = u^{−1}(uπ^{n}) ∈ Υ. Recall that n ≤ 0. We cannot have n = 0, because then uπ^{n} ∈ Λ. Thus π^{−1} = π^{n}π^{|n|−1} ∈ Υ, and hence Υ contains all negative powers of π. It now suffices to obtain the contradiction Υ = Δ. Choose any nonzero element vπ^{m} ∈ Δ (v ∈ Γ^{×}). If m > 0, then vπ^{m} ∈ Γπ = 𝔪 ⊆ Λ ⊆ Υ. The remaining situation is m ≤ 0, in which case vπ^{m} = (vπ)π^{m−1} ∈ ΛΥ = Υ. This completes our contradiction, hence the proof of the theorem in the unsplit case.

Next let Λ be strictly split Dedekind-like. Then Γ = Γ_{1} ⊕ Γ_{2} where each Γ_{i} is a DVR; and hence Δ = Δ_{1} ⊕ Δ_{2} where each Δ_{i} is the quotient division ring of Γ_{i}. Since Γ_{i} is a DVR, it is a maximal subring of Δ_{i} by Claim 1 above. Let Υ be any ring strictly between Λ and Δ that is a finitely generated (left) Λ-module. Then each projection Υ_{i} of Υ in Δ_{i} is again a finitely generated Λ-module containing the projection Γ_{i} of Λ in Δ_{i}. We claim that Υ_{i} = Γ_{i}.

By maximality of Γ_{i} in Δ_{i}, the only other possibility is that Υ_{i} = Δ_{i}. But since Δ_{i} is a division ring we have 𝔪Δ_{i} = Δ_{i}, and so Nakayama’s Lemma implies that ΛΔ_{i} is not finitely generated, a contradiction.

Since each Υ_{i} = Γ_{i} and Γ = Γ_{1} ⊕ Γ_{2}, we have Λ ⊂ Υ ⊆ Γ. But since Γ/Λ is a simple Λ-module — in fact a 1-dimensional k-vector space by diagram (1.1.1) and the fact that Γ̄ is a 2-dimensional k-vector space — this implies that Υ = Γ, completing the proof.

In the commutative case our definition of “unsplit Dedekind-like” agrees with the definition given in [KL2]. For commutative complete local rings Λ, the definition of “strictly split Dedekind-like” becomes the same as the definition of “split Dedekindlike” given in [KL2]. In the incomplete local case the latter is more general than the former (because idempotents cannot necessarily be lifted modulo the radical).

The reason for this slight refinement of terminology is that we do not require completeness in the present paper. The only property we need, that completeness would provide — the Krull-Schmidt Theorem — is already a consequence of our refined definitions [Lemma 1.3]. Not having to worry about the complications of completions will simplify a few of our proofs.

Lemma 1.3. The Krull-Schmidt Theorem holds for finitely generated modules over unsplit and strictly split Dedekind-like rings.

Proof. Note that Λ has no nonzero nilpotent ideals, because Λ is contained in the direct sum of one or two integral domains. We may therefore assume that the local ring R over which Λ is an algebra has no nilpotent elements [LO, Notation 2.1]. Therefore the total quotient ring R_{Q} of R is the localization that inverts all elements of R that lie outside of every minimal prime ideal.

The consequence of this that we need is that — in the terminology of [LO, p. 3391] — Γ is a maximal R-order in Λ_{Q}. Also, Λ ≠ Γ, and Λ is an indecomposable ring. Under these circumstances, a special case of [LO, Theorem 1.1 and Remark 2.2] is that Λ satisfies the Krull-Schmidt Theorem for finitely generated modules if and only if primitive idempotents of Γ remain primitive in Γ/ rad(Γ); and this last condition is clearly satisfied by unsplit and completely split Dedekind-like rings.

The main results of this paper give an explicit description of the isomorphism classes of finitely generated Λ-modules and considerable information about the morphisms in this category — when Λ is Dedekind-like — except if Λ is unsplit Dedekind-like and the 2-dimensional field extension k ⊂ F is inseparable. In this situation, we do not know whether Λ is tame or wild or neither.

Our proof of tameness of Λ occupies sections 2–8. Moreover the statements of our main theorems (but not the proofs) in §7 and §8 are designed so that they can be read immediately after this introduction. In §9 we show how to apply the previous results to obtain the structure of finitely generated modules over Klein rings and over homomorphic images of Dedekind-like rings. The paper ends with some examples, in §10. Because of the length and complexity of our tameness proof, we outline the basic steps below.

In §2 we introduce the basic abstract concept that allows us to deal with module structure over rings that are not algebras over fields. Assuming Γ-modules to be known, we define a “best approximation” to Λ-modules and morphisms by means of Γ-modules and morphisms, calling this approximation a “separated cover”. The basic difficulty here — a difficulty that does not arise when studying only lattices — is that not all Λ-modules are contained in Γ-modules. It turns out to be a triviality that separated covers of finitely generated Λ-modules exist. The main result of this section is that separated covers of Λ-modules have an “almost functorial” property: every map f: M′ → M of Λ-modules can be lifted to a map θ: S′ → S between their separated covers. Moreover, if f is one-to-one or onto, so is θ.

In this section, Λ is a much more general ring than in the rest of the paper. Λ is still the pullback of a diagram of the form (1.1.1), but we do not require the noetherian rings Λ and Γ to be module-finite over a commutative ring. Also, we allow Λ̄ to be any semisimple artinian ring, and we do not require that ker ρ ⊆ rad Γ. In addition we only require ker Γ to be a left (rather than 2-sided) conductor ideal. In fact, such additional assumptions would not make anything easier. In the present paper we apply the results of this section to the situation in which Λ is split or unsplit Dedekind-like and Γ is its normalization, and to homomorphic images of such rings Λ. The existence and uniqueness of separated covers — when Λ is not local and ker ρ ⊈ rad Γ — will be crucial in the third paper of this series.

Earlier versions of the results in this section were used to find all finitely generated modules over over a class of rings that includes commutative split Dedekindlike rings [L1, L2, L3] and the integral group ring ℤG, where G is a nonabelian group of order pq (p, q distince primes) [K]. These earlier versions, however, do not apply to unsplit Dedekind-like rings. But the basic idea of separated covers (called “separated representations” in this earlier work) applies to nonfinitely generated modules as well as finitely generated ones in much of this previous work.

In §3, we return to (residue-commutative) unsplit Dedekind-like rings. We use separated covers to reformulate the problem of classifying isomorphism classes of finitely generated Λ-modules as a matrix problem involving k and F, the residue fields of Λ and its normalization Γ, respectively. In this matrix problem, each Λ-module is represented by a pair of matrices (A, B) with entries in F; and two modules turn out to be isomorphic if and only if their corresponding matrix pairs can be transformed into each other by means of left multiplication by matrices in GL(m, k) and right multiplication by matrices in a certain subgroup of GL(n, F). A precise statement of these matrix operations is the last result of this section.

In §4 we do the analogous reduction to a matrix problem in the strictly split case. Here we deal with matrix 4-tuples (A_{1}, A_{2}, B_{1}, B_{2}) with entries in the field k, instead of pairs as in §3, and the matrix operations are more complicated. But the matrix problem is easier to solve because all matrix operations involve the single field k.

In §5, we recall the solution of this “strictly split” matrix problem, from [KL1]. The results of these two sections were previously obtained in [L1, L3], so we omit most of the details, concentrating on making the terminology and notation consistent with that of the present paper.

In §6, we solve the matrix problem in the unsplit case. When Λ is a k-algebra, we can tensor it by F. This converts the k-F matrix problem, in the unsplit case, to the split matrix problem over F alone. It can then be solved by using the main result in the split case — with F in place of k. The details of this change of scalars argument require our hypothesis that F be separable k.

Of course, one of the main purposes of this series of papers is to not require that Λ be a k-algebra. To deal with this more general situation, we take the unsplit k-F matrix problem out of context, allowing us to reinterpret the problem as a problem about modules over k-algebras. We then solve the problem by reducing it to the split case, getting exactly the same answer as if Λ had been a k-algebra. The main results of this section give the answer to this k-F matrix problem.

In §7 and §8 we use the solution of the matrix problems given in §6 and §5, respectively, to give our final description of Λ-module structure in the unsplit and strictly split cases, respectively. As mentioned earlier, it is possible to read the results of these two sections (but not the proofs) independently of the intervening sections. In fact, we encourage readers to do this, viewing these statements of results as a continuation of this introduction.

In §9, we show that, if Λ is a Klein ring, then Λ is quasi-Frobenius, with a unique minimal ideal, modulo which Λ is a homomorphic image of a split Dedekindlike ring. Thus, it follows that Klein rings must be tame. We also show how to apply the previous results (about modules over Dedekind-like rings) to modules over homomorphic images of Dedekind-like rings. Although these descriptions follow from the previous results, some work is required: our description of Λ-modules involves both Λ and Γ; and using this to describe Λ/I-modules is complicated by the fact that the ideal I of Λ might not be an ideal of Γ.

In §10, we establish three new examples of Dedekind-like rings: the quadratic order ℤ[\sqrt{n}] for every square-free integer n, the ring of integer quaternions ℤ[i, j, k], and all subrings of square-free index in ℤ^{n}, where n is any positive integer. (More precisely, all completions of these rings are either maximal orders or Dedekind-like.)

Remark 1.4 (Typographical error in [L1]). The structure of Λ-modules of finite length, in the strictly split case, was worked out in [L1]. Unfortulately, a nonrepeatedness condition was omitted when block cycle indecomposable modules were discussed in the introduction to that paper, [L1, p. 68]. However, the theorem is correctly stated in the paper itself. (See [L1, Theorem 8.2 and the preceeding paragraph].) This omission is doubly unfortunate because it was also made in [NR], although subsequently corrected in [NRSB].

Since the rings Λ considered in the present paper vary from section to section, each section begins by stating what class of rings Λ can represent. When the setting is not left-right symmetric, “module” means “left module” unless otherwise stated.

2. Separated Covers and Almost Functorial Property

Notation and Definitions 2.1. The situation considered in this section is more general than in the rest of this paper. Consider the following commutative diagram

Λ ⊆ Γ (_{Γ}Γ noetherian)
↓ ↓ρ  ^{↓ ↓}ρ (ker = C)
Λ̄ ⊆ Γ̄ (Λ̄ semisimple artinian, Λ̄Γ̄ finitely generated)

(2.1.1)

where (2.1.2) (i) Λ and Γ are rings, with Γ left noetherian,

(ii) ρ: Γ → → Γ̄ is a surjective left Γ-module homomorphism that becomes a ring homomorphism when restricted to Λ, (and hence its kernel, C, is a 2-sided ideal of Λ but a left ideal of Γ),

(iii) ρ(Λ) = Λ̄,

(iv) the ring Λ̄ is semisimple artinian,

(v) the Λ̄-module Γ̄ is finitely generated, and

(vi) Λ = {x ∈ Γ | ρ(x) ∈ Λ̄} is the pullback of this diagram.

The above notation remains fixed throughout this section. We call diagram (2.1.1) a left conductor square for Λ and Γ, and we call C a left conductor ideal for Λ and Γ. We note:

(2.1.3) The ring Λ is left noetherian.

To see this, first note that_{Λ}Γ is finitely generated since_{Λ}(Γ̄/Λ̄) is finitely generated. Let L be any left ideal of Λ. Since _{Γ}Γ is noetherian, _{Γ}(CL) is finitely generated. Since _{Λ}Γ is finitely generated, so therefore is _{Λ}(CL). Thus it suffices to show that Λ(L/CL) is finitely generated. Since C is a 2-sided ideal of Λ, this last Λ-module is a module over the artinian ring Λ̄ = Λ/C; and it is a submodule of the finitely generated Λ̄-module (ΓL)/(CL), hence it is itself finitely generated.

The purpose of this section is to show how all finitely generated left Λ-modules and their homomorphisms can built from Γ-modules and homomorphisms. In the situations in which we apply these results, C is a 2-sided ideal of Γ. But the main results might have applications elsewhere; for example, Λ can be the idealizer of any semimaximal left ideal C of Γ.

Since the hypotheses on Λ are not left-right symmetric, the term module means “left module” unless otherwise stated. We include some reminders of this near the beginning of the section.

We call a (left) Λ-module S separated — Γ-separated if additional precision is required — if S is a Λ-submodule of some left Γ-module, say X. In this situation ΓS denotes the Γ-submodule of X generated by S. Unless X is specified, ΓS is not well-defined, not even up to isomorphism [Example 4.2]. However, there is always a canonical choice for ΓS, namely Γ ⊗_{Λ} S. This follows from: (2.1.4) The Λ-module S is separated (if and) only if the natural map S → Γ ⊗_{Λ} S

is one-to-one. To prove this, let X be any Γ-module containing S. Then the we can consider the composite map S → Γ ⊗ S → X, where the second arrow denotes the map ω ⊗ s → ωs. The nontrivial half of (2.1.4) holds since the composite map is the identity on S.

The following simple properties hold. (2.1.5) Let S, S′ be separated Λ-modules, regarded as Λ-submodules of ΓS = Γ ⊗ S

and ΓS′ = Γ ⊗ S′, respectively. Then

(i) Every Λ-homomorphism f: S → S′ can be uniquely extended to a Γhomomorphism ΓS → ΓS′ (namely to 1 ⊗ f). (ii) If B is any left ideal of Γ contained in Λ, then BS is a Γ-submodule of S.

Caution. Statement (ii) becomes false, without the hypothesis that S is separated. The point is that we have γ(bs) = (γb)s provided that both sides are defined, but unless S is contained in some Γ-module, the left-hand side is undefined. For a simple example of a non-separated module, see Example 2.9.

We define a separated cover of a (left) Λ-module M to be a surjective Λ-module homomorphism ϕ: S → → M in which S is separated and “as close as possible to M” in the following sense. If S → S′ → → M is any factorization of ϕ, and S′ is a separated Λ-module, then the map S → S′ is one-to-one (and therefore passing from S to S′ gets no closer to M).

The starting point of our theory is the following triviality, which uses the fact that Λ is left noetherian.

Proposition 2.2. Every finitely generated (left) Λ-module has a separated cover.

Proof. Let ϕ′: F → → M be a homomorphism of any finitely generated separated (e.g. free) Λ-module onto the given Λ-module M.

Since the Λ-module F is noetherian, it has a submodule H that is maximal with respect to the property that H ⊆ ker(ϕ′) and F/H is a separated Λ-module. Then the Λ-module homomorphism ϕ: S = F/H → → M is easily seen to be a separated cover of M.

Lemma 2.3. Let ϕ: S → → M be a Λ-module surjection, with S separated. Then the following two conditions are equivalent.

(i) ϕ is a separated cover of M. (ii) (a) ker(ϕ) has no nonzero Γ-submodules [and hence C ker(ϕ) = 0, so that

ker(ϕ) is canonically a Λ̄-module]; and (b) ker(ϕ) ⊆ CS.

Proof. Let X = Γ ⊗ S, which contains S by (2.1.4).

(ii) ⇒ (i). Consider any factorization ϕ: S →^{θ} S′ → → M with S′ separated, and suppose that ker(θ) contains a nonzero element s. Then s ∈ ker(ϕ) ⊆ CS, so there is an expression s = ∑_{i} c_{i}s_{i} with each c_{i} ∈ C and s_{i} ∈ S.

On the other hand, ker(ϕ) contains no nonzero Γ-submodules. Therefore, for some ω ∈ Γ, the element ωs of X is not an element of ker(ϕ). But ωs = ∑_{i}(ωc_{i})s_{i}, an element of S since C is a left ideal of Γ contained in Λ. Therefore ϕ(ωs) is defined, and is nonzero since ωs ∉ ker(ϕ). It follows that θ(ωs) ≠ 0. However, θ can be viewed as a Γ-homomorpism ΓS → ΓS′ by (2.1.5). This yields the contradiction 0 ≠ θ(ωs) = ωθ(s) = 0, proving (i).

(i) ⇒ (ii)(a). Let H be any Γ-submodule of ker(ϕ). Then there is a factorization ϕ: S → → S/H → → M. S/H is a separated Λ-module, since X/H is a Γ-module containing S/H. Hence, by the definition of separated cover, the map S → → S/H must be one-to-one; that is, H = 0 as desired.

Since C ker(ϕ) is a Γ-submodule of ker(ϕ), the previous paragraph shows that C ker(ϕ) = 0, as claimed in (ii)(a).

Note that we have not yet used the standing hypothesis that the ring Λ̄ is semisimple. The consequence of the semisimplicity of Λ̄ that we need below is that every Λ̄-module is semisimple.

(i) ⇒ (ii)(b). Suppose, by way of contradiction, that ker(ϕ) ⊈ CS. Since C ker(ϕ) = 0, ker(ϕ) is a Λ̄-module and hence is a semisimple Λ-module. Therefore, it has a simple submodule U that is not contained in CS. Let ν be the natural homomorphism of S onto S/CS. Since the Λ-module S/CS is annihilated by C, it is a Λ̄-module and hence is semisimple. Therefore there is a projection map π: S/CS → → ν(U). Moreover, since U is simple and not contained in CS, ν is one-to-one on U. Therefore ν^{−1}: ν(U) → U is a well-defined Λ-module map. The composition of the maps

S ⟶^{ν} S/CS ⟶^{π} ν(U) ⟶^{ν^{−1}} U

is a map S → U that equals the identity on U. Therefore U is a direct summand of S, say S = U ⊕T. Then S/U ≅ T is again a separated Λ-module. Since U ⊆ ker(ϕ) there is a factorization ϕ: S → → S/U → → M. Since ϕ is a separated cover, we obtain the contradiction U = 0, proving that statement (ii)(b) holds.

Lemma 2.4. Let ϕ: S → → M be a separated cover. Then ϕ is a minimal epimorphism (i.e., if T is a Λ-submodule of S such that ϕ(T) = M, then T = S).

Proof. Let T be as above. Then T + ker(ϕ) = S. Since ker(ϕ) is a Λ̄-module [Lemma 2.3], it is semisimple. Therefore we have ker(ϕ) = (T ∩ ker(ϕ)) ⊕ K for some submodule K, and therefore T ⊕ K = S. This yields a factorization ϕ: S → → S/K ≅ T → → M with S/K ≅ T separated, showing that K = 0 and therefore T = S.

Proposition 2.5. If ϕ: S → → M is a separated cover and M is finitely generated, then so is S.

Proof. Choose pre-images s_{1}, …, s_{n} ∈ S of some finite set of generators of M. Then ϕ(∑_{i} Λs_{i}) = M. Since ϕ is a minimal epimorphism we have ∑_{i} Λs_{i} = S, as desired.

Theorem 2.6 (Almost functorial property). Let f: N → M be a Λ-module homomorphism, and let ϕ′, ϕ be separated covers. Then f can be lifted to a Λhomomorphism θ such that the following diagram commutes.

\begin{matrix}S′ & −−→^{θ} & S \\ \begin{matrix}↓ & ↓ \\ ϕ′ \end{matrix} & & ^{↓ ↓}ϕ \\ N & ⟶^{f} & M \end{matrix} (2.6.1)

If f is one-to-one or onto, then any such θ has the same property.

Proof. Since S′/CS′ is a module over the semisimple artinian ring Λ̄, it has a decomposition S′/CS′ = ⊕_{i} Λx̄′_{i} where each Λx̄′_{i} ≅ Λ̄ē_{i} for some idempotent element ē_{i} ∈ Λ̄. We can choose each x̄′_{i} such that

(2.6.2)  ē_{i} → x̄′_{i} under the isomorphism Λ̄ē_{i} ≅ Λ̄x̄′_{i}

Choose a pre-image e_{i} ∈ Λ of each ē_{i} and x′_{i} ∈ S′ of x̄′_{i}. Let {c_{j}s′_{j}} (with c_{j} ∈ C) be a set of generators of the Λ-module CS′. Then choose elements x_{i}, s_{j} related to the previously chosen elements as follows.

fϕ′(x′_{i}) = ϕ(x_{i}) and fϕ′(s′_{j}) = ϕ(s_{j}) (2.6.3)

Since ē_{i} is idempotent we have ē_{i}x̄′_{i} = x̄′_{i} and therefore the elements e_{i}x′_{i} together with c_{j}s′_{j} generate S′. If we can define a Λ-homomorphism θ that sends each e_{i}x′_{i} → e_{i}x_{i} and c_{j}s′_{j} → c_{j}s_{j} then we have lifted f. Let

y′ = ∑_{i} λ_{i}e_{i}x′_{i} + ∑_{j} δ_{j}c_{j}s′_{j} and y = ∑_{i} λ_{i}e_{i}x_{i} + ∑_{j} δ_{j}c_{j}s_{j} (2.6.4)

where the sums are finite and each λ, δ ∈ Λ. To see that θ is well-defined it suffices to show that y′ = 0 ⇒ y = 0. Since ker(ϕ) contains no nonzero Γ-submodules, by Lemma 2.3, it suffices to show that Γy ⊆ ker(ϕ). Thus we suppose that y′ = 0 and choose ω ∈ Γ. Our objective to show that ϕ(ωy) is defined and equal to zero.

We must be careful about two things: elements of N and M cannot be multiplied by ω, if these Λ-modules are not contained in Γ-modules; and consequently the Λhomomorphisms ϕ′, ϕ need not be extendable to Γ-homomorphisms.

Reading the statement y′ = 0 modulo CS′ and using the expression in (2.6.4) for y′ yields 0 = ∑_{i} λ_{i}ē_{i}x̄′_{i}. Directness of the sum ⊕_{i}Λx̄′_{i} then yields λ_{i}ē_{i}x̄′_{i} = 0 for all i. Hence, by the isomorphism in (2.6.2), we have λ̄_{i}ē_{i} = 0. Therefore each λ_{i}e_{i} ∈ C. Since C is a left ideal of Γ contained in Λ, we have ωλ_{i}e_{i}x′_{i} ∈ CS′ ⊆ S′, and therefore ϕ′(ωλ_{i}e_{i}x′_{i}) is defined. Similarly ϕ′(ωδ_{j}c_{j}s′_{j}) is defined. Therefore y′ = 0 yields

0 = fϕ′(ωy′) = ∑_{i} fϕ′(ωλ_{i}e_{i}x′_{i}) + ∑_{j} fϕ′(δ_{j}c_{j}s′_{j}) (2.6.5)

Since each ωλ_{i}e_{i} ∈ C ⊆ Λ, relations (2.6.3) show that

fϕ′(ωλ_{i}e_{i}x′_{i}) = ωλ_{i}e_{i}fϕ′(x′_{i}) = ωλ_{i}e_{i}ϕ(x_{i}) = ϕ(ωλ_{i}e_{i}x_{i})

Similarly we have fϕ′(ωδ_{j}c_{j}s′_{j}) = ϕ(ωδ_{j}c_{j}s_{j}). Making these replacements in (2.6.5) now shows that ϕ(ωy) = 0, completing the proof that θ is well-defined.

Suppose that f is onto. Then commutativity of diagram (2.6.1) shows that ϕθ(S′) = M. Since ϕ is a minimal epimorphism [Lemma 2.4] we have θ(S′) = S, as claimed in the theorem.

On the other hand, suppose that f is one-to-one. Then fϕ′ is a separated cover of f(N) by S′. Therefore the factorization fϕ′ = ϕθ: S′ → → θ(S′) → → f(N) shows (by the definition of “separated cover”) that θ is one-to-one.

Corollary 2.7. Let ϕ: S → → M be a separated cover and f: S′ → → M any surjective Λ-homomorphism such that S′ is a separated Λ-module. Then f factors through ϕ, that is, there is a Λ-homomorphism θ such that f factors as follows.

f : S′ →^{θ} S → →^{ϕ} M (2.7.1)

Proof. The Corollary follows from Theorem 2.6 if we take N = S′ and note that the identity map ϕ′ on S′ is a separated cover.

Corollary 2.8. Let ϕ: S → → M be a separated cover such that ker(ϕ) ≠ 0. Then M is not a separated module.

Proof. Suppose that M ⊆ S′ for some finitely generated separated Λ-module S′. Since the identity map S′ → S′ is a separated cover, the almost functorial property shows that the inclusion map i: M → S′ can be lifted to a Λ-monomorphism θ: S → S′ such that θ = iϕ. But then θ(ker ϕ) = iϕ(ker ϕ) = 0. Since θ is a monomorphsm, this shows that ker(ϕ) = 0, as desired.

Example 2.9 (Non-separated modules). For the two simplest examples, let Λ be an unsplit Dedekind-like ring with normalization Γ. Then 𝔪/𝔪^{2} is Γ-isomorphic to Γ/𝔪 = F, and therefore this simple Γ-module has length 2 as a Λ-module. Therefore there is a Λ-module K strictly contained in 𝔪 and strictly containing 𝔪^{2}.

The natural maps Γ/𝔪^{2} → → Γ/K and Λ/𝔪^{2} → → Λ/K are separated covers [Lemma 2.3 with C = 𝔪] and have nonzero kernels. Therefore the Λ-modules Γ/K and Λ/K are not separated modules [Corollary 2.8].

Remark 2.10. Before proceeding to the next section, we comment on the reason for the word “separated”. Let ϕ: S → → M be a separated cover. If Λ is strictly split Dedekind-like, ker(ϕ) is always an amalgamation relation. (See diagrams (8.3.1), in which the passage from S to M is always given by what we call “bottom-gluing” relations.) Thus a Λ-module is “separated” if no such amalgamation relation has been imposed. In the unsplit case, ker(ϕ) is often, but not always an amalgamation. (See diagrams (7.4.1) and (7.5.2), in which the passage from S to M involves both bottom gluing and bottom reduction.) Thus our terminology is slightly misleading in the context of unsplit Dedekind-like rings.

3. Isomorphism as Matrix Problem, Unsplit Case

The main results of this section transform the problem of describing isomorphism classes of Λ-modules into a matrix problem over the fields k and F, for the case that (Λ, 𝔪, k) is an unsplit Dedekind-like ring with normalization (Γ, 𝔪, F).

Notation 3.1. Throughout this section Λ is a pullback ring, as specified in diagram (1.1.1) and display (1.1.2).

For a Λ-submodule S of a Γ-module X we write ΓS = Γ ⊗_{Λ} S to mean that the natural surjection of the right-hand side onto the left-hand side is a bijection.

Beginning in Notation 3.6 below, we shall assume that Λ is in fact unsplit Dedekind-like. But first we collect a few useful facts that hold in the current, more general context. Recall that a separated Λ-module is any Λ submodule of some Γ-module.

Lemma 3.2. Let S ≠ 0 be a separated, finitely generated Λ-module and ΓS a Γmodule that S generates. Then ΓS = Γ ⊗_{Λ} S if and only if there is a positive integer n such that S/𝔪S ≅ k^{(n)} (free k-module of rank n) and ΓS/𝔪S ≅ Γ̄^{(n)} (free Γ̄-module of rank n).

Proof. Let τ: Γ ⊗_{Λ} S → → ΓS be the natural surjection.

Suppose first that S/𝔪 ≅ k^{(n)} and ΓS/𝔪S ≅ Γ̄^{(n)}. The surjection τ induces a surjection τ̄: (Γ⊗S)/𝔪(Γ⊗S) → → ΓS/𝔪S. We claim that τ̄ is a bijection. Tensoring the short exact sequence 𝔪 ↪ Γ → → Γ̄ by S and using right-exactness of ⊗ yields the following chain of isomorphisms.

(3.2.1) \frac{Γ ⊗ S}{𝔪(Γ ⊗ S)} ≅ \frac{Γ}{𝔪} ⊗_{Λ} S ≅ \frac{Γ}{𝔪} ⊗_{Λ} \frac{S}{𝔪S} ≅ Γ̄ ⊗_{k} k^{(n)} ≅ Γ̄^{(n)}

Since τ̄ is a Γ-module surjection of the left-hand side of (3.2.1) onto ΓS/𝔪S ≅ Γ̄^{(n)}, and Γ̄ is a Γ̄-module of finite length, we see that τ̄ is a bijection, as claimed.

Now we lift this assertion to τ itself. Take x ∈ ker(τ). Let x̄ be the image of x in the left-hand side of (3.2.1). Since τ̄ is an isomorphism, we have x̄ = 0, that is, x ∈ 𝔪(Γ ⊗ S). Therefore there is an expression x = ∑_{i} m_{i} ⊗ s_{i} (m_{i} ∈ 𝔪, s_{i} ∈ S). Since 𝔪 ⊆ Λ we have x = 1 ⊗ (∑_{i} m_{i}s_{i}). But x ∈ ker(τ) implies that ∑_{i} m_{i}s_{i} = 0. Therefore x = 0; and so τ is an injection, hence a bijection.

Conversely, suppose that ΓS = Γ ⊗ S. Then by right exactness of ⊗, as in the first part of this proof, we have ΓS/𝔪S ≅ Γ̄ ⊗ (S/𝔪S). Since S/𝔪S is a module over the field Λ/𝔪 = k we have S/𝔪S ≅ k^{(n)} for some n, and hence Γ̄ ⊗_{Λ} (S/𝔪S) ≅ Γ̄ ⊗_{k} k^{(n)} ≅ Γ̄^{(n)} as desired.

Example 3.3. Caution. Suppose Λ ≠ Γ. Since Γ is a ring, we have Γ · Γ = Γ. However, we do not have Γ · Γ = Γ ⊗_{Λ} Γ. If this equality held, the previous lemma, with S = Γ, would require Γ/𝔪Γ to be Λ-isomorphic to k^{(n)} for some n and Γisomorphic to Γ̄^{(n)} for this same n. This would imply that k = Γ̄, which never holds when Λ ≠ Γ.

This fact will cause some inconvenience in the remainder of this paper.

Corollary 3.4. Let X be a finitely generated Γ-module. Then X has a Λ-submodule S such that X = Γ ⊗_{Λ} S if and only if X/𝔪X is a free Γ̄-module.

Proof. The “only if” assertion is an immediate consequence of Lemma 3.2.

Conversely, suppose that X/𝔪X ≅ Γ̄^{(n)} for some integer n. Choose n elements x_{1}, …, x_{n} ∈ X whose images in Γ̄^{(n)} are a free Γ̄-basis of Γ̄^{(n)}, and let S = ∑_{i} Λx_{i}. Since 𝔪 ⊆ rad Γ, Nakayama’s lemma shows that ΓS = X.

Since S/𝔪S is a k = Λ/𝔪-vector space generated by n elements, we have S/𝔪S ≅ k^{(m)} for some m ≤ n. It now suffices, by Lemma 3.2, to show that m = n. So suppose that m < n. Then the Γ̄-module Γ(S/𝔪S) = ΓS/𝔪S ≅ Γ̄^{(n)} would be generated by m < n elements, which is impossible for a free module of rank n over an artinian ring.

Lemma 3.5. Let Y be a k-subspace of some finitely generated Γ̄-module, and suppose Λ ≠ Γ.

(i) If Γ̄Y = Γ̄ ⊗_{k} Y, then Y contains no nonzero Γ̄-submodule. (ii) If Y contains no nonzero Γ̄-module and dim_{k}(Γ̄) = 2 [which holds whenever

Λ is Dedekind-like], then Γ̄Y = Γ̄ ⊗_{k} Y.

Proof. (i) Let d = dim_{k}(Γ̄). Since Λ ≠ Γ we have d > 1. Let G be the largest Γ̄-submodule of Y. Then Y = G ⊕ V for some k-subspace V of Y. Let y, g, v be the k-dimensions of Y, G, V respectively, so that y = g + v.

We have dim_{k}(Γ̄Y) ≤ dim_{k}(G) + dim_{k}(Γ̄V) ≤ g + dv. Also, dim_{k}(Γ̄ ⊗ Y) = dy = dg + dv. Since dim_{k}(Γ̄Y) = dim_{k}(Γ̄ ⊗ Y) by (i), we have dg ≤ g. But then d > 1 implies that g = 0 as desired.

(ii) Let ε ∈ Γ̄ − k. Since dim_{k} Γ̄ = 2 we have Γ̄ = k + kε. Hence 1 and ε are klinearly independent and ε^{2} ∈ k + kε. We want to show that the natural surjection τ: Γ̄⊗_{k} Y → → Γ̄Y is a monomorphism. Let x = ∑_{i} γ_{i} ⊗y_{i} ∈ ker(τ). We may assume that the y_{i} are linearly independent over k. Write γ_{i} = α_{i} + β_{i}ε (α_{i}, β_{i} ∈ k).

Then 0 = τ(x) = (∑_{i} α_{i}y_{i})+ε(∑_{i} β_{i}y_{i}) which we can write in the form 0 = u+εv with u, v ∈ Y. But then ku + kv = kεv + kv is a Γ̄-submodule of Y. By assumption this Γ̄-submodule must be 0, and hence u = v = 0. Linear independence of the y_{i} over k therefore implies that every α_{i} and β_{i} equals zero, and therefore x = 0, as desired.

Notation 3.6. (Matrix setup, unsplit case) Let Λ be a (residue-commutative) unsplit Dedekind-like ring with normalization Γ, as in Notation 1.1. Thus Γ̄ = F, a 2-dimensional field extension of k.

The map ρ: Γ → → F = Γ̄ in pullback diagram (1.1.1) induces a Γ-linear map Γ/𝔪^{t} → → F for every t, which we again call ρ. In fact, we denote any direct sum of such maps — from the direct sum of any n such modules to F^{(n)} — by ρ. Moreover, we write 𝔪^{∞} = 0, and therefore Γ = Γ/𝔪^{∞}.

When t ≠ ∞, we have F ≅ 𝔪^{t−1}/𝔪^{t} via the Γ-isomorphism γ +𝔪 → γπ^{t−1} +𝔪^{t}, where π is the distinguished Γ-generator of 𝔪 mentioned in Notation 1.1. Thus — given this choice of π — every element of F has a distinguished image in Γ/𝔪^{t}, which we often regard as an identification; and the set of all such distinguished images of elements of F defines the distinguished copy of F in Γ/𝔪^{t}. We note:

(3.6.1) When t ≠ ∞, ≠ 1, the distinguished copy of F in Γ/𝔪^{t} satisfies ρ(F) = 0.

Let X be the nonzero (external) direct sum of Γ-modules displayed below.

X = ⊕_{ν=1}^{n} Γ/𝔪^{t_{ν}} (3.6.2)

where every t_{ν} is a positive integer or ∞. Since Γ is a DVR, every finitely generated indecomposable left Γ-module is isomorphic to such a module X [J, Chap. 3, Theorem 19].

The map ρ yields the Γ-linear surjection to F^{(n)} shown in (3.6.3).

ρ: X → → F^{(n)} F^{(n−e_{∞})} ⊆ X (3.6.3)

We call coordinate ν of F^{(n)} the coordinate corresponding to the the ν^{th} summand Γ/𝔪^{t_{ν}} of X.

Let e_{∞} be the number of summands Γ/𝔪^{t_{ν}} of X of infinite length t_{ν}. Each of the n − e_{∞} summands Γ/𝔪^{t_{ν}} of finite length contains a distinguished copy of F. This yields the inclusion shown in (3.6.3). Thus every coordinate of F^{(n−e_{∞})} has a corresponding summand Γ/𝔪^{t_{ν}} of X, namely the summand containing this copy of F.

A matrix setup X (for Λ unsplit Dedekind-like) is any direct sum of the form X in (3.6.2), together with the corresponding surjection and inclusion displayed in (3.6.3).

Notation 3.7. (Matrices A, B; associated Λ-modules) Let X be a matrix setup (unsplit case). Let A be an n × n matrix over F = Γ̄, and let B be a q × (n − e_{∞}) matrix over F, for some q ≥ 0. We attach a length label to each column of A and B as follows. For 1 ≤ ν ≤ n we define the length label of column ν of A to be the length t_{ν} of the corresponding summand Γ/𝔪^{t_{ν}} of X. For 1 ≤ j ≤ n − e_{∞} we define the length label of column j of B to be the length of the summand of X corresponding to coordinate j of F^{(n−e_{∞})}.

Thus every column of B has a corresponding column in A, namely the column of A with the same corresponding summand of X; and every column of A whose length label is finite has a corresponding column in B.

We require A and B to have the following properties.

(3.7.1) A is invertible; the rows of B are linearly independent over F; and every column of B whose length label is 1 consists of zeros.

Thus right multiplication by A and B define k-linear monomorphisms

A: k^{(n)} ↪ F^{(n)} B: k^{(q)} ↪ F^{(n−e_{∞})} (3.7.2)

We call (A, B) a matrix pair associated with X.

Although we write the matrices in the ordered pair (A, B) side by side, we always think of B as being written underneath A, with each column of B written under its corresponding column in A. Thus columns of A with length label ∞ have no column of B written under them, and corresponding columns in A and B have the same length label.

Using the maps in (3.7.2), let S(A) be the Λ-module

S(A) = {x ∈ X | ρ(x) ∈ im(A) = k^{(n)}·A} (3.7.3)

and define the Λ-module M(A, B) by

(3.7.4)  M(A, B) = S(A)/ im(B)  where im(B) = k^{(q)}·B

If q = 0 we interpret this to mean that M(A, B) = S(A).

This definition makes sense — that is, im(B) ⊆ S(A) — because ρ(im(B)) = 0 by (3.6.1) and the requirement about columns of zeros in (3.7.1).

We note that Γ·S(A) = X, since A is invertible. Thus S(A) is the pullback of the commutative square in diagram (3.7.5) below.

\begin{aligned}S(A) & ⊂ & Γ·S(A) = X \\ ^{↓ ↓}ρ & & ^{↓ ↓}ρ \\ k^{(n)}A & ⊂ & F^{(n)} \end{aligned} (3.7.5)

Note that ker(ρ) = 𝔪X = 𝔪S(A). (The second equality holds because 𝔪Γ = 𝔪.)

Definition 3.8 (Display operation). A display operation on a matrix pair (A, B) (associated with a matrix setup X) is a permutation of the columns of A, together with the same permutation of their length labels and corresponding columns of B (so that corresponding columns of A and B remain corresponding columns after being moved).

Note that the matrix pair (A, B) — together with its length labels — determines its associated matrix setup X. Similarly, the length labels of the matrix pair (A′, B′) that results from a display operation applied to (A, B) determine another matrix setup X′; and the indecomposable summands of X′ can be obtained from those of X by means of the permutation involved in the display operation. Thus we see:

(3.8.1) Let (A′, B′) — including its length labels — be obtained by performing a display operation on (A, B). Then M(A′, B′) ≅ M(A, B), where the module on the left is computed with respect to the correspondingly altered matrix setup.

Theorem 3.9. Let X be a matrix setup (unsplit case) with associated Γ-module X, and let (A, B) be a matrix pair associated with X. Also, let S = S(A). Then

(i) ΓS = Γ ⊗_{Λ} S. (ii) The natural surjection S → → M(A, B) is a separated cover of M(A, B).

Moreover,

(iii) Every finitely generated Λ-module is isomorphic to M(A, B) for some (A, B)

with respect to some matrix setup X. (iv) Let X and X′ be the Γ-modules associated with matrix setups X and X′

respectively, and let M(A, B) and M(A′, B′) be Λ-modules computed with respect to X and X′ respectively. If M(A, B) ≅ M(A′, B′) as Λ-modules then X ≅ X′ as Γ-modules.

Proof. (i) This follows from Lemma 3.2.

(ii) By Lemma 2.3 (with C = 𝔪) it suffices to show: (a) im(B) contains no nonzero Γ̄-submodules, and (b) im(B) ⊆ 𝔪S. Consider Γ̄· im(B), the Γ̄-module generated by the rows of B. We claim that Γ̄· im(B) = Γ̄ ⊗_{k} im(B).

Since the right-hand side maps onto the left-hand side, it suffices to show that both sides have the same F -dimension. Since the rows of B are F -linearly independent, by (3.7.1), the F -dimension of the left-hand side is the number of rows of B. On the other hand, F -independence of the rows of B implies that they form a k basis of im(B) = k^{(q)}B, and hence an F -basis of F ⊗_{k} im(B); and hence the F -dimension of F ⊗_{k} im(B) again equals the number of rows of B, proving the claim.

The claim, together with Lemma 3.5(i), shows that im(B) contains no nonzero Γ̄-submodules, establishing (a).

To establish (b) it suffices to show that im(B) is contained in the sum of all those summands Γ/𝔪^{t} of X whose length is finite but greater than 1. The “finite” part is part of the definition of a matrix setup; and the “greater than 1” part follows from the fact that every column of B with length label 1 consists of zeros, by (3.7.1).

(iii) Let M be any finitely generated Λ-module. Then M has a separated cover ϕ: S → → M [Proposition 2.2]. Since S is a separated Γ-module, it can be regarded as a Λ-submodule of the Γ-module X = ΓS = Γ ⊗_{Λ} S [by (2.1.4)]. Since Γ is a DVR and _{Γ}X is finitely generated, we can take X to be as shown in (3.6.2). This gives us the top row of the commutative square in (3.7.5) that we are building.

Decomposition (3.6.2) of X, together with the map ρ in (3.6.3) yield an F - linear identification ΓS/𝔪S = F^{(n)}. Since ΓS = Γ ⊗_{Λ} S we have a Λ-isomorphism (equivently, k-isomorphism) S/𝔪S ≅ k^{(n)} [Lemma 3.2]. In addition, the relation ΓS = Γ ⊗_{Λ} S, reduced modulo 𝔪S yields F (S/𝔪S) = F ⊗_{k} (S/𝔪S). Therefore the n-dimensional k-submodule S/𝔪S of F^{(n)} = ΓS/𝔪S is k-generated by an F -basis of F^{(n)}. This yields an invertible n × n-matrix A over F such that S/𝔪S = k^{(n)}A.

To complete the proof that S = S(A) it suffices to prove that S is the pullback of this diagram; and for this it suffices to show that S ⊇ ker(ρ) = 𝔪X. But 𝔪X = 𝔪ΓS = 𝔪S, which is contained in S since 𝔪 ⊆ Λ.

It now suffices to prove that ker(ϕ) = im(B) for a suitable B. Since ϕ is a separated cover, Lemma 2.3 shows that ker(ϕ) is a k-submodule of X; that is, 𝔪· ker(ϕ) = 0. Thus, in the current notation,

ker(ϕ) ⊆ ⊕{𝔪^{t_{ν}−1}/𝔪^{t_{ν}} | t_{ν} ≠ ∞} = F^{(n−e_{∞})} (3.9.1)

We claim that, for some non-negative integer q, ker(ϕ) is generated — as a k-vector space — by q elements of F^{(n−e_{∞})} that are linearly independent over F. For this, it suffices (by a dimension argument) to show that F · ker(ϕ) = F ⊗_{k} ker(ϕ). But ϕ is a separated cover, so Lemma 2.3 shows that ker(ϕ) has no nonzero F -submodules. Since Λ is unsplit Dedekind-like, we have dim_{k}(F) = 2, so the desired equality follows by Lemma 3.5(ii).

In view of the claim, there is a q × (n − e_{∞}) matrix B over F whose rows are F -linearly independent and form a k-basis of ker(ϕ). One last appeal to Lemma 2.3 shows that ker(ϕ) ⊆ 𝔪S = 𝔪ΓS = 𝔪X. This shows that (3.9.1) can be refined to

ker(ϕ) ⊆ ⊕{𝔪^{t_{ν}−1}/𝔪^{t_{ν}} | t_{ν} ≠ ∞, ≠ 1} ⊆ F^{(n−e_{∞})}

from which it follows that all columns of B with length label 1 must be zero. Thus we now have ker(ϕ) = k^{(n)}B were B is as in (3.7.1), as desired.

(iv) By uniqueness of the separated cover [Theorem 2.6 with N = M and f = 1], the Λ-isomorphism class of S(A) is determined by that of M(A, B). Statement (iv) now follows from the fact that, in any matrix setup, X = Γ ⊗_{Λ} S(A), by statement (i) above.

The final step in transforming the problem of classifying isomorphism classes of finitely generated Λ-modules into a matrix problem is to transform the condition M(A, B) ≅ M(A′, B′) into a purely matrix-theoretic statement about the two matrix pairs (A, B) and (A′, B′). In view of statements (iii) and (iv) of the previous theorem, we can restrict our attention to the situation that both matrix pairs arise from the same matrix setup.

The notation A[cols j] denotes the submatrix of A consisting of the j-labeled columns.

Theorem 3.10 (Matrix operations, unsplit case). Let (A, B) and (A′, B′) be matrix pairs associated with a matrix setup X (where Λ is unsplit Dedekind-like). Then M(A, B) ≅ M(A′, B′) if and only if (A′, B′) can be obtained from (A, B) by a finite sequence of the following operations.

(i) (a) Left multiply A by an invertible matrix over k. (b) Left multiply B by an invertible matrix over k.

(ii) For any length label j right-multiply A[cols j] by an invertible matrix Q over

F and, if j ≠ ∞, simultaneously right-multiply B[cols j] by Q. (iii) (a) For any i > j add an F -scalar multiple of a i-labeled column of A to a

j-labeled column of A. (“Sweep toward smaller lengths in A.”)

(b) For any i < j add any F -scalar multiple of a i-labeled column of B to a

j-labeled column of B. (“Sweep toward larger lengths in B.”)

We call these matrix operations “k-F sweeping-similarity” operations.

Proof. The matrix operations in the theorem are stated in a way that is convenient for the use of the theorem and our eventual statement of the canonical form of (A, B), but not for the proof of the present theorem. Therefore we begin the proof by changing notation.

Note that it is possible to perform display operations, rearranging the summands Γ/𝔪^{t} of X in descending order; that is, such that the decomposition (3.6.2) of X takes the form

X = (Γ/𝔪^{∞})^{(e_{∞})} ⊕ (Γ/𝔪^{t})^{(e_{t})} ⊕ … ⊕ (Γ/𝔪^{i})^{(e_{i})} ⊕ … ⊕ (Γ/𝔪)^{(e_{1})}

where ∞ > t > ⋯ > i > ⋯ > 1 (3.10.1)

where any particular block, e.g. the ∞ block, might not actually be present. (Recall that the notation (e_{t}) denotes the multiplicity of Γ/𝔪^{t} in X.) We can do this in such a way that the summands of X in each length-block retain their original order; that is, such that every matrix of the form A[cols j] and B[cols j] is exactly as is was before the display operations. Note that none of the operations mentioned in the theorem is affected by this display operation.

After doing this display operation we can restate these matrix operations as relations of the form

A′ = P AQ_{1} B′ = P BQ_{2} (3.10.2)

in which Q_{1} and Q_{2} have certain block triangular forms, due to the fact that all sweeping in A is to the right while all sweeping in B is to the left. We describe this block triangular form precisly in the next subsection, and then state and prove the theorem in the altered notation.

Notation 3.11 (X triangular form). Let X be a matrix setup (unsplit case) whose summands are arranged in decreasing order, as in (3.10.1). As usual, let n denote the number of individual summands Γ/𝔪^{j} of X. We can use the sequence of length labels to label both the rows and columms of any n × n matrix, and we can use the labels ≠ ∞ to label both the rows and colums of any (n − e_{∞}) × (n − e_{∞}) matrix. This partitions the matrix into blocks. If, for example, Q is a matrix being partitioned in this way, Q[rows i, cols j] denotes the e_{i} × e_{j} submatrix consisting of the intersection of the i-labeled rows with the j-labeled colums of Q.

We say that an n×n matrix Q_{1} is X upper triangular if Q_{1}[rows i, cols j] = 0 when i < j. (In interpreting this, remember that the length labels occur in decreasing order.) For example, see the matrix Q_{1} in (3.11.1), noting that we have supressed multiplicities, writing each distinct length label only once.

Similarly, we say that an (n − e_{∞}) × (n − e_{∞}) matrix Q_{2} is in X lower triangular if Q_{2}[rows i, cols j] = 0 if i > j. (Again see the example in (3.11.1).)

Finally, we call any block of the form Q[rows i, cols i] a main-diagonal block.

Q_{1} = Q_{2} = (3.11.1)

\begin{matrix}∞ & 6 & 4 \\ \begin{matrix}∞ \\ 6 \\ 4 \end{matrix} & [\begin{matrix}Q_{1}[∞, ∞] & ∗ & ∗ \\ 0 & Q_{1}[6, 6] & ∗ \\ 0 & 0 & Q_{1}[4, 4] \end{matrix} ] \end{matrix}  \begin{matrix}6 & 4 \\ {6 \atop 4} & [ \begin{matrix}Q_{2}[6, 6] & 0 \\ ∗ & Q_{2}[4, 4] \end{matrix} ] \end{matrix}

We reformulate and prove Theorem 3.10 in the new notation.

Theorem 3.12. Let (A, B) and (A′, B′) be matrix pairs associated with a matrix setup X (Λ unsplit Dedekind-like) whose summands occur in descending order, as in (3.10.1). Then M(A, B) ≅ M(A′, B′) if and only if there exist matrix relations A′ = P_{1}AQ_{1} and B′ = P_{2}BQ_{2} in which the following conditions hold.

(i) P_{1} and P_{2} are invertible, with entries in k. (ii) Q_{1} is invertible and X upper triangular, with entries in F. (iii) Q_{2} is invertible and X lower triangular, with entries in F. (iv) Q_{2}[rows i, cols i] = Q_{1}[rows i, cols i] whenever i ≠ ∞; that is, every diagonal

block of Q_{2} equals the corresponding diagonal block of Q_{1}.

Remark. The matrices in (3.11.1) illustrate the triangular forms of the matrices Q_{1} and Q_{2} in this theorem. In addition, the matrices in the theorem satisfy Q_{2}[rows i, cols i] = Q_{1}[rows i, cols i] for i ≠ ∞.

Proof. Let X be the Γ-module associated with X. Also, let K(A, B) denote the kernel of the natural homomorphism S(A) → → M(A, B).

Claim 1: M(A, B) ≅ M(A′, B′) if and only if there is a Γ-automorphism τ of X such that τ(S(A)) = S(A′) and τ(K(A, B)) = K(A′, B′) (as illustrated below).

\begin{aligned}K(A, B) & ⊂ & S(A) & → → & M(A, B) \\ ^{(≅)↓τ} & & ^{(≅)↓τ} & & ^{(≅)↓σ} \\ K(A′, B′) & ⊂ & S(A′) & → → & M(A′, B′) \end{aligned} (3.12.1)

First let σ: M(A, B) ≅ M(A′, B′) be given. Since S(A) → → M(A, B) is a separated cover [Theorem 3.9(ii)], the almost functorial property of separated covers [Theorem 2.6] yields a Λ-isomorphism τ such that both squares in (3.12.1) commute. Also, since X = ΓS(A) = Γ ⊗_{Λ} S(A) [Theorem 3.9(i)], we can extend τ to a Γ-automorphism of X, namely 1⊗τ. The converse assertion of the claim is obvious.

The effect of Claim 1 is that we have no further need for M(A, B) and M(A′, B′) in the rest of this proof. We replace the condition M(A, B) ≅ M(A′, B′) by the condition that the automorphism τ exists.

Now suppose that the τ exists. To see how P_{1} and Q_{1} arise, we build the following commutative cube. Let the back (inner) square be the pullback diagram that defines A, and the front (outer) square be the pullback diagram that defines A′. Then insert τ in the two places shown.

\begin{aligned}S(A′) & & & ⊆ & & & X \\ & ↖τ & & & & ↗ τ \\ & & S(A) & ⊆ & X \\ ρ & & ρ & & ρ & & ρ \\ & & k^{(n)}A & ⊆ & F^{(n)} \\ & ↙ Q^{1} & & & & ↘Q^{1} \\ k^{(n)}A′ & & & ⊆ & & & F^{(n)} \end{aligned} (3.12.2)

The top square commutes, by the hypothesis on τ in this half of the proof. Since ρ is a surjection and ker(ρ) = 𝔪X, there is a unique (necessarily invertible) matrix Q_{1} over F such that right multiplication by Q_{1} makes the right-hand square commute. Since τ takes S(A) to S(A′), and the images of S(A) and S(A′) under ρ are k^{(n)}A and k^{(n)}A′ respectively, the left-hand square is a restriction of the right-hand square, and therefore commutes.

Therefore k^{(n)}AQ_{1} = k^{(n)}A′. Since A′ is invertible, this can be rewritten k^{(n)}AQ_{1}(A′)^{−1} = k^{(n)}. Thus right multiplication by AQ_{1}(A′)^{−1} is a k-linear automorphism of k^{(n)}, and hence equals right multiplication by a unique invertible matrix over k that we call P_{1}^{−1}. But then AQ_{1}(A′)^{−1} = P_{1}^{−1}, which is equivalent to A′ = P_{1}AQ_{1}, as desired.

To see how P_{2} and Q_{2} arise, we build the next diagram, starting with its top and bottom rows, as shown.

\begin{aligned}k^{(q)} & ⟶^{B} & F^{(n−e_{∞})} & ⊆ & X \\ ↓ ; P_{2}^{−1} & & ^{↓Q_{2}} & & ^{↓τ} \\ k^{(q)} & ⟶^{B′} & F^{(n−e_{∞})} & ⊆ & X \end{aligned} (3.12.3)

Since F^{(n−e_{∞})} is the Γ-socle of X, the Γ-automorphism τ takes F^{(n−e_{∞})} isomorphically onto itself. Therefore the restriction of τ to this submodule induces an isomorphism (given by an invertible matrix Q_{2}) from F^{(n−e_{∞})} to F^{(n−e_{∞})} making the right-hand square of (3.12.3) commute.

We have τ(K(A, B)) = (A′, B′) by (3.12.1). Since K(A, B) = k^{(q)}B and K(A′, B′) = k^{(q)}B′, this and the second square in (3.12.3) yield k^{(q)}BQ_{2} = k^{(q)}B′. Since right multiplication by B′ is one-to-one, this defines a one-to-one k-linear map k^{(q)} → k^{(q)} which is therefore also surjective and equals right multiplication by a unique invertible matrix over k that we call P_{2}^{−1}, making the left-hand square in (3.12.3) commute. Commutativity of the first square in (3.12.3) is precisely the desired relation B′ = P_{2}BQ_{2}.

To complete the proof of this half of the theorem we need to show that the existence of τ implies the stated block triangular forms of Q_{1} and Q_{2}. Recall the structure of X (below), where n = ∑_{i} e_{i}.

X = Γ^{(e_{∞})} ⊕ (Γ/𝔪^{t})^{(e_{t})} ⊕ … ⊕ (Γ/𝔪^{i})^{(e_{i})} ⊕ … ⊕ (Γ/𝔪)^{(e_{1})}
↓τ = (τ_{μν})
X = Γ^{(e_{∞})} ⊕ (Γ/𝔪^{t})^{(e_{t})} ⊕ … ⊕ (Γ/𝔪^{i})^{(e_{i})} ⊕ … ⊕ (Γ/𝔪)^{(e_{1})}

(3.12.4)

We can view τ as right multiplication by a matrix (τ_{μν}) each of whose entries is a Γ-homomorphism

(3.12.5)  τ_{μν}: Γ/𝔪^{i} → Γ/𝔪^{j}  (μ ∈ block i, ν ∈ block j)

where “μ ∈ block i” signifies that summand μ of X is one of the e_{i} summands equal to Γ/𝔪^{i}.

The rest of the proof uses the following properties of our (possibly noncommutative) DVR Γ: 𝔪 is a 2-sided ideal, Γ/𝔪^{t} is a uniserial left Γ-module, and it has finite length when t ≠ ∞. Since ρ is a ring homomorphism of Γ onto F = Γ/𝔪 and has kernel 𝔪, we identify ρ with the natural homomorphism Γ → Γ/𝔪. Commutativity of the division ring F is not used. (Commutativity will become important when we use the theory of similarity of matrices over F to find the canonical form of matrix pairs.)

We note that, since 𝔪 is a left ideal and τ_{μν} is left Γ-linear, τ_{μν} equals right multiplication by an element of Γ. That is, there exists γ ∈ Γ such that, for all x ∈ Γ:

τ_{μν}(x + 𝔪^{i}) = x τ_{μν}(1 + 𝔪^{i}) = x(γ + 𝔪^{j}) (3.12.6)

Moreover, since 𝔪 is also a right ideal, τ_{μν} induces the endomorphism of the left F -vector space F = Γ/𝔪 given by right multiplication by ρ(γ) = γ + 𝔪. Finally, when i ≠ ∞, the homomorphism τ_{μν} must take the unique minimal submodule 𝔪^{i−1}/𝔪^{i} (our distinguished copy of F) to a module of length ≤ 1. We break up the remaining details into three cases.

Case 1: i < j (below the main-diagonal blocks). Here Γ/𝔪^{i} is uniserial of finite length i and therefore its image under τ_{μν} is uniserial of length ≤ i. Therefore im(τ_{μν}) ⊆ 𝔪/𝔪^{j}, and so the induced map Γ/𝔪 → Γ/𝔪 equals zero. In other words, the (μ, ν)-entry of Q_{1} is zero whenever (μ, ν) belongs to a block Q_{1}[rows i, cols j] that lies below the main-diagonal blocks, as claimed in statement (ii).

Case 2: ∞ ≠ i > j (above the main-diagonal blocks). Here τ_{μν} cannot be oneto-one, and therefore takes the unique simple submodule 𝔪^{i−1}/𝔪^{i} of Γ/𝔪^{i} to zero. Therefore, in this case, the (μ, ν)-entry of Q_{2} equals zero as claimed in (iii).

Case 3: i = j ≠ ∞ (main-diagonal block). We have already noted that the endomorphism of Γ/𝔪 induced by τ_{μν} equals right multiplication by ρ(γ). But since τ_{μν} equals right multiplication by γ ∈ Γ, and since 𝔪 is a 2-sided ideal, it follows that τ_{μν} induces an endomorphism of the socle 𝔪^{i−1}/𝔪^{i} of Γ/𝔪 given by right multiplication by the same element ρ(γ), as desired.

This completes the proof of the “only if” part of the theorem.

Conversely, suppose that A′ = P_{1}AQ_{1} and B′ = P_{2}BQ_{2} and (i)–(iv) hold. We want to prove that a Γ-automorphism τ of X exists satisfying Claim 1, that is, such that τ(S(A)) = S(A′) and τ(K(A, B)) = K(A′, B′). We define the map τ by defining each τ_{μν} in (3.12.5), considering two cases.

Case A. i ≥ j (on or above main diagonal). Let x be the (μ, ν)-entry of Q_{1}. Then x = ρ(γ) for some γ ∈ Γ. Since i ≥ j, right multiplication by γ followed by reduction modulo 𝔪^{j} is a well-defined left Γ-homomorphism that we use for τ_{μν} in (3.12.5).

Case B. i < j (below main diagonal). Using our distinguished Γ-generator π of 𝔪, right multiplication by π^{j−i} yields a Γ-embedding of Γ/𝔪^{i} into Γ/𝔪^{j}. Let x be the (μ, ν)-entry of Q_{2}, and let γ be any element of Γ such that ρ(γ) = x. Then right multiplication by π^{j−i} · γ is a well-defined left Γ-linear map that we use for τ_{μν} in (3.12.5).

Claim 2: With the given definition of τ, the right-hand square in diagram (3.12.2) commutes. It suffices to check commutativity of each smaller square, in which τ is replaced by some τ_{μν} in (3.12.5) and Q_{1} is replaced by its (μ, ν)-entry. Except if the block containing (μ, ν) is strictly below the main-diagonal blocks, this is true by the definition in Case A. For the below-diagonal blocks, the map induced by τ_{μν} is zero regardless of our choice of τ_{μν}, as shown in the proof of Case 1 above; so the claim, for below-diagonal blocks, follows from the block upper triangular form of Q_{1}.

Claim 3: The endomorphism τ is an automophism. The map induced by τ modulo 𝔪 — i.e. right multiplication by Q_{1} — is a surjection. Therefore, by Nakayama’s lemma, τ itself is a surjection. Since X is noetherian, if follows that τ is also an injection.

Claim 4. The endomorphism τ induces right multiplication by Q_{2}; that is, the right-hand square of diagram (3.12.3) commutes. As in the proof of Claim 2, it suffices to check each τ_{μν} individually. For (μ, ν) strictly below the main-diagonal blocks this is true by the definition in Case B. For (μ, ν) strictly above the maindiagonal blocks, the map induced by τ_{μν} is zero regardless of our choice of τ_{μν}, as shown in the proof of Case 2 above. This leaves only the case that (μ, ν) belongs to some main-diagonal block, say [rows i, cols i].

Let γ and x = ρ(γ) be as in Case A of our definition. Then x is the (μ, ν)-entry of Q_{1}, and hence, by condition (iv) in the statement of the theorem, x is the (μ, ν)- entry of Q_{2}. The desired commutativity of the second square of diagram (3.12.3) is now immediate.

Now we are ready to prove that τ((S(A)) = S(A′). First note that, in the bottom left of diagram (3.12.2), right multiplication by Q_{1} takes k^{(n)}A onto k^{(n)}A′ because A′ = P_{1}AQ_{1} and P_{1} is an invertible matrix over k. Therefore the bottom square commutes. The inner and outer squares commute, being pullback squares by definition (3.7.5), and the right-hand square commutes by Claim 2. Therefore the upper left map τ takes the pullback S(A) of the inner square onto the pullback S(A′) of the outer square, as claimed.

Finally, we show that τ(K(A, B)) = K(A′, B′). The left-hand square in (3.12.3) commutes since B′ = P_{2}BQ_{2}, and the right-hand square commutes by Claim 4. Therefore τ takes im(B) = K(A, B) onto im(B′) = K(A′, B′).

Notation 3.13. We write (A, B) ≅ (A′, B′) to indicate matrix pairs obtained from each other by means of display operations [Definition 3.8] and k-F sweeping similarity operations [Theorem 3.10].

It is easy to see that, when such an isomorphism holds, we can do the display operations before the sweeping-similarity operations.

The following result summarizes much of the content of this section.

Theorem 3.14. (In the unsplit case:) Every finitely generated Λ-module is isomorphic to some M(A, B). Moreover, M(A, B) ≅ M(A′, B′) if and only if (A, B) ≅ (A′, B′).

Proof. The first assertion is Theorem 3.9(iii). Next, suppose that M(A, B) ≅ M(A′, B′). Then X ≅ X′, where X and X′ are the Γ-modules associated with the matrix setups given by the row of length labels in (A, B) and in (A′, B′) [Theorem 3.9(iv)]. Since the Krull-Schmidt Theorem holds for finitely generated modules over principal ideal domains, X and X′ have the same summands, except for order of occurrence. Therefore some permutation of the indecomposable summands of X transforms them to the indecomposable summands of X′. After performing the corresponding display operations on (A, B) we may assume that X = X′, whence Theorem 3.10 states that k-F sweeping-similarity operations transform (A, B) to (A′, B′).

The converse implication holds because neither display operations [(3.8.1)] nor sweeping-similarity operations [Theorem 3.10] change the isomorphism class of M(A, B).

4. Isomorphism as Matrix Problem, Strictly Split Case

The main results of this section transform the problem of describing isomorphism classes of Λ-modules into a matrix problem, for the case that Λ is a strictly split Dedekind-like ring. Since this problem was solved in [L2] (and in [NR], corrected in [NRSB]), we omit most details, which are very similar to the corresponding details in the unsplit case considered in the previous section. The emphasis here is on phrasing these results in the terminology of the present paper in such a way that it will be useful later in this paper and in the forthcoming paper on the non-local situation.

The principal differences between results in the split and unsplit cases are due to the fact that, in the split case Γ is the direct sum of two DVRs, while in the unsplit case, Γ is a single DVR. Therefore, in the split case, there are twice as many matrices to manipulate as in the unsplit case.

Notation 4.1. Throughout this section (Λ, 𝔪, k) is a (residue commutative) strictly split Dedekind-like ring, as in Notation 1.1. Thus the normalization of Λ is Γ = Γ_{1} ⊕ Γ_{2} where each (Γ_{i}, 𝔪_{i}, k) is a (residue commtative) DVR and 𝔪 = 𝔪_{1} ⊕ 𝔪_{2}.

Following the style of Notation 3.6, we use the notation ρ not only for the map Γ → → Γ̄ = Γ/𝔪 = k⊕k in pullback diagram (1.1.1), but also for the maps Γ_{i}/𝔪_{i}^{t} → → k induced by the original map ρ, and for direct sums of such maps. Moreover, we let 𝔪_{i}^{∞} = 0, so that Γ_{i} = Γ_{i}/𝔪_{i}^{∞}.

Example 4.2. Note that the two direct summands k of Γ̄ are isomorphic rings but not isomorphic Γ-modules (because their annihilators are different). The field k has two Γ-module structures. When necessary we distinguish between these nonisomorphic Γ-modules by using the notation (k, 0) and (0, k). However — and this is very important in the rest of this paper — we have (k, 0) ≅ (0, k) as Λmodules (equivalently, as k-vector spaces).

Let X be any of the nonisomorphic Γ-modules (k, 0), (0, k), (k, k) = (k, 0)⊕(0, k). Each of these has a Λ-submodule S that is Λ-isomorphic to k and such that X = ΓS, namely X itself in the first two cases, and the diagonal submodule diag(k) = {(x, x) ∈ k ⊕ k} in the third case. However only S = diag(k) satisfies ΓS = Γ ⊗_{Λ} S [Lemma 3.2].

This shows that the notation ΓS is not well-defined unless one specifies the Γ-module X inside of which it is computed.

Notation 4.3 (Matrix setup, strictly split case). Let t ≠ ∞. Then, making the identifications k = Γ_{i}/𝔪_{i} (i = 1, 2) via ρ, we have k ≅ 𝔪_{i}^{t−1}/𝔪_{i}^{t} (i = 1, 2) as kvector spaces via γ_{i} + 𝔪_{i} → γ_{i}π_{i}^{t−1} + 𝔪_{i}^{t}, where π_{i} is the distinguished Γ-generator of 𝔪_{i} mentioned in Notation 1.1. Thus every element of k has a distinguished image in each 𝔪_{i}^{t−1}/𝔪_{i}^{t}, which we often regard as an identification; and the set of all such distinguished images defines the distinguished copy of k in Γ_{i}/𝔪_{i}^{t}.

Let X be the nonzero (external) direct sum of Γ-modules as displayed below.

X = X_{1} ⊕ X_{2} where

X_{1} = ⊕_{ν=1}^{n} Γ_{1}/𝔪_{1}^{s_{ν}} and X_{2} = ⊕_{ν=1}^{n} Γ_{2}/𝔪_{2}^{t_{ν}} (4.3.1)

where each s_{ν} and t_{ν} is a positive integer or ∞. Since each Γ_{i} is a DVR, every finitely generated Γ_{i}-module is isomorphic to a module of the form X_{i} (i = 1, 2) [J, Chap. 3, Theorem 19]. However, note that the same number n of indecomposable summands occurs in both X_{1} and X_{2} in (4.3.1). The reason for this restriction is the following fact, which we will have many occasions to use.

(4.3.2) The Γ-module Y = (⊕_{ν=1}^{m} Γ_{1}/𝔪_{1}^{s_{ν}}) ⊕ (⊕_{ν=1}^{n} Γ_{2}/𝔪_{2}^{t_{ν}}) has a Λ-submodule S such that Y = Γ ⊗_{Λ} S if and only if m = n.

This follows immediately from Corollary 3.4 since — in the notation of Example 4.2 — the right hand side, when reduced modulo 𝔪 = 𝔪_{1} ⊕ 𝔪_{2}, becomes Γ-isomorphic to (k, 0)^{(m)} ⊕ (0, k)^{(n)}, while Γ/𝔪 ≅ (k, k).

The map ρ yields a pair of Γ_{i}-linear surjections to k^{(n)} shown in (4.3.3).

ρ: X_{i} → → k^{(n)} (i = 1, 2)  k^{(n−d_{∞})} ⊆ X_{1}  k^{(n−e_{∞})} ⊆ X_{2} (4.3.3)

where k really denotes (k, 0) when i = 1, and (0, k) when i = 2. We call coordinate ν of k^{(n)} the coordinate corresponding to the the ν^{th} summand Γ_{i}/𝔪_{i}^{t_{ν}} of X_{i}.

Let d_{∞}, e_{∞}, respectively, be the number of indecomposable summands of X_{1}, X_{2} of infinite length. Each of the n − d_{∞} summands of finite length contains a distinguished copy of k. This yields the first inclusion shown in (4.3.3). Thus every coordinate of k^{(n−d_{∞})} has a corresponding summand Γ_{1}/𝔪_{1}^{s_{ν}} of X_{1}, namely the summand containing this copy of k. Analogous statements apply to the second displayed inclusion.

A matrix setup X (for Λ strictly split Dedekind-like) is any direct sum of the form X in (4.3.1), together with the corresponding surjections and inclusions displayed in (4.3.3). (This replaces the somewhat differently stated, but equivalent, “matrizing choices” in [L1], [L2].)

Notation 4.4 (Matrices (A_{1}, A_{2}, B_{1}, B_{2}); associated Λ-modules). Let X be a matrix setup (strictly split case). Let A_{1}, A_{2} be n × n matrices over k; and let B_{1}, B_{2} respectively be q × (n − d_{∞}) and q × (n − e_{∞}) matrices over k (for some q ≥ 0). We attach a length label to each column of these matrices as in the unsplit case: We define the length label of column ν of A_{1} to be the length s_{ν} of the corresponding indecomposable direct summand of X_{1}, and we define the length label of column j of B_{1} to be the length of the the summand of X_{1} corresponding to coordinate j of k^{(n−d_{∞})}.

Thus every column of B_{1} has a corresponding column in A_{1}, namely the column with the same corresponding summand (of finite length) of X_{1}. Analogous definitions apply to A_{2}, B_{2}, X_{2}, n − e_{∞}. We require these matrices to satisfy:

(4.4.1) A_{1} and A_{2} are invertible; the rows of each B_{i} are linearly independent over k; and the columns of each B_{i} with length label 1 consist of zeros.

Consequently, right multiplication by (A_{1}, A_{2}) and by (B_{1}, B_{2}) are k-linear monomorphisms:

(A_{1}, A_{2}): k^{(n)} ↪ k^{(n)} ⊕ k^{(n)} (B_{1}, B_{2}): k^{(q)} ↪ k^{(n−d_{∞})} ⊕ k^{(n−e_{∞})} (4.4.2)

We call (A_{1}, A_{2}, B_{1}, B_{2}) a matrix 4-tuple associated with X.

As in the unsplit case, we write these four matrices side by side, but we always think of each B_{i} as being written underneath A_{i}, with each column of B_{i} written under the corresponding column of A_{i} (and no column of B_{i} written under columns of A_{i} with infinite length labels).

Using the maps in (4.4.2), let S(A_{1}, A_{2}) be the Λ-module

S(A_{1}, A_{2}) = {x ∈ X | ρ(x) ∈ im(A_{1}, A_{2}) = k^{(n)}·(A_{1}, A_{2})} (4.4.3)

and define the Λ-module M(A_{1}, A_{2}, B_{1}, B_{2}) by

M(A_{1}, A_{2}, B_{1}, B_{2}) = S(A_{1}, A_{2})/ im(B_{1}, B_{2})  (im(B_{1}, B_{2}) = k^{(q)}·(B_{1}, B_{2})) (4.4.4)

If q = 0 we interpret this to mean that M(A_{1}, A_{2}, B_{1}, B_{2}) = S(A_{1}, A_{2}).

We note that Γ·S(A_{1}, A_{2}) = X, since A_{1} and A_{2} are invertible. Thus S(A_{1}, A_{2}) is the pullback of the commutative square in diagram (4.4.5) below.

\begin{aligned}S(A_{1}, A_{2}) & ⊂ & Γ·S(A_{1}, A_{2}) = X \\ ^{↓ ↓}ρ & & ^{↓ ↓}ρ \\ k^{(n)}(A_{1}, A_{2}) & ⊂ & (k, k)^{(n)} \end{aligned} (4.4.5)

The similarity between (4.4.5) and (3.7.5) can be enhanced by noting that we can consider the n × 2n matrix (A_{1}, A_{2}) over k to be an invertible n × n matrix over Γ̄ = k ⊕ k, and just calling it A. This will occasionally be useful, although it will more often be useful to focus on the individual matrices A_{1} and A_{2}. However, we cannot consider (B_{1}, B_{2}) to be a matrix over Γ̄ because the number of columns of B_{1} need not equal the number of columns of B_{2}.

Definition 4.5 (Display operation, split case). We define a display operation on a matrix 4-tuple (A_{1}, A_{2}, B_{1}, B_{2}) (associated with a matrix setup X) to be a permutation of the columns of each A_{i}, together with the same permutation of their length labels and corresponding columns of B_{i} (so that corresponding columns of A_{i} and B_{i} remain corresponding columns after being moved).

As in the unsplit case [Definition 3.8], this corresponds to rearranging the indecomposable summands of the module X associated with X. Therefore:

(4.5.1) Let (A′_{1}, A′_{2}, B′_{1}, B′_{2}) — including its length labels — be obtained by performing a display operation on (A_{1}, A_{2}, B_{1}, B_{2}). Then M(A′_{1}, A′_{2}, B′_{1}, B′_{2}) ≅ M(A_{1}, A_{2}, B_{1}, B_{2}), where the module on the left is computed with respect to the correspondingly altered matrix setup.

Theorem 4.6. Let X be a matrix setup (strictly split case) with associated Γmodule X, and let (A_{1}, A_{2}, B_{1}, B_{2}) be a matrix 4-tuple associated with X. Also, let S = S(A_{1}, A_{2}). Then

(i) ΓS = Γ ⊗_{Λ} S. (ii) The natural surjection S → → M(A_{1}, A_{2}, B_{1}, B_{2}) is a separated cover. Moreover, (iii) Every finitely generated Λ-module is isomorphic to M(A_{1}, A_{2}, B_{1}, B_{2}) for

some matrix 4-tuple (A_{1}, A_{2}, B_{1}, B_{2}) with respect to some matrix setup X. (iv) Let X and X′ be the Γ-modules associated with matrix setups X and X′ respec-

tively, and let M(A_{1}, A_{2}, B_{1}, B_{2}) and M(A′_{1}, A′_{2}, B′_{1}, B′_{2}) be Λ-modules associated with X and X′ respectively. If M(A_{1}, A_{2}, B_{1}, B_{2}) ≅ M(A′_{1}, A′_{2}, B′_{1}, B′_{2}) as Λ-modules then X ≅ X′ as Γ-modules.

Proof. The proof is a minor modification of that of Theorem 3.9, so we omit most details except where the two situations differ.

(i) Use Lemma 3.2, remembering that ker ρ = 𝔪S = 𝔪X.

(ii) The proof of Theorem 3.9(ii) works in the current context except for the proof of statement (a): im(B_{1}, B_{2}) contains no nonzero Γ̄-submodules. If (x, y) is an element of any Γ̄-module, then that module also contains (x, 0) and (0, y) (since Γ̄ = k ⊕k). But since each of B_{1} and B_{2} has linearly independent rows, right multiplication by these matrices is a one-to-one map. Therefore (x, 0) ∈ im(B_{1}, B_{2}) implies x = 0, and the analogous statement holds for (0, y).

(iii) Let M be a finitely generated Λ-module. As in the proof of Theorem 3.9(iii) we take a separated cover ϕ: S → → M, where S is a Λ-submodule of X := ΓS = Γ ⊗_{Λ} S. We need to prove that S = S(A_{1}, A_{2}) and ker(ϕ) = im(B_{1}, B_{2}) for appropriate matrices A_{1}, A_{2}, B_{1}, B_{2}. In fact, we do this with A_{1} and A_{2} equal to identity matrices. (However, we will not always have each A_{i} = I, later in this paper.)

Since the Λ-module S is finitely generated [Proposition 2.5], so is the Γ-module X = Γ ⊗_{Λ} S. Since Γ_{1}, Γ_{2} are DVRs, we can take X to be as displayed as in (4.3.1), provided that we can show that both of X_{1} and X_{2} have the same number of indecomposable direct summands. This is done in (4.3.2). We use the inclusion S ⊂ X as the top row of pullback square (4.4.5) that we are building.

For some n we have S/𝔪S ≅ k^{(n)} and X/𝔪X ≅ Γ̄^{(n)} as Λ- and Γ-modules, respectively, by Lemma 3.2. We use the inclusion map k^{(n)} ⊂ Γ̄^{(n)} for the bottom row. Define the right hand vertical map to be our usual map ρ, and define the left-hand vertical map to be the restriction of ρ to S. Since 𝔪S = 𝔪ΓS = 𝔪X, we have that S is the pulback of this diagram and, in fact, S = S(I_{n}, I_{n}) as desired.

Now we show that ker(ϕ) has the required form, im(B_{1}, B_{2}). By Lemma 2.3, ker(ϕ) is a k-module. Choose a k-linear identification ker(ϕ) = k^{(q)}, for some q. The projection of ker(ϕ) in each X_{i} (i = 1, 2) is again a k-submodule of X_{i}. Therefore the projection of ker(ϕ) in X_{1} is contained in (k, 0)^{(n−d_{∞})}. Moreover, Lemma 2.3 together with the fact that we are in the strictly split case yields ker(ϕ) ⊆ 𝔪S = 𝔪X = 𝔪_{1}X_{1} ⊕ 𝔪_{2}X_{2}. Since 𝔪_{1} annihilates the left Γ_{1}-module Γ_{1}/𝔪_{1}, we see that the projection of ker(ϕ) in any indecomposable summand of X_{1} of length 1 is zero. An analogous statement, with d replaced by e, holds for the projection in X_{2}. This gives us two q-rowed matrices B_{1}, B_{2} over k such that ker(ϕ) = k^{(q)}(B_{1}, B_{2}), and every column with length label 1 consists of zeros. Moreover, right multiplication by (B_{1}, B_{2}) from k^{(q)} to ker(ϕ) is one-to-one.

All that remains to be shown now is that each of these matrices has linearly independent rows; that is, right multipication by each of B_{1} and B_{2} is one-to-one. Suppose that (say) xB_{1} = 0. Then x·(B_{1}, B_{2}) = (0, x·B_{2}) is contained in the Γ-socle of X_{2}. Note that every Λ-submodule of X_{2} is a Γ-module. Therefore the Λ-module kx · (B_{1}, B_{2}) = (0, x·B_{2}) of ker(ϕ) is a Γ-submodule. By Lemma 2.3, it follows that x·(B_{1}, B_{2}) = (0, 0). Since right multiplication by (B_{1}, B_{2}) is one-to-one, we now have x = 0, as desired. This completes the proof of (iii).

(iv) This is exactly the same as the proof of Theorem 3.9(iv).

The final main result of this section determines when M(A_{1}, A_{2}, B_{1}, B_{2}) ≅ M(A′_{1}, A′_{2}, B′_{1}, B′_{2}) in terms of the matrices involved. Because of Theorem 4.6(iv) we can restrict our attention to the situation that both matrices arise from the same associated matrix setup.

Theorem 4.7 (Matrix Operations, strictly split case). Suppose that (A_{1}, A_{2}, B_{1}, B_{2}) and (A′_{1}, A′_{2}, B′_{1}, B′_{2}) are matrix 4-tuples over k associated with the matrix setup X (where Λ is strictly split Dedekind-like with residue field k). Then M(A_{1}, A_{2}, B_{1}, B_{2}) ≅ M(A′_{1}, A′_{2}, B′_{1}, B′_{2}) if and only if (A′_{1}, A′_{2}, B′_{1}, B′_{2}) can be obtained from (A_{1}, A_{2}, B_{1}, B_{2}) by a finite sequence of the following operations.

(i) (a) Simultaneously left multiply A_{1} and A_{2} by an invertible matrix over k. (b) Simultaneously left multiply B_{1} and B_{2} by an invertible matrix over k.

(ii) (a) For any length label j right-multiply A_{1}[cols j] by an invertible matrix Q

over k and, if j ≠ ∞, simultaneously right-multiply B_{1}[cols j] by Q.

(b) For any length label j right-multiply A_{2}[cols j] by an invertible matrix Q over k and, if j ≠ ∞, simultaneously right-multiply B_{2}[cols j] by Q.

(iii) (a) For any i > j add a k-scalar multiple of an i-labeled column of A_{1} to a

j-labeled column of A_{1}. (“Sweep toward smaller lengths in A_{1}.”)

(b) For any i > j add a k-scalar multiple of an i-labeled column of A_{2} to a j-labeled column of A_{2}. (“Sweep toward smaller lengths in A_{2}.”)

(c) For any i < j add a k-scalar multiple of an i-labeled column of B_{1} to a j-labeled column of B_{1}. (“Sweep toward larger lengths in B_{1}.”)

(d) For any i < j add a k-scalar multiple of an i-labeled column of B_{2} to a j-labeled column of B_{2}. (“Sweep toward larger lengths in B_{2}.”)

We call these matrix operations “k-k sweeping similarity” operations.

Proof. We omit the proof, which is similar to the proof of Theorem 3.10 and is given in detail in [L3, Theorems 4.6, 4.7], albeit it with many changes of notation. (See next Remark.)

Remark 4.8 (Comparison with notation in [L3]). The translation of isomorphism of modules to a matrix problem is carried out in [L3, §4] for commutative rings called “Dedekind-like.” In the local case, this is what we call “split Dedekind-like” in [KL2], and is slightly more general than what we call “strictly split Dedekindlike” in the present paper. But in the complete local case, the two are the same. Many of the difficulties one has in reading [L3] stem from the fact that the local case was not completely detached from the global case, as in the present paper.

The four matrices that we call A_{1}, A_{2}, B_{1}, B_{2} are called C^{−1}, D^{−1}, A, B respectively in [L3], and matrices act via left multiplication there, rather than via right multiplication, as in the present paper. Instead of making the direct module-versus-matrix connection, as done in the present paper, [L3] first connects a given module with a diagram D that gives a detailed picture of the separated cover (called a “separated representation” there) of the module [L3, §2] and then converts isomorphism of diagrams to a matrix problem [L3, §4]. The nature of the matrices in a matrix setup is given in [L3, Proposition 4.5], and our present Matrix Operations Theorem 4.7 is given in [L3, 4.6, 4.7]. In the strictly split local case, the case considered in the present paper, one can ignore the “global condition” in [L3, 4.7] since it becomes trivial locally.

We return to this global condition, in greater generality, in the third paper of the present series.

Notation 4.9. We write (A, B, C, D) ≅ (A′, B′, C′, D′) to indicate matrix 4-tuples (including thie length labels) obtained from each other by means of display operations [Definition 4.5] and k-k sweeping similarity operations [Theorem 4.7]. It is easy to see that, when such an isomorphism holds, we can do the display operations before the sweeping-similarity operations.

As in the unsplit case [Theorem 3.14], one proves:

Theorem 4.10. (In the strictly split case:) Every finitely generated Λ-module is isomorphic to some M(A, B, C, D). Moreover, M(A, B, C, D) ≅ M(A′, B′, C′, D′) if and only if (A, B, C, D) ≅ (A′, B′, C′, D′).

5. Solution of Matrix Problem, Strictly Split Case

In this section we describe the canonical form to which matrix 4-tuples (which arise in the split case) can be reduced, using display operations [Definition 4.5] and sweeping-similarity operations [Theorem 4.7]. As in the previous section, this was solved in [L3], so our object in the present section is to focus on establishing consistency with the notation of the present series. We then refer to [KL1] for a better proof of the reduction to canonical form than was in [L3].

Our canonical forms of indecomposable 4-tuples are of two types that we call “deleted cycle” and “block cycle” 4-tuples.

Throughout this section Λ is strictly split Dedekind-like, and all matrices have entries in the residue field k of Λ.

Definitions 5.1 (Direct sums). We define the direct sum

(A_{1}, A_{2}, B_{1}, B_{2}) = (A′_{1}, A′_{2}, B′_{1}, B′_{2}) ⊕ (A′′_{1}, A′′_{2}, B′′_{1}, B′′_{2})

of matrix 4-tuples to be the expected matrix 4-tuple: A_{1} = [ \begin{matrix}A′_{1} & 0 \\ 0 & A′′_{1} \end{matrix} ] with length labels those of A′_{1} followed by those of A′′_{1}, and the remaining three matrices in this 4-tuple defined similarly.

For example, consider the following matrix 4-tuple, where the numbers 3, 6, 3, ∞, 5 are the length labels of the columns of A_{1} as well as the corresponding columns of B_{1}, and an analogous comment applies to A_{2} and B_{2}.

\begin{aligned}& 3 6 3 ∞ 5 & & 6 4 1 6 ∞ \\ A_{1} = & [\begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}] & A_{2} = & [\begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}] \\ B_{1} = & [\begin{matrix}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] & B_{2} = & [\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ] \end{aligned} (5.1.1)

This 4-tuple is the direct sum of the two 4-tuples whose A-matrices are identity matrices of appropriate sizes and whose B-matrices, including their length labels, are

(B′_{1}, B′_{2}) = ( \begin{matrix}3 & 6 & 3 \\ [ & 0 1 0 0 0 1 & ] \end{matrix} , \begin{matrix}6 & 4 & 1 \\ [ & 1 0 0 0 1 0 & ] \end{matrix} ) (B′′_{1}, B′′_{2}) = ( \begin{matrix}∞ & 5 \\ [ & 1 & ] \end{matrix} , \begin{matrix}6 & ∞ \\ [ & 1 & ] \end{matrix} ) (5.1.2)

Note that B′′_{1} and B′′_{2} are 1 × 1 matrices, with no actual columns labeled by ∞.

Definitions 5.2 (Canonical forms). We call our first canonical form a deleted cycle 4-tuple. It is determined by:

(5.2.1) A pair of label sequences i_{1}, i_{2}, …, i_{d} and j_{1}, j_{2}, …, j_{d} (d ≥ 1), where each term is a positive integer or ∞, but only i_{1} and j_{d} are allowed to equal ∞ or 1.

The deleted cycle matrix 4-tuple determined by these sequences is defined as follows. Set A_{1} = A_{2} = I_{d}, with the length labels of A_{1} and A_{2} given respectively by the two sequences in (5.2.1). The matrices B_{1} and B_{2} are as displayed in (5.2.2).

(B_{1}, B_{2}) = ( \begin{matrix}i_{1} & … \\ [ & Z I_{d−1} & ] \end{matrix} , \begin{matrix}… & j_{d} \\ [ & I_{d−1} Z′ & ] \end{matrix} ) (5.2.2)

Here, the dots over the first I_{d−1} indicate that its columns are labeled by i_{2}, …, i_{d} and the dots over the second I_{d−1} indicate that its columns are labeled by j_{1}, …, j_{d−1}. Column Z consists of zeros if the label i_{1} is finite, but the column is not present if i_{1} = ∞. Similarly, Z′ is a column of zeros if the label j_{d} is finite, but is not present if j_{d} = ∞.

If, for example, i_{1} is finite and j_{d} infinite, then the sizes of B_{1} and B_{2} are (d − 1) × d and (d − 1) × (d − 1) respectively, and Z′ indicates that the ∞-labeled column of A_{2} has no corresponding column in B_{2}.

As a second example, the 4-tuple displayed in (5.1.1) is the direct sum of the two deleted cycle 4-tuples whose B-matrices are displayed in (5.1.2).

Our next canonical form requires the following technical definition.

(5.2.3) Let I, J be a pair of finite sequences of the same length, say d. We call the pair I, J repetition-free if there exists no pair of strictly shorter sequences I′, J′ of length, say c < d, such that I and J consist of d/c repetitions of I′ and J′, respectively.

For example, the pair of sequences 3, 2, 3, 3, 2, 3 and 6, 5, 4, 6, 5, 4 is not repetitionfree, since we can double the first three terms in each, but the pair 3, 2, 3, 3, 2, 3 and 6, 5, 6, 5, 6, 5 is repetition-free.

We call our other canonical form a block cycle 4-tuple. It is determined by:

(5.2.4) (i) A pair of label sequences i_{1}, i_{2}, …, i_{d} and j_{1}, j_{2}, …, j_{d} (d ≥ 1), where each term is a positive integer not equal to 1 (never ∞!), and the pair of sequences is repetition-free (as defined above).

(ii) An invertible matrix L (over k), indecomposable under similarity, called the blocking matrix.

We display in (5.2.5) the block cycle 4-tuple determined by the data (5.2.4), and then explain it.

\begin{aligned}& i_{1} i_{2} i_{3} … i_{d} & & j_{1} j_{2} j_{3} … j_{d} \\ A_{1} = & [ \begin{matrix}L & 0 & 0 & … & 0 \\ 0 & I_{m} & 0 & … & 0 \\ 0 & 0 & I_{m} & … & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & … & I_{m} \end{matrix}] & A_{2} = & [\begin{matrix}I_{m} & 0 & 0 & … & 0 \\ 0 & I_{m} & 0 & … & 0 \\ 0 & 0 & I_{m} & … & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & … & I_{m} \end{matrix}] \\ B_{1} = & [\begin{matrix}0 & I_{m} & 0 & … & 0 \\ 0 & 0 & I_{m} & … & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & … & I_{m} \\ I_{m} & 0 & 0 & … & 0 \end{matrix}] & B_{2} = & [\begin{matrix}I_{m} & 0 & 0 & … & 0 \\ 0 & I_{m} & 0 & … & 0 \\ 0 & 0 & I_{m} & … & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & … & I_{m} \end{matrix}] \end{aligned} (5.2.5)

Let the size of L be m × m; then all four matrices have size dm × dm. Matrices A_{1} and A_{2} have the displayed block diagonal form, with all blocks of size m × m, the upper left-hand block of A_{1} equals L, and all other diagonal blocks equal I_{m}.

Each term of each label sequence appears m times as a length label (although we display it only once here), thus labeling all m columns of the block over which it is displayed. The matrix B_{1} has identity blocks on the super-diagonal, one identity block in the lower left-hand corner, and zeros elsewhere. Thus — as suggested by the name “block cycle” — B_{1} is a block permutation matrix whose associated permutation is a cycle. We also have B_{2} = I_{dm} = A_{2}. We emphasize that ∞ and 1 never occur in the label sequences of a block cycle 4-tuple.

Recall [Notation 4.9] that two matrix 4-tuples are called isomorphic if one can be transformed into the other by means of display operations and k-k sweeping similarity operations. Let I, J and I′, J′ be two pairs of finite sequences, all of the same length, say d. We say that I, J equals I′, J′ modulo simultaneous cyclic permutations if there is a cyclic permutation σ of d symbols such that σ(I) = I′ and σ(J) = J′. Finally, we call a matrix 4-tuple indecomposable if it is not isomorphic to the direct sum of two matrix 4-tuples of strictly smaller size.

Theorem 5.3. (i) Every deleted cycle 4-tuple is indecomposable. Two deleted cycle 4-tuples are isomorphic if and only if they are equal — which holds if and only if they have the same pair of label sequences.

(ii) Every block cycle 4-tuple is indecomposable. Two block cycle 4-tuples are isomorphic if and only if their pairs of label sequences are equal modulo simultaneous cyclic permutations and their blocking matrices are similar.

(iii) A deleted cycle 4-tuple is is never isomorphic to a block cycle 4-tuple.

Since only the similarity class of L is important, we note that L can always be chosen to be the companion matrix of some power g(X)^{e} of some irreducible polynomial g(X) ≠ X over k. Recall that “display operations” were defined in 4.5.

Theorem 5.4. Every matrix 4-tuple is isomorphic to a direct sum of deleted cycle and block cycle 4-typles. Moreover, two direct sums of deleted cycle and block cycle 4-tuples are isomorphic if and only if they contain the same deleted cycle summands and isomorphic block cycle summands, with the same multiplicities but ignoring order of occurrence.

Proof. These two theorems are a special case of [KL1, 1.3–1.7]. So we limit ourselves to a few comments relating the notation here to the notation there. First we note that the sweeping-similarity operations in Theorem 4.7 form a set of sweepingsimilarity operations, as described in [KL1, p.68], with one change: all sweeping in that paper is done to the right. But this is just a matter of notation, only affecting the way in which the problem is displayed. We note that, in our situation, no actual row-sweeping occurs; that is, all rows of A_{1} belong to a single row-block, and the same is true of the other three matrices.

Having made these observations, one then applies the aforementioned theorems from [KL1].

6. Solution of Matrix Problem, Unsplit Case

Throughout this section we assume that (Λ, 𝔪, k) is unsplit Dedekind-like. As usual, F denotes the residue field of the normalization Γ of Λ. In this section we always make the assumption (not invoked earlier in the paper) that the 2dimensional field extension F of k is separable. Our objective is to find a canonical form for indecomposable matrix pairs p = (A, B) (entries in F), in this situation.

Since F is Galois of dimension 2 over k, any matrix 4-tuple g over the field F has a natural conjugate ḡ (with appropriate length labels). Given a matrix pair p over the field F, our plan is to associate with p the matrix 4-tuple f(p) = (A, Ā, B, B̄). Our solution of the problem then falls into four parts.

(i) Show that p ≅ p′ if and only if f(p) ≅ f(p′) (Theorem 6.3). This reduces isomorphism in the unsplit case to isomorphism in the strictly split case, whose canonical forms were summarized in the previous section. But our 4-tuples now have entries in F instead of k.

(ii) For each self-conjugate indecomposable 4-tuple g we describe a pair p such that f(p) ≅ g, and for each non-self-conjugate indecomposable 4-tuple g we describe a pair p such that f(p) ≅ g ⊕ ḡ. We obtain seven types of canonical forms p in this way, and it is easy to see that they are all indecomposable pairs.

Our proof that every self-conjugate indecomposable 4-tuple has the form f(p), for some matrix pair p, uses a “Hilbert Theorem 90” for matrices, due to Ballantine.

(iii) We show that the list of canonical forms in (ii) is complete. The critical theorem is that a 4-tuple g is a “package” — that is, g ≅ f(p) for some pair p — if and only if g ≅ ḡ. The proof of this seems to require the list of seven canonical forms listed in (ii).

An easy consequence of all of this is that the Krull-Schmidt Theorem holds for matrix pairs.

(iv) The final main theorem in this section describes the precise extent to which our canonical forms are uniquely determined by their isomorphism class.

The results of the present section are purely matrix-theoretic results involving any pair of fields k and F, with F separable of degree 2 over k. In other words, Λ and Γ are really irrelevant. In fact we use this observation at one critical point in the proof of part (i). But by keeping Λ and Γ in the background (rather than omitting them), we avoid the necessity of introducing new terminology.

Notation 6.1. Since F is separable of dimension 2 over k, there is precisely one non-identity automorphism of F that equals the identity on k. We denote the image of α ∈ F, under this automorphism, by ᾱ, the conjugate of α.

Let p = (A, B) be a matrix pair associated with a matrix setup X (unsplit case) [Notation 3.7], and recall that every column of A and B has an attached length label, a positive integer or ∞. Let f(p) = (A, Ā, B, B̄), the matrix 4-tuple in which “bar” indicates that we take the conjugate of every entry of the matrix. The length labels of f(p) are obtained by keeping the original length labels of A and B, and giving Ā and B̄ the same length labels as A and B, respectively. We call f(p) the matrix 4-tuple associated with p. It is straightforward to verify that the length labels of f(p) are a legitimate set of length labels for some matrix setup in the strictly split case [Notation 4.3], and f(p) is a legitimate matrix 4-tuple associated with that setup [Notation 4.4].

Recall [Notation 3.13] that, for matrix pairs p and p′, we write p ≅ p′ if p′ can be obtained from p by display operations and k-F sweeping-similarity operations [Theorem 3.10].

Let g = (C_{1}, C_{2}, D_{1}, D_{2}) and g′ = (C′_{1}, C′_{2}, D′_{1}, D′_{2}) be matrix 4-tuples with entries in F (associated with suitable matrix setups for a strictly split Dedekindlike ring with residue field F). We write g ≅ g′ if g′ can be obtained from g by display operations and F -F sweeping-similarity operations, the the analog over F of the k-k sweeping similarity operations of Theorem 4.7.

The conjugate of the matrix 4-tuple g = (C_{1}, C_{2}, D_{1}, D_{2}) is the 4-tuple ḡ = (C̄_{2}, C̄_{1}, D̄_{2}, D̄_{1}) obtained by interchanging C_{1} with C_{2}, together with their length labels, doing the same with D_{1} and D_{2}, and then taking the conjugates of all four matrices. We call the matrix 4-tuple g self-conjugate if g ≅ ḡ. We call the matrix 4-tuple g a package if g ≅ f(p) for some matrix pair p. We call a package g indecomposable if it is not the direct sum of two packages.

¿From the definitions, it is evident that every package is self-conjugate. The converse of this is also true, as we show in Theorem 6.15.

We always view F ⊗_{k} F as an F -algebra by means of the scalar product γ·(α ⊗ β) = γα ⊗ β.

Lemma 6.2. (i) F ⊗_{k}F ≅ F ⊕F as F -algebras via the map defined by τ(α⊗β) =

(αβ, αβ̄). (ii) F ⊗_{k} F [X] ≅ F [X] ⊕ F [X] as F -algebras (X an indeterminate) via the exten-

sion τ(α ⊗ f) = (αf, αf̄) of the map in (i), where f̄ is the result of replacing every coefficient of f by its conjugate over k.

Proof. (i) This well-known fact is easily proved by first choosing any ε ∈ F − k and noting that F = k[ε] since F has dimension 2 over k. Since, in addition, F is separable over k, the minimal polynomial of ε over k is m(X) = (X −ε)(X −ε̄) and has distinct linear factors. Therefore F ⊗_{k}F ≅ F ⊗k[X]/(m(X)) ≅ F [X]/(m(X)) ≅ F [X]/(X − ε) ⊕ F [X]/(X − ε̄), and the result follows easily.

(ii) Use (i), and note that F ⊗_{k} (F [X]) ≅ (F ⊗_{k} F)[X] ≅ (F ⊕ F)[X] ≅ F [X] ⊕ F [X].

Theorem 6.3. Let p and p′ be matrix pairs. Then p ≅ p′ if and only if f(p) ≅ f(p′).

Proof. This proof is an adaptation a familiar “extension of scalars” argument. The main difficulty to be surmounted is that Λ need not be an algebra over a field.

First note that the matrix assertions in the theorem ultimately depend only on the pair of fields k and F. Hence we can choose Λ to be any convenient unsplit Dedekind-like ring such that the bottom row of its pullback diagram (1.1.1) is the inclusion k ⊂ F. Let Γ = F [X]_{(X)}, the localization of the polynomial ring F [X] that inverts all monic irreducible polynomials except X, and Λ = k + X·F [X]_{(X)}. Then Λ is an unsplit Dedekind-like ring with normalization Γ, and 𝔪 = X·F [X]_{(X)} is the maximal ideal of both Λ and Γ. Moreover, the pullback diagram for Λ is the first square displayed in (6.3.1), with ker(ρ) = 𝔪.

\begin{aligned}Λ & ⊂ & Γ \\ ^{↓ ↓}ρ & & ^{↓ ↓}ρ \\ k & ⊂ & F \end{aligned} \begin{matrix}F ⊗_{k} Λ & ⟶^{τ} & F ⊗_{k} Γ = Γ ⊕ Γ \\ \begin{matrix}↓ & ↓ \\ 1 ⊗ ρ \end{matrix} & & \begin{matrix}↓ & ↓ \\ 1 ⊗ ρ \end{matrix} \\ F & ⟶^{τ} & F ⊗_{k} F = F ⊕ F \end{matrix} (6.3.1)

We claim that F ⊗_{k}Λ is the strictly split Dedekind-like ring whose pullback diagram is the second square displayed in (6.3.1). The equality in the upper right-hand corner refers to the identification α ⊗ f = (αf, αf̄) from Lemma 6.2(ii). The map labeled τ in the top row is 1 ⊗ i, where i is the inclusion map, and can therefore be identified with the map τ(α ⊗ f) = (αf, αf̄). Similarly we regard the bottom row, via Lemma 6.2(i), to be our choice of 1⊗i and the restriction τ(α) = α⊗1 = (α, α) of the map τ in the top row. This square is a pullback diagram because tensoring a pullback diagram by a flat module again yields a pullback diagram [K, Proposition 2.10] or [L3, Lemma 6.1].

When the indicated identifications in the second square are made, the rightmost vertical map becomes the natural coordinate-by-coordinate map with kernel 𝔪⊕𝔪. This completes the proof of the claim.

Now we have a pair Λ, Γ to use in a standard extension of scalars argument. Since the argument involves a lot of details, we describe only the main steps. Let M, N be finitely generated Λ-modules. Then we claim:

(6.3.2) M ≅ N ⟺ F ⊗_{k} M ≅ F ⊗_{k} N (as F ⊗_{k} Λ-modules).

The proof of the nontrivial (⇐) assertion is that, since F is a 2-dimensional k-vector space, the second isomorphism implies that M ⊕ M ≅ N ⊕ N as Λ-modules; and the Krull-Schmidt Theorem holds over strictly split Dedekind-like rings [Lemma 1.3].

Let Λ′ = F ⊗_{k} Λ and Γ′ = F ⊗_{k} Γ = Γ ⊕ Γ. When it is necessary to distinguish between the two direct summands of Γ′, we write Γ′ = Γ_{1} ⊕Γ_{2}. (Note that Γ_{1} ≅ Γ_{2} as F -algebras and Γ-modules, but not as Γ′-modules, being annihilated by (0, Γ) and (Γ, 0), respectively.) The length labels of our given matrix pair p determine the Γ-module X in some matrix setup X for Λ, and the length labels of f(p) determine the Γ′-module X′ = X_{1} ⊕ X_{2} in some matrix setup X′ for Λ′. In fact, one verifies with the help of Lemma 6.2(ii), that for every indecomposable summand Y = Γ or Y = Γ/𝔪^{i} in the definition of X, we have F ⊗_{k} Y = Y_{1} ⊕ Y_{2} where Y ≅ Y_{1} ≅ Y_{2} as Γ-modules, (but not as Γ′-modules).

It now suffices to show that

(6.3.3) F ⊗_{k} M(A, B) ≅ M(A, Ā, B, B̄) [as (F ⊗_{k} Λ)-modules],

where p = (A, B) and hence f(p) = (A, Ā, B, B̄). For it then follows from (6.3.2) that M(p) ≅ M(p′) if and only if M(f(p)) ≅ M(f(p′)), after which our two theorems on modules versus matrices 3.14 and 4.10 (unsplit and split cases, respectively) imply that p ≅ p′ if and only if f(p) ≅ f(p′).

The proof of (6.3.3) consists of reviewing the definitions of the maps in the definitions of M(A, B) and M(A, Ā, B, B̄) and noting that, for every map θ that arises in the definition of M(A, B), the map 1 ⊗ θ is the direct sum of two corresponding maps in the definition of M(A, Ā, B, B̄), and for every entry β of A (respectively B), the tensor product 1 ⊗ β corresponds to a pair of entries (β, β̄) of the matrices A and Ā (respectively B and B̄), under the map τ of Lemma (6.2)(i).

Lemma 6.4. Given ε ∈ F − K, let Y be an invertible m × m matrix over the field F, and consider the conjugate pair of 2m × 2m matrices T and T̄ below.

T = [\begin{matrix}Y & I_{m} \\ εY & ε̄I_{m} \end{matrix}] ,  T̄ = [\begin{matrix}Ȳ & I_{m} \\ ε̄Ȳ & εI_{m} \end{matrix}] (6.4.1)

Then there is an invertible matrix P over F such that P T and P T̄ equal the following two matrices, respectively.

[\begin{matrix}Y & 0 \\ 0 & I_{m} \end{matrix}] , [\begin{matrix}0 & I_{m} \\ Ȳ & 0 \end{matrix}] (6.4.2)

(We often refer to the pair T and T̄ as “basic tiles”.)

Proof. We prove the lemma by performing simultaneous row operations on the matrices in (6.4.1). Subtracting ε times the first m rows from the last m rows in both matrices yields:

[\begin{matrix}Y & I_{m} \\ 0 & δI_{m} \end{matrix}] [\begin{matrix}Ȳ & I_{m} \\ δȲ & 0 \end{matrix}] (6.4.3)

where δ = ε̄ − ε ≠ 0, since F is separable over k. Dividing the last m rows by δ in both matrices in (6.4.3) and then subtracting the last m rows from the first m rows yields the desired matrices (6.4.2).

The following simple lemma helps to motivate our systematic enumeration of canonical forms for matrix pairs (but does not show that the resulting list is complete).

Lemma 6.5. Let p be a matix pair such that f(p) is either an indecomposable self-conjugate 4-tuple, or else f(p) ≅ g ⊕ ḡ where g is indecomposable and nonself-conjugate. Then p is indecomposable (equivalently, f(p) is an indecomposable package).

Proof. Only the case f(p) ≅ g ⊕ ḡ is nontrivial. So suppose that p ≅ q ⊕ r, and hence f(p) ≅ f(q) ⊕ f(r). Since g is indecomposable, so is ḡ. Since the Krull-Schmidt Theorem holds for matrix 4-tuples [Theorem 5.4], we must therefore have, say, f(q) ≅ g. But, since f(q) is a package, it is obviously self-conjugate. Therefore g ≅ ḡ, a contradiction.

The following notation will be fundamental throughout this section.

Notation 6.6 (Sequence manipulation). Given the finite sequence I = {x_{1}, …, x_{d}}, we let μ be the mirror image permutation that reverses the order of I; that is, μ(I) = {x_{d}, x_{d−1}, …, x_{1}}. We let ν be the unit forward rotation, that is, the cyclic permutation defined by ν(I) = {x_{2}, x_{3}, …, x_{d}, x_{1}}. (In fact the permutations μ and ν depend upon the length d of the sequence, but this will always be clear from context.) A cycle is any cyclic permutation of the form ν^{t}, where t is an integer. Thus, ν^{t}(I) = {x_{t+1}, x_{t+2}, …, x_{t−1}, x_{t}} (reading subscripts modulo t). We frequently use the following, easily verified relation.

ν^{t}μ = μν^{−t} (6.6.1)

For finite sequences I and J we use the notation {I, J} for the concatenation of I and J, that is, the sequence consisting of the terms of I followed by the terms of J.

Definition 6.7 (Pairs in canonical form arising from deleted cycle 4-tuples). For each pair p = (A, B) that we describe here, we shall show that f(p) is isomorphic to the direct sum of one or two deleted cycle 4-tuples [Theorem 6.9]. The name that we use for each individual form is chosen to coincide with the name of the corresponding type of module, described in §7. (The names themselves will seem more natural after reading the module descriptions in §7.)

All blank spaces in our matrix displays — except ellipsis dots — represent zeros; and ε denotes any element of F − k (hence F = k[ε]).

The data from which each matrix pair is constructed is a pair of label sequences

I = {i_{1}, i_{2}, …, i_{d}}  and  J = {j_{1}, j_{2}, …, [j_{d}]}. (6.7.1)

in which each term is a positive integer or ∞. The term j_{d} is in brackets because, in some types, it is not present.

Remark. Note that we use a pair of label sequences to define the deleted cycle matrix pairs below, even though a matrix pair (A, B) needs only a single label sequence for labeling the columns of both A and B. As we shall see below, the two label sequences I and J are woven together to form a single label sequence for the matrix pairs to be defined. The reason for splitting into two sequences I and J in this definition is that these separate sequences will play an important role in determining the form of the associated matrix 4-tuple. (See Theorem 6.9 below.)

Nonreduced pair. Label j_{d} occurs. The label sequences must satisfy:

(6.7.2) (i) Only labels i_{1} and j_{d} can equal ∞ or 1.

(ii) The pair of label sequences I and J must be unsymmetrical, in the sense that J ≠ μ(I) (the mirror image of I).

We first display, then explain, the form of such a pair.

\begin{matrix}i_{1} & j_{1} & i_{2} & j_{2} & i_{3} & … & j_{d−1} & i_{d} & j_{d} \\ A = & [ \begin{matrix}1 & 1 & & & … \\ ε & ε̄ & & & … \\ & & 1 & 1 & … \\ & & ε & ε̄ & … \\ . & . & . \\ & & & & … & 1 & 1 \\ & & & & … & ε & ε̄ \end{matrix} ] \\ B = & [ \begin{matrix}Z & 1 & 1 & & & … & & & Z \\ Z & ε & ε̄ & & & … & & & Z \\ Z & & & 1 & 1 & … & & & Z \\ Z & & & ε & ε̄ & … & & & Z \\ ⋮ & & & & & ... & & & ⋮ \\ Z & & & & & … & 1 & 1 & Z \\ Z & & & & & … & ε & ε̄ & Z \end{matrix} ] \end{matrix} (6.7.3)

The row of length labels at the top, comes from the pair of label sequences. Each column of Z’s is a column of zeros except if the length label is ∞, in which case that column is not present in B (because no column of B is allowed to have an infinite length label). Note that A is a direct sum of 2 × 2 basic tiles (6.4.1), while B is a modified such direct sum. The basic tiles that appear in A always overlap the tiles that appear in B in such a way that every column except the first and last of (A, B) contains the first (respectively second) column of some tile in A and the second (respectively first) column of some tile in B.

We shall show, for nonreduced pairs p, that f(p) is isomorphic to the direct sum of two non-self-conjugate deleted cycle 4-tuples [Theorem 6.9]. The reason for nonsymmetry condition (6.7.2)(ii) is that, when it is violated, f(p) becomes isomorphic to the direct sum of two self-conjugate 4-tuples, and p itself is isomorphic to the direct sum of two isomorphic indecomposable pairs [Proposition 6.19].

Bottom-reduced pair. Label j_{d} occurs. We require the label sequences to satisfy:

(6.7.4) Only label i_{1} can equal ∞ or 1.

(There is no symmetry restriction.) The matrices are the same as in the non-reduced case, except that B has the following additional row at its bottom.

\begin{matrix}i_{1} & j_{1} & i_{2} & j_{2} & i_{3} & … & i_{d} & j_{d} \\ (B :) & [ Z 1 ] \end{matrix} (6.7.5)

where Z denotes zero except if i_{1} = ∞, in which case that column of B is not present. As usual, all entries in this row that are not explicitly shown equal 0. In the extreme case d = 1, the new bottom row consists only of Z (if it is present) and 1.

Top-reduced pair. In this case, label j_{d} is not present. We require the label sequences to satisfy (6.7.4), and there is no symmetry restriction. The matrix pair takes the following form, in which the previously discussed conventions apply.

\begin{matrix}i_{1} & j_{1} & i_{2} & j_{2} & i_{3} & … & i_{d−1} & j_{d−1} & i_{d} \\ A = & [ \begin{matrix}1 & 1 & & & … \\ ε & ε̄ & & & … \\ & & 1 & 1 & … \\ & & ε & ε̄ & … \\ . & . & . \\ & & & & … & 1 & 1 \\ & & & & … & ε & ε̄ \\ & & & & … & & & 1 \end{matrix} ] \\ B = & [ \begin{matrix}Z & 1 & 1 & & & … \\ Z & ε & ε̄ & & & … \\ Z & & & 1 & 1 & … \\ Z & & & ε & ε̄ & … \\ ⋮ & & & & & ... \\ Z & & & & & … & 1 & 1 \\ Z & & & & & … & ε & ε̄ \end{matrix} ] \end{matrix} (6.7.6)

In the extreme case d = 1, A is the one-by-one identity matrix (i.e. no basic tiles occur) and B is the one-by-one zero matrix except if i_{1} = ∞, in which case the matrix B does not occur.

We shall show [Theorem 6.9] that, if p is either of the reduced pairs, then f(p) is a self-conjugate deleted cycle 4-tuple. First, we need to determine which deleted cycle 4-tuple is isomorphic to the conjugate of a given deleted cycle 4-tuple.

Lemma 6.8. Let g and h be the deleted cycle 4-tuples whose label sequences are:

\begin{aligned}g: & H & = & \{i_{1}, i_{2}, …, i_{e}\} & and & K & = & \{j_{1}, j_{2}, …, j_{e}\} \\ h: & μ(K) & = & \{j_{e}, j_{e−1}, …, j_{1}\} & and & μ(H) & = & \{i_{e}, i_{e−1}, …, i_{1}\} \end{aligned} (6.8.1)

Then ḡ ≅ h. In particular, the deleted cycle 4-tuple g is self-conjugate if and only if K = μ(H).

Proof. We have g = (I_{e}, I_{e}, [Z I_{e−1}], [I_{e−1} Z]) with length labels as displayed. Therefore we have ḡ = (I_{e}, I_{e}, [I_{e−1} Z], [Z I_{e−1}]), and the label sequences of ḡ are the pair K and H. We transform ḡ to h.

Let h′ be the 4-tuple obtained by the display operation that replaces the pair of label sequences by μ(K) and μ(H), respectively. This reverses the order of columns of each A_{i}, together with their corresponding columns in the B-matrices. Then h′ ≅ ḡ and the label sequences are now as desired. The first and second matrices of h′ are now the matrix R_{e} obtained by reversing the columns of I_{e}. The third and fourth matrices are now [Z R_{e−1}] and [R_{e−1} Z]. Therefore, left multiplying the first and second matrices of h′ by R_{e}^{−1} changes these two matrices to I_{e} and left-multiplying the third and fourth matrices by R_{e−1}^{−1} changes them to [Z I_{e−1}] and [I_{e−1} Z] respectively. In other words, we have now changed h′ to h, as desired.

The last statement now follows from Theorem 5.3.

Theorem 6.9. Let p = (A, B) be any of the types of matrix pairs listed below, constructed from the pair of label sequences

I = {i_{1}, …, i_{d−1}, i_{d}} and J = {j_{1}, …, j_{d−1}, [j_{d}]} (6.9.1)

Then f(p) is as described below, where μ is the mirror image permutation as in Notation 6.6.

(i) p is non-reduced. Then f(p) is isomorphic to g ⊕ ḡ where g is the non-self-conjugate deleted cycle 4-tuple whose pair of label sequences is I and J.

(ii) p is bottom-reduced. Then f(p) is isomorphic to the deleted cycle 4-tuple g ≅ ḡ

whose pair of label sequences is {I, μ(J)} and {J, μ(I)}. (iii) p is top-reduced. Then f(p) is isomorphic to the deleted cycle 4-tuple g ≅ ḡ

whose pair of label sequences is {I, μ(J)} and {J, μ(I)}. (Recall that, in this case, the label j_{d} does not occur.)

In all cases, p is an indecomposable matrix pair (equivalently, f(p) is an indecomposable package). Conversely, if g is a non-self-conjugate deleted cycle 4-tuple, then there is some non-reduced matrix pair p with f(p) ≅ g⊕ḡ, and if g is a self-conjugate deleted cycle 4-tuple, then there is some bottom reduced or top-reduced matrix pair p with f(p) ≅ g.

Proof. (ii) Let g and p be as stated. Our plan is to use display operations and F -F sweeping similarity operations to transform f(p) to g [see Notation 4.9 and Theorem 4.10].

Choose a pair of rows of A and Ā whose nonzero columns consist of a basic tile and its conjugate, as displayed in (6.4.1). The assertion of Lemma 6.4 is that we can transform this pair of tiles to the form displayed in (6.4.2), by simultaneously leftmultiplying these two rows of A and Ā by an invertible matrix P over k. Since this can be accomplished by left-multiplying A and Ā themselves by a block diagonal matrix of the form diag(P, I, I, …, I), it is a sweeping similarity operation. Do this for every pair of basic tiles in f(p), calling the result h.

The form of h, when d = 3 for example, is displayed in (6.9.2) below with our usual convention that non-displayed entries equal zero.

(6.9.2) \begin{matrix}i_{1} & j_{1} & i_{2} & j_{2} & i_{3} & j_{3} \\ A_{1} = & [ \begin{matrix}1 & 0 \\ 0 & 1 \\ & & 1 & 0 \\ & & 0 & 1 \\ & & & & 1 & 0 \\ & & & & 0 & 1 \end{matrix} ] \\ B_{1} = & [ \begin{matrix}Z & 1 & 0 \\ Z & 0 & 1 \\ Z & & & 1 & 0 \\ Z & & & 0 & 1 \\ Z & & & & & 1 \end{matrix} ] \end{matrix}  \begin{matrix}i_{1} & j_{1} & i_{2} & j_{2} & i_{3} & j_{3} \\ A_{2} = & [ \begin{matrix}0 & 1 \\ 1 & 0 \\ & & 0 & 1 \\ & & 1 & 0 \\ & & & & 0 & 1 \\ & & & & 1 & 0 \end{matrix} ] \\ B_{2} = & [ \begin{matrix}Z & 0 & 1 \\ Z & 1 & 0 \\ Z & & & 0 & 1 \\ Z & & & 1 & 0 \\ Z & & & & & 1 \end{matrix} ] \end{matrix}

Note that certain columns of A_{1} also occur in A_{2}, but in different locations. More precisely:

(6.9.3) For every index κ, the i_{κ}-labeled column of A_{1} equals the j_{κ}-labeled column of A_{2}, and the j_{κ}-labeled colum of A_{1} equals the i_{κ}-labeled copy of A_{2}.

Note that statement (6.9.3) remains true after an arbitrary simultaneous column permutation of A_{1}, B_{1}, together with the row of labels above them. In other words, (6.9.3) remains true after arbitrary display operations. Let h′ be the result of doing the display operation that arranges the label sequences as follows.

{I, μ(J)} and {J, μ(I)} (6.9.4)

Then (6.9.3) implies that the A_{1} and A_{2}-matrices of h′ are equal (ignoring length labels). Call this matrix C. It is invertible, in fact, a permutation matrix. Leftmultiplying A_{1} and A_{2} by C^{−1} changes h′ to the form h′′ = (I, I, B′′_{1}, B′′_{2}).

Now consider the B-matrices. The B-matrices of h′′ are the same as those in (6.9.2) except for having their columns permuted by a display operation. We can therefore find the form of any column of the B-matrices of h′′ by finding the column with the same length label in the corresponding matrix in (6.9.2). Moreover, we can simultaneously left-multiply B′′_{1} and B′′_{2} by any permutation matrix. In other words, we can arbitrarily permute the order of occurrence of the rows of (B′′_{1}, B′′_{2}). Permute the rows of these two matrices so that the new B_{2} takes the form [I Z], with the newly relabeled columns. In our example B_{2} is as shown below, and the length labels of B_{1} are also shown in their new order.

(6.9.5) \begin{matrix}i_{1} & i_{2} & i_{3} & j_{3} & j_{2} & j_{1} \\ B_{1} = & [  ???  ] \end{matrix}  \begin{matrix}j_{1} & j_{2} & j_{3} & i_{3} & i_{2} & i_{1} \\ B_{2} = & [ \begin{matrix}1 & & & & & Z \\ & 1 & & & & Z \\ & & 1 & & & Z \\ & & & 1 & & Z \\ & & & & 1 & Z \end{matrix} ] \end{matrix}

This can be done because, if we ignore the Z-column of B_{2} in (6.9.2), B_{2} becomes a permutation matrix.

It now suffices to show that the new B_{1}, with its newly rearranged labels and rearranged rows, equals [Z I]. For this purpose, return to (6.9.2). The j_{1}-, j_{2}- and j_{3}-labeled columns of B_{2} have their nonzero entries in the same rows as the i_{2}-, i_{3}-, and j_{3}-labeled columns, respectively, of B_{1}. Therefore the same is true in the new (B_{1}, B_{2}). In other words, the first, second, and third rows of the new B_{1} have their nonzero entries in colums 2,3,4 respectively, as desired. The remaining columns are treated similarly. Also, the (i_{1}-labeled) Z-column occurs in the first column of the new B_{1}, as desired.

(iii) This is essentially the same as the proof of (ii), so we omit the details.

(i) This is similar to the proof of (ii). We sketch enough of the proof to show why we get a direct sum of two 4-tuples here, and to establish the following stronger statement that we need later.

(6.9.6) We have f(p) ≅ g ⊕ ḡ, even without the non-symmetry restriction (6.7.2)(ii) in the definition of a nonreduced pair.

Note that the only differences between a nonreduced pair and a bottom reduced pair are the presence of an extra row at the bottom of the latter, and the fact that nonreduced pairs are allowed to have the last column length-labeled by ∞.

Begin by reducing all pairs of basic tiles in f(p), as in the proof of (ii), calling the result h as before. To visualize the result in the case d = 3, delete the bottom row of the B-matrices in (6.9.2), and change the entries the last column of B_{1} and B_{2} to Z′. Continue the reduction until A_{1} = I = A_{2} (ignoring length labels) and — in the case d = 3 — we reach the situation displayed in (6.9.5), the only difference being that the j_{3}-labeled column of B_{2} is now the Z′-column. (Note that this column is now in the middle of the matrix!) Similarly the i_{1}- and j_{3}-labeled columns of B_{1} now have entries Z and Z′ respectively. The remaining entries of B_{1} are the same as in the bottom-reduced case, and this completes our proof-sketch of this case.

Supplementary statements. Indecomposability is a special case of Lemma 6.5. Completeness of our list follows easily from Lemma 6.8 and the definition of deleted cycle 4-tuples.

The next step is to do for block cycles what we just did for deleted cycles.

Definition 6.10 (Pairs in canonical form arising from block cycle 4-tuples). The pairs arising in these definitions are constructed from a blocking matrix U — always invertible, of size that we call the block size and usually write m × m — and a pair of label sequences:

I = {i_{1}, i_{2}, …, i_{d}}  and  J = {j_{1}, j_{2}, …, [j_{d}]}. (6.10.1)

in which each term is a positive integer ≠ 1 (never ∞). The term j_{d} is in brackets because, in some types, it is not present. Moreover, each label in (6.10.1) is the length label of m consecutive columns of A and B, namely the m columns in the block over which it appears in the diagrams that define the individual forms.

As in Definition 6.7, we weave together the two label sequences I and J to form a single label sequence for each of the matrix pairs defined below. The two label sequences I and J reappear when we consider the associated matrix 4-tuples in Theorem 6.14 below.

We shall show [Theorem 6.14] that if p is any of the “reduced” pairs below, then f(p) is a self-conjugate block cycle 4-tuple. In the remaining case, cycle pairs, f(p) is the direct sum of two conjugate, but non-self-conjugate, block cycle 4-tuples.

Bottom-bottom reduced pairs. Label j_{d} occurs. We require:

(6.10.2) (i) UŪ^{−1} is indecomposable under similarity.

(ii) The concatenated sequence {I, μ(J)} is repetition-free; that is, it does not consist of repetitions of some strictly shorter sequence.

We display, then explain, the form of these matrices.

\begin{matrix}i_{1} & j_{1} & i_{2} & j_{2} & i_{3} & … & j_{d−1} & i_{d} & j_{d} \\ A = & [ \begin{matrix}I & I & & & … \\ Iε & Iε̄ & & & … \\ & & I & I & … \\ & & Iε & Iε̄ & … \\ . & . & . \\ & & & & … & I & I \\ & & & & … & Iε & Iε̄ \end{matrix} ] \\ B = & [ \begin{matrix}& I & I & & & … \\ & Iε & Iε̄ & & & … \\ & & & I & I & … \\ & & & Iε & Iε̄ & … \\ . & . & . \\ & & & & & … & I & I \\ & & & & & … & Iε & Iε̄ \\ 0 & & & & & … & & & U^{−1} \\ I & & & & & … & & & 0 \end{matrix} ] \end{matrix} (6.10.3)

Here I = I_{m} with m the block size. The matrix A is a direct sum of one or more basic tiles (defined in Lemma 6.4, and always with Y = I). The matrix B is the direct sum of basic tiles (one fewer than A) and two exceptional tiles, one equal to the inverse U^{−1} of the blocking matrix and the other equal to I. (We use the inverse U^{−1} rather than the blocking matrix U in the matrix B in order to achieve uniformity in later formulas, for example, for L in the first three parts of Theorem 6.14.) The basic tiles in B always overlap those in A, as in the canonical forms that come from deleted cycle 4-tuples. Here, as before, entries not shown (except for ellipsis dots) are zero. As mentioned above, each term of the label sequences is the length label of all m rows in the block over which it appears.

The extreme case is d = 1, in which case B contains no basic tiles.

Bottom-top reduced pairs. Label j_{d} does not occur. Again we require (6.10.2); the pair is as displayed below.

\begin{matrix}i_{1} & j_{1} & i_{2} & j_{2} & i_{3} & … & i_{d−1} & j_{d−1} & i_{d} \\ A = & [ \begin{matrix}I & I & & & … \\ Iε & Iε̄ & & & … \\ & & I & I & … \\ & & Iε & Iε̄ & … \\ . & . & . \\ & & & & … & I & I \\ & & & & … & Iε & Iε̄ \\ & & & & … & & & U \end{matrix} ] \\ B = & [ \begin{matrix}& I & I & & & … \\ & Iε & Iε̄ & & & … \\ & & & I & I & … \\ & & & Iε & Iε̄ & … \\ . & . & . \\ & & & & & … & I & I \\ & & & & & … & Iε & Iε̄ \\ I & & & & & … \end{matrix} ] \end{matrix} (6.10.4)

In the extreme case d = 1 no basic tiles occur.

Top-top-reduced pairs. These are the same as bottom-bottom-reduced pairs except that A is interchanged with B and the matrix U^{−1} is replaced by the blocking matrix U. In particular, we require (6.10.2).

Cycle pairs. Label j_{d} occurs. Here we require:

(6.10.5) (i) The blocking matrix U is indecomposable under similarity.

(ii) The blocking matrix U is not similar to Ū^{−1} or, for all cycles ν^{t}, J ≠ ν^{t}μ(I), where ν is the unit forward rotation and μ is the mirror image permutation [Notation 6.6].

(iii) The pair of sequences I and J is repetition-free in the sense of (5.2.3).

These pairs have the form displayed in diagram (6.10.6) below. Thus, no exceptional tiles occur here. Note that both A and B are direct sums of basic tiles, but one of the tiles “wraps” from the end to the begining of B. The blocking matrix U always occurs in exactly one basic tile.

\begin{matrix}i_{1} & j_{1} & i_{2} & j_{2} & i_{3} & … & j_{d−1} & i_{d} & j_{d} \\ A = & [ \begin{matrix}U & I & & & … \\ Uε & Iε̄ & & & … \\ & & I & I & … \\ & & Iε & Iε̄ & … \\ . & . & . \\ & & & & … & I & I \\ & & & & … & Iε & Iε̄ \end{matrix} ] \\ B = & [ \begin{matrix}& I & I & & & … \\ & Iε & Iε̄ & & & … \\ & & & I & I & … \\ & & & Iε & Iε̄ & … \\ . & . & . \\ I & & & & & … & I \\ Iε̄ & & & & & … & Iε \end{matrix} ] \end{matrix} (6.10.6)

In order to prove that our list of canonical forms yields all isomorphism classes of self-conjugate 4-tuples, we need the following matrix version of Hilbert’s Theorem 90. It follows immediately from Ballantine [Ba, Theorem 2.1], using the fact that every matrix over a field is similar to its transpose.

Lemma 6.11. Let F be a 2-dimensional separable extension field of a field k. Then the following two assertions about an invertible matrix L over F are equivalent.

(i) L is similar to L̄^{−1}. (ii) L is similar to UŪ^{−1} for some invertible symmetric matrix U over F.

Also, we need to determine the block cycle 4-tuple that is isomorphic to the conjugate of a given block cycle 4-tuple.

Lemma 6.12. Let g and h be the block cycle 4-tuples with blocking matrices L and L̄^{−1} respectively, and with label sequences:

\begin{matrix}g : & H = \{i_{1}, i_{2}, …, i_{e}\} & and & K = \{j_{1}, j_{2}, …, j_{e}\} \\ h : & μ(K) = \{j_{e}, j_{e−1}, …, j_{1}\} & and & μ(H) = \{i_{e}, i_{e−1}, …, i_{1}\} \end{matrix} (6.12.1)

Then ḡ ≅ h. In particular, g ≅ ḡ if and only if L is similar to L̄^{−1} and K = ν^{t}μ(H) (equivalently, H = ν^{t}μ(K)) for some cycle ν^{t}.

Proof. Let the size of L be m × m. We have g = (A, I, P, I) where, as displayed in (5.2.5) with d in place of the present e, A is the me × me block diagonal matrix whose upper left main-diagonal block is L and whose other main diagonal blocks equal I_{m}, and P is a block-permutation matrix. Then ḡ = (I, Ā, I, P), where the length labels are moved in exactly the same way that the matrices were moved. Let g′ be obtained from ḡ by reversing the order of the column blocks in all four matrices — together with their length labels — but keeping the columns within each block in their original order. Since this is a display operation we have g′ ≅ ḡ [Notation 4.9 and Theorem 4.10].

Let R be the block permutation matrix whose e nonzero blocks each equal I_{m}, and are located on the diagonal that runs from the lower left corner to the upper right corner of R. Then we have g′ = (R, ĀR, R, P R), and the label sequences for g′ are those of h, displayed in (6.12.1). Let g′′ be the 4-tuple obtained by leftmultiplying all four matrices of g′ by R. Then g′′ ≅ g′ by Theorem 4.7, and we have g′′ = (I, RĀR, I, RP R).

Let g′′′ be obtained by left multiplying the first and second matrices of g′′ by (RĀR)^{−1} and the third and fourth matrices by (RP R)^{−1}. Then g′′′ ≅ g′′ and we have g′′′ = (RĀ^{−1}R, I, RP^{−1}R, I).

Note that RP^{−1}R = P. [This is essentially formula (6.6.1).]

Next consider the form of RĀR. Ā is the block diagonal matrix whose diagonal blocks are L̄, I, I, …, I. Therefore RĀ^{−1}R is the block diagonal matrix whose diagonal blocks are I, I, …, I, L̄^{−1}. Thus, when e = 5, for example, our 4-tuple g′′′ is as shown in (6.12.2) below.

\begin{matrix}j_{5} & j_{4} & j_{3} & j_{2} & j_{1} \\ A_{1} = & [ \begin{matrix}I & 0 & 0 & 0 & 0 \\ 0 & I & 0 & 0 & 0 \\ 0 & 0 & I & 0 & 0 \\ 0 & 0 & 0 & I & 0 \\ 0 & 0 & 0 & 0 & L̄^{−1} \end{matrix} ] \\ B_{1} = & [ \begin{matrix}0 & I & 0 & 0 & 0 \\ 0 & 0 & I & 0 & 0 \\ 0 & 0 & 0 & I & 0 \\ 0 & 0 & 0 & 0 & I \\ I & 0 & 0 & 0 & 0 \end{matrix} ] \end{matrix}  \begin{matrix}i_{5} & i_{4} & i_{3} & i_{2} & i_{1} \\ A_{2} = & [ \begin{matrix}I & 0 & 0 & 0 & 0 \\ 0 & I & 0 & 0 & 0 \\ 0 & 0 & I & 0 & 0 \\ 0 & 0 & 0 & I & 0 \\ 0 & 0 & 0 & 0 & I \end{matrix} ] \\ B_{2} = & [ \begin{matrix}I & 0 & 0 & 0 & 0 \\ 0 & I & 0 & 0 & 0 \\ 0 & 0 & I & 0 & 0 \\ 0 & 0 & 0 & I & 0 \\ 0 & 0 & 0 & 0 & I \end{matrix} ] \end{matrix} (6.12.2)

Therefore, to complete the proof it now suffices to move L̄^{−1} to the upper left corner of A_{1}. This is accomplished by the following sequence of multiplications, which are F -F sweeping-similarity operations: Left multiply the last row block of A_{1} and A_{2} by L̄; then right multiply the i_{1}-labeled columns of A_{2} and B_{2} by L̄^{−1}; then left multiply the last row block of B_{1} and B_{2} by L̄; then right multiply the j_{e}-labeled columns of A_{1} and B_{1} by L̄^{−1}. The resulting matrix 4-tuple is the given h, and so h ≅ ḡ.

Supplementary statement. Suppose that g ≅ ḡ. By the part of the lemma already proved, ḡ is isomorphic to a block cycle 4-tuple with blocking matrix L̄^{−1} and pair of label sequences μ(K) and μ(H). But by the uniqueness theorem for block cycle 4-tuples [Theorem 5.3], the pair of label sequences of any block cycle 4-tuple isomorphic to g must have the form ν^{t}(H) and ν^{t}(K) for some cycle ν^{t}. This yields ν^{t}(K) = μ(H), and hence K = ν^{−t}μ(H), as desired. Moreover, also by the uniqueness theorem for block cycle 4-tuples, the blocking matrix L̄^{−1} of ḡ must be similar to the blocking matrix L of g.

Conversely, suppose that L is similar to L̄^{−1} and K = ν^{t}μ(H) for some cycle ν^{t}. Then formula (6.6.1) yields H = ν^{t}μ(K). Therefore, by the uniqueness theorem for block cycle 4-tuples [Theorem 5.3], the block cycle 4-tuple g, with blocking matrix L and label sequences H and K, is isomorphic to the block cycle 4-tuple with blocking matrix L and label sequences ν^{−t}(H) = μ(K) and ν^{−t}(K) = μ(H), which is in turn isomorphic to ḡ, since L is similar to L̄^{−1}.

The following lemma simplifies the way to visualize self-conjugate block cycle 4-tuples.

Lemma 6.13. Let g be a self-conjugate block cycle 4-tuple. Then g is isomorphic to a block cycle 4-tuple whose pair of label sequences has one of the forms:

(a) H and μ(H), or

(b) H and νμ(H) where μ is the mirror image permutation and ν is the unit forward rotation.

Let e be the number of terms in H. If e is even then only one of (a) and (b) applies to g. If e is odd then (a) always applies.

Proof. By Lemma 6.12, g has a pair of label sequences of the form H and ν^{t}μ(H) for some cycle ν^{t}. We consider two cases, according to the parity of t.

Case (a): t = 2s for some integer s. Then K = ν^{2s}μ(H) = ν^{s}μν^{−s}(H) and hence ν^{−s}(K) = μν^{−s}(H). Therefore, by the uniqueness theorem for block cycle 4-tuples, possibility (a) holds if we replace H and K by the sequences ν^{−s}(H) and ν^{−s}(K), respectively.

Case (b): t = 2s + 1 for some integer s. Proceed similarly to case (a).

Now suppose that e is odd. Then the cyclic group generated by ν has odd order, and therefore contains a square root of ν. Say ν = (ν^{u})^{2}. Then ν^{t} = ν^{2ut} and case (a) applies.

Finally, suppose (by way of contradiction) that e is even and both possibilities (a) and (b) hold. Thus some pair of isomorphic block cycle 4-tuples have label sequences of the form H and μ(H), and H′ and νμ(H′), respectively. By the uniqueness theorem for block cycle 4-tuples, there is a cycle ν^{t} such that ν^{t}(H) = H′ and ν^{t}μ(H) = νμ(H′). Therefore, ν^{t}μ(H) = νμν^{t}(H). Simplifying, using (6.6.1), yields H = ν^{2t−1}(H). Since e is even, ν^{2t−1} ≠ 1. Thus H consists of some number, say r, of repetitions of some strictly shorter sequence; and hence so does μ(H). But then the pair of sequences H and μ(H) is not repetition-free [see (5.2.3)], contrary to the definition of a block cycle 4-tuple.

Theorem 6.14. Let p = (A, B) be any of the types of matrix pairs listed below, with blocking matrix U and constructed from the pair of label sequences

I = {i_{1}, …, i_{d−1}, i_{d}} and J = {j_{1}, …, j_{d−1}, [j_{d}]} (6.14.1)

Then f(p) is as described below, where μ is the mirror image permutation, and ν is the unit forward rotation, as in Notation 6.6.

(i) (p bottom-bottom reduced) Then f(p) is isomorphic to the self-conjugate block cycle 4-tuple whose blocking matrix is L = UŪ^{−1} and whose pair of label sequences is {I, μ(J)} and {J, μ(I)}.

(ii) (p bottom-top reduced) Then f(p) is isomorphic to the self-conjugate block cycle 4-tuple whose blocking matrix is L = UŪ^{−1} and whose pair of label sequences is {I, μ(J)} and {J, μ(I)}. (Recall that, in this case, the label j_{d} does not occur.)

(iii) (p top-top reduced) Then f(p) is isomorphic to the self-conjugate block cycle 4-tuple whose blocking matrix is L = UŪ^{−1} and whose pair of label sequences is {J, μ(I)} and ν({I, μ(J)}).

(iv) (p a cycle pair) Then f(p) is isomorphic to the direct sum g ⊕ ḡ where g is the non-self-conjugate block cycle 4-tuple whose blocking matrix is L = U and whose pair of label sequences is I and J.

In all cases, p is an indecomposable matrix pair (equivalently, f(p) is an indecomposable package). Conversely, if g is a non-self-conjugate block cycle 4-tuple, then there is some cycle pair p with f(p) ≅ g ⊕ ḡ, and if g is a self-conjugate block cycle 4-tuple, then there is some bottom-bottom reduced, bottom-top reduced or top-top-reduced matrix pair p with f(p) ≅ g.

Proof. Write f(p) = (A, Ā, B, B̄). We do situation (i) in detail, and then the others more briefly. Situation (i) itself is a slightly more complicated version of what was done in the deleted cycle case; and we refer to that case for steps that are very similar.

(i) p is bottom-bottom reduced, as in (6.10.3). As in the deleted cycle situations we first reduce every pair of basic tiles, as described in Lemma 6.4, and call the resulting 4-tuple h. The situation d = 3 is shown in (6.14.2).

\begin{matrix}i_{1} & j_{1} & i_{2} & j_{2} & i_{3} & j_{3} \\ A_{1} = & [ \begin{matrix}I & 0 \\ 0 & I \\ & & I & 0 \\ & & 0 & I \\ & & & & I & 0 \\ & & & & 0 & I \end{matrix} ] \\ B_{1} = & [ \begin{matrix}& I & 0 \\ & 0 & I \\ & & & I & 0 \\ & & & 0 & I \\ 0 & & & & & U^{−1} \\ I & & & & & 0 \end{matrix} ] \end{matrix}  \begin{matrix}i_{1} & j_{1} & i_{2} & j_{2} & i_{3} & j_{3} \\ A_{2} = & [ \begin{matrix}0 & I \\ I & 0 \\ & & 0 & I \\ & & I & 0 \\ & & & & 0 & I \\ & & & & I & 0 \end{matrix} ] \\ B_{2} = & [ \begin{matrix}& 0 & I \\ & I & 0 \\ & & & 0 & I \\ & & & I & 0 \\ 0 & & & & & Ū^{−1} \\ I & & & & & 0 \end{matrix} ] \end{matrix} (6.14.2)

Note that statement (6.9.3) is valid here, too; and it obviously remains valid after arbitrary display operations. Apply the display operation — in block form — that changes the pair of label sequences to the desired sequences {I, μ(J)} and {J, μ(I)}, and call the resulting 4-tuple h′. By (6.9.3) we have h′ = (C, C, B′_{1}, B′_{2}) for some C. Left multiplying the A-matrices of h′ by C^{−1} converts it to h′′ = (I, I, B′_{1}, B′_{2}).

Next, change the pair (U^{−1}, Ū^{−1}) to (ŪU^{−1}, I) = (L^{−1}, I), by left-multiplying the row-block containing these matrices by Ū. Then right-multiply the j_{d}-labeled columns of A_{1} and B_{1} by L, changing L^{−1} to I and changing the corresponding nonzero entry in the j_{d}-labeled column-block of A_{1} to L. The result of this is that, after a change of notation, h′′ takes the form (D, I, B_{1}, B_{2}) where D is a blockdiagonal matrix one of whose diagonal blocks equals L and whose other diagonal entries equal I, and B_{1} and B_{2} are block-permutation matrices.

Now consider the pair of block-permutation matrices (B_{1}, B_{2}). We can arbitrarily permute the row blocks of this pair by simultaneously left-multiplying B_{1} and B_{2} by a suitable permutation matrix; we multiply by B_{2}^{−1}, so that B_{2} is replaced by an identity matrix. The result of all of the above transformations is illustrated below, in the case d = 3. (Ignore the entries of B_{1} until we find them.)

(6.14.3) \begin{matrix}i_{1} & i_{2} & i_{3} & j_{3} & j_{2} & j_{1} \\ A_{1} = & [ \begin{matrix}I \\ & I \\ & & I \\ & & & L \\ & & & & I \\ & & & & & I \end{matrix} ] \\ B_{1} = & [ \begin{matrix}& I \\ & & I \\ & & & I \\ & & & & I \\ & & & & & I \\ I \end{matrix} ] \end{matrix}  \begin{matrix}j_{1} & j_{2} & j_{3} & i_{3} & i_{2} & i_{1} \\ A_{2} = & [ \begin{matrix}I \\ & I \\ & & I \\ & & & I \\ & & & & I \\ & & & & & I \end{matrix} ] \\ B_{2} = & [ \begin{matrix}I \\ & I \\ & & I \\ & & & I \\ & & & & I \\ & & & & & I \end{matrix} ] \end{matrix}

Next we find the locations of the nonzero entries of B_{1}. We first look for the nonzero entries of the first three row-blocks. Consider column-blocks j_{1}, j_{2}, and j_{3} of B_{2}. We ask: which column-blocks of B_{1} have their nonzero entries in the same rows as the nonzero entries of the first three column-blocks of B_{2}? After consulting (6.14.2) we see that we want the i_{2}, i_{3}, and j_{3}-labeled column-blocks of B_{1}. Thus, returning to (6.14.3), we see that the first three row-blocks of B_{1} have their nonzero entries in the second, third, and fourth column-blocks of B_{1}, respectively, as shown. By proceeding in this fashion, we eventually find that B_{1} is the block form of the cyclic permutation matrix required for a block cycle 4-tuple.

In order to complete the proof, two things are needed: (a) move L to the upper left corner of A_{1}; and (b) check that the pair of label sequences {I, μ(J)} and {J, μ(I)} is repetition-free, as defined in (5.2.3) (as required in the definition of a block cycle 4-tuple).

(a) Move L. It suffices to slide L up one level; that is, move it to the next higher I, for we can repeat this often enough to get L to the right location. We accomplish this by the following four sweeping-similarity operations. Right-multiply the column-block of A_{1} and B_{1} containing L by L^{−1}, thus replacing the nonzero entry in that column-block of B_{1} by L^{−1}. Then use a left multiplication of the row block of (B_{1}, B_{2}) containing L^{−1} to put L into that row-block of B_{2}. Then use a right multiplication to put L^{−1} into A_{2}, and finally a left multiplication to put L into A_{1}.

(b) By hypothesis (6.10.2)(ii), the sequence {I, μ(J)} is repetition-free, which is formally stronger than the statement that the pair of sequences {I, μ(J)} and {J, μ(I)} is repetition-free. (Actually, since the sequence {J, μ(I)} is just {I, μ(J)} written backwards, the two types of “repetition-free” are equivalent here.)

(ii) p is bottom-top reduced, as in (6.10.4). We omit the details of this reduction, which is similar to that in (i).

(iii) p is top-top reduced. Thus the matrix pair p is obtained from the bottombottom reduced pair p′ in (6.10.3) by interchanging A with B and replacing U^{−1} by U. All of steps in the reduction of f(p′) done in part (i) can be done with the roles of A and B reversed. Therefore, by the result of (i), f(p) is isomorphic to the first matrix 4-tuple displayed in (6.14.4) with its pair of label sequences above it.

\begin{matrix}\{I, μ(J)\} & \{J, μ(I)\} \\ P & I \\ D & I \end{matrix} → \begin{matrix}\{J, μ(I)\} & \{I, μ(J)\} \\ RP R & I \\ RDR & I \end{matrix} → \begin{matrix}ν^{−1}\{J, μ(I)\} & \{I, μ(J)\} \\ I & I \\ RDRP & I \end{matrix} (6.14.4)

Here P is the block permutation matrix displayed as B_{1} in (5.2.5). This matrix, acting on the right performs the inverse ν^{−1} of the unit forward column rotation. The matrix D is a block diagonal matrix [see A_{1} in (5.2.5)] whose upper left-hand block is U^{−1}Ū and whose other diagonal blocks each equal I_{m}, by the result of part (i) with U in place of U^{−1}.

Let R (“reversal matrix”) be the matrix formed by reversing the order of the columns of the d × d identity matrix and then replacing each nonzero entry by I_{m} and each zero entry by a zero matrix.

To continue the reduction of f(p), perform the display operation that reverses the order of the column blocks of P and D, together with their block labels, while leaving the order of the columns within each block unchanged; and do the same with the two identity matrices. Then left-multiply all 4 matrices by R, obtaining the second displayed 4-tuple.

Note that RP R = P^{−1} [this is essentially formula (6.6.1)]. Therefore rightmultiplying the left-hand pair of matrices and its row of column labels by P is a display operation. It yields the third 4-tuple in (6.14.4).

Note that RDR = D_{1}, the block diagonal matrix obtained by moving its nonidentity bock U^{−1}Ū to the lower right-hand corner. Moreover, D_{1}P = P D_{2}, where D_{2} is the block diagonal matrix obtained from D_{1} by moving U^{−1}Ū to a different block on the main diagonal. Right-multiplying this column-block of P D_{2} and the identity matrix above it by (U^{−1}Ū)^{−1} is a sweeping-similarity operation and replaces the last 4-tuple in (6.14.4) by (D_{2}^{−1}, I, P, I) without moving any blocklabels. Note that D_{2}^{−1} is the block diagonal matrix obtained from D_{2} by inverting its non-identity diagonal block U^{−1}Ū. This diagonal block of D_{2}^{−1} therefore equals Ū^{−1}U.

Call this last 4-tuple h′. Then h′ is the block cycle 4-tuple in the statement of the theorem except that: (a) Its blocking matrix is in the wrong block of A_{1};

(b) the blocking matrix of h is Ū^{−1}U instead of UŪ^{−1}; and (c) the pair of label sequences is not yet exactly correct.

(a) As in the proof of part (i) [“Move L”] we can slide Ū^{−1}U up to the upper left corner of A_{1} without changing the isomorphism class of the 4-tuple. The resulting 4-tuple is now a block cycle matrix (though not yet the correct one).

(b) Reversing factors in a product of two invertible matrices does not change its similarity class, and replacing the blocking matrix of a block cycle 4-tuple within its similarity class does not change the isomorphism class of the 4-tuple [Theorem 5.3(ii)]. Doing this yields the correct blocking matrix. Call this latest 4-tuple h′′.

(c) Performing any rotation simultaneously to both of the label sequences of any block cycle matrix leaves the isomorphism class of the matrix unchanged [Theorem 5.3(ii)]. Note that the pair of label sequences of h′′ is the same as that of the last 4-tuple displayed in (6.14.4). Therefore applying the rotation ν to these label sequences yields the desired sequences, and completes the proof of (iii).

(iv) p is a cycle pair, as in (6.10.6). Unlike the previous cases, the reduction of basic tiles in this case uses the full strength of Lemma 6.4, because U is located in a basic tile. Otherwise, the details are just a slight modification of the bottom-bottom reduced case, and we omit them. As in the non-reduced deleted cycle situation, the reduction in this case actually proves:

(6.14.5) We have f(p) ≅ g⊕ḡ, even without the non-symmetry restriction (6.10.5)(ii) in the definition of a cycle pair.

Supplementary statements. Indecomposability is an immediate consequence of Lemma 6.5.

To see that our list is exhaustive, let g be any block cycle 4-tuple, with blocking matrix L and pair of label sequences I′ and J′.

Consider, first, the case that g ≅ ḡ. Then L is similar to L̄^{−1} [Lemma 6.12]. Therefore, by Ballantine’s Hilbert Theorem 90 [Lemma 6.11], there is a matrix U such that UŪ^{−1} is similar to L. This gives us the blocking matrix U to use in situations (i)–(iii) of the theorem. Again, since g ≅ ḡ, we can choose g, within its isomorphism class, such that J′ = μ(I′) or J′ = νμ(I′) [Lemma 6.13]. Moreover, if the length e of each of these sequences is odd, then we can choose J′ = μ(I′). This yields three cases, corresponding to the enumeration of situations in the statement of the theorem:

Case (i): e is even and J′ = μ(I′). Since e is even we can define sequences I and J of length d = e/2 by I′ = {I, μ(J)}. Take p to be the bottom-bottom reduced pair of (i), with blocking matrix U and label sequences I and J. Then the block cycle 4-tuple yielded by situation (i) has blocking matrix L and label sequences {I, μ(J)} = I′ and {J, μ(I)} = μ(I′) = J′, as desired.

Case (ii): e is odd and J′ = μ(I′). Since e is odd, we can define sequences I and J of lengths that we call d and d−1 respectively, such that I′ = {I, μ(J)}. Take p to be the bottom-top reduced pair of (ii), with blocking matrix U and label sequences I and J. Then the block cycle 4-tuple yielded by situation (ii) has blocking matrix L and the desired label sequences I′ and J′, as in case (i).

Case (iii): e is even and J′ = νμ(I′). Define sequences I and J of length d = e/2 by I′ = {J, μ(I)}. Take p to be the top-top reduced pair of (iii), with blocking matrix U and label sequences I and J. Then the block cycle 4-tuple yielded by situation (iii) has blocking matrix L and label sequences I′ = {J, μ(I)} and ν({I, μ(J)}) = νμ(I′) = J′, as desired.

This leaves the case that g is non-self-conjugate; and here we take p to be the cycle pair of (iv).

Theorem 6.15. Let g be a matrix 4-tuple. The following are equivalent: (i) g is a package. (ii) g ≅ ḡ. (iii) g ≅ ⊕_{i} f(p_{i}), where each p_{i} is one of the seven canonical forms in Definitions

6.7 and 6.10.

Proof. We already noted that (i) ⇒ (ii) follows from the definitions. Also, (iii) ⇒ (i) is immediate since each f(p_{i}) is a package.

Thus, the only non-trivial implication is (ii) ⇒ (iii), so suppose g ≅ ḡ. By the structure theorems for decompositions of 4-tuples [Theorems 5.3 and 5.4] we have

(6.15.1)  g ≅ ⊕_{j} g_{j}  and therefore  ḡ ≅ ⊕_{j} ḡ_{j}

where each g_{j} is either a deleted or block cycle 4-tuple, and the multiplicity of each isomorphism class of deleted or block cycle 4-tuple that occurs in the first decomposition is determined by the isomorphism class of g. Since the conjugate of every deleted cycle or block cycle 4-tuple is again (isomorphic to) a deleted or block cycle 4-tuple, respectively [Lemmas 6.8 and 6.12] and g ≅ ḡ, it follows from Theorem 5.4 that, for each index j, ḡ_{j} occurs (up to isomorphism) in the decomposition (6.15.1) of g exactly as often as g_{j} occurs. Therefore g has a decomposition of the form

g ≅ ⊕_{i} h_{i} (6.15.2)

where each h_{i} is either a self-conjugate g_{j} or the direct sum g_{j} ⊕ḡ_{j} for some g_{j} such that g_{j} ≅ ¯g̸_{j}. Each h_{i} is therefore isomorphic to f(p_{i}) for some deleted or block cycle pair p_{i} by Theorems 6.9 and 6.14, completing the proof.

Corollary 6.16. Suppose g is a matrix 4-tuple. Then g is an indecomposable package if an only if g is either:

(i) A self-conjugate, indecomposable 4-tuple; or (ii) The direct sum of two conjugate, non-self-conjugate, indecomposable 4-tuples.

Proof. This follows from Theorem 6.15 and the fact that, for all of our canonical forms p, f(p) has the form (i) or (ii) [Theorems 6.9 and 6.14].

Corollary 6.17. Every indecomposable matrix pair is isomorphic to one of the seven canonical forms in Definitions 6.7 and 6.10. Moreover, if

⊕_{i=1}^{m} p_{i} ≅ ⊕_{j=1}^{n} q_{j} (6.17.1)

where each p_{i} and each q_{j} is indecomposable, then m = n and, after suitable renumbering, p_{i} ≅ q_{i} for each index i.

Proof. If p is a matrix pair, then f(p) is self-conjugate, so by Theorem 6.15, we can write f(p) ≅ ⊕_{i} f(p_{i}), where each p_{i} is a one of our canonical forms. By Theorem 6.3, it follows that p ≅ ⊕_{i} p_{i}, from which the first assertion of the corollary follows.

For the uniqueness claim, suppose that we are given an isomorphism of direct sums as in (6.17.1). Then by Theorem 6.3 we get that ⊕_{i=1}^{m} f(p_{i}) ≅ ⊕_{j=1}^{n} f(q_{j}). It suffices to show that some q_{i}, say q_{1}, satisfies p_{1} ≅ q_{1} and ⊕_{i≠1} p_{i} ≅ ⊕_{i≠1} q_{i}. Write each f(p_{i}) and f(q_{j}) as the direct sum of one or two indecomposable terms, as described in Theorems 6.9 and 6.14.

Let x be one of the one or two indecomposable direct summands of f(p_{1}). By the Krull-Schmidt Theorem for 4-tuples [Theorem 5.4], some indecomposable direct summand on the right-hand side, say of f(q_{1}), must be isomorphic to x. If x is not a package, then x ⊕ x̄ is the unique indecomposable package containing x as a direct summand [by Corollary 6.16], and is therefore isomorphic to both f(p_{1}) and f(q_{1}). Otherwise x itself is a package, and indecomposability of f(p_{1}) and f(q_{1}) again implies that f(p_{1}) ≅ f(q_{1}). In either case the Krull-Schmidt Theorem for 4-tuples then implies that ⊕_{i≠1} f(p_{i}) ≅ ⊕_{i≠1} f(q_{i}), and therefore p_{1} ≅ q_{1} and ⊕_{i≠1} p_{i} ≅ ⊕_{i≠1} q_{i} [Theorem 6.3], completing the proof.

There remains the question of uniqueness of the seven canonical forms in Definitions 6.7 and 6.10. The next theorem establishes the extent of this uniqueness.

Theorem 6.18. Let p and q be matrix pairs in canonical form, with pairs of label sequences I and J, and I′ and J′, respectively, and blocking matrices U and V, respectively, if relevant. Then p ≅ q if and only if they are of the same type (i)– (vii) below, and the conditions listed for their type hold [with μ and ν as in Notation 6.6].

(i) (Non-reduced pair) Either I′ = I and J′ = J, or I′ = μ(J) and J′ = μ(I). (ii) (Bottom-reduced pair) I′ = I and J′ = J. (iii) (Top-reduced pair) I′ = I and J′ = J. (iv) (Bottom-bottom reduced pair) Either I′ = I and J′ = J, or I′ = μ(J) and

J′ = μ(I); and (in either case) V V̄^{−1} is similar to UŪ^{−1}.

(v) (Bottom-top reduced pair) I′ = I and J′ = J; and V V̄^{−1} is similar to UŪ^{−1}. (vi) (Top-top reduced pair) Either I′ = I and J′ = J, or I′ = μ(J) and J′ = μ(I);

and (in either case) V V̄^{−1} is similar to UŪ^{−1}. (vii) (Cycle pair) Either I′ = ν^{t}(I) and J′ = ν^{t}(J) (for some t) and V is similar

to U; or I′ = ν^{t}μ(J) and J′ = ν^{t}μ(I) (for some t) and V is similar to Ū^{−1}.

Proof. All of the cases considered in this proof use our description of f(p) in terms of p [Theorems 6.9 and 6.14] and the fact that p ≅ q ⟺ f(p) ≅ f(q) [Theorem 6.3].

Uniqueness of types: We claim that the type, (i)–(vii), of p is determined by the isomorphism class of p. It suffices to show that this type is determined by the isomorphism class of f(p). We use the following facts: f(p) is the direct sum of one or two deleted cycle or block cycle 4-tuples; deleted cycle and block cycle 4-tuples are indecomposable [Theorem 5.3]; the Krull-Schmidt Theorem holds for direct sums of 4-tuples [Theorem 5.4]; and no deleted cycle 4-tuple is isomorphic to any block cycle 4-tuple [Theorem 5.3]. The following enumeration of cases therefore distinguishes among types of 4-tuples.

Types (i)–(iii) are the types for which f(p) is a direct sum of deleted cycle 4tuples. Among these types, type (i) yields two summands while types (ii) and (iii) each yield only one summand. In both of these last two types the first label sequence of f(p) is {I, μ(J)}. In type (ii) this has even length because I and J have the same length, but in type (iii) the length is odd because j_{d} is missing from J.

Types (iv)–(vii) are the types built from block cycle 4-tuples. Among these, type (vii) is a direct sum of two block cycle 4-tuples while (iv)–(vi) each consist of a single block cycle 4-tuple. Among types (iv)–(vi), the label sequences of f(p) have even length in types (iv) and (vi) because I and J have the same length, and odd length in type (v) because j_{d} is missing from J. Thus, it remains to distinguish between types (iv) and (vi), the most subtle of the distinctions.

Recall that both label sequences I and J of p have the same length, say d, in each of situations (iv) and (vi). The label sequences of f(p) are

{I, μ(J)} and {J, μ(I)}  in type (iv)
{J, μ(I)} and ν({I, μ(J)})  in type (vi)

(6.18.1)

After writing H = {I, μ(J)} and K = μ(H) these sequences become

H and μ(H)  in type (iv)
K and νμ(K)  in type (vi)

(6.18.2)

Note that H and K both have even length e = 2d. Therefore, by Lemma 6.13, a self-conjugate block cycle 4-tuple with one of the pairs of label sequences displayed in (6.18.2) can never be isomorphic to one with the other pair of label sequences. This shows that no pair of type (iv) can be isomorphic to a pair of type (vi), and completes our proof of the uniqueness of types.

We now proceed to the remaining uniqueness properties of each individual type. In view of the uniqueness of types, we may assume, for the rest of this proof that p and q are of the same type (i)–(vii).

Types (i)–(iii). Here we frequently use, without explicitly mentioning it, that two deleted cycle 4-tuples are isomorphic if an only if their pairs of label sequences are the same [Theorem 5.3].

Type (i): non-reduced. Here f(p) ≅ g ⊕ ḡ where g is the deleted cycle 4-tuple with label sequences I and J, and hence ḡ is isomorphic to the deleted cycle 4-tuple with label sequences μ(J) and μ(I) [Lemma 6.8]. Similarly f(q) ≅ h ⊕ h̄ where the label sequences of h are I′ and J′, and h̄ is isomorphic to the deleted cycle 4-tuple with label sequences μ(J′) and μ(I′). If h ≅ g then I′ = I and J′ = J, the first possibility stated in part (i) of the theorem. Otherwise, since g, ḡ, h, h̄ are all indecomposable 4-tuples and the Krull-Schmidt Theorem holds for 4-tuples, we have h ≅ ḡ; and therefore I′ = μ(J) and J′ = μ(I), the second possibiity stated in part (i) of the theorem.

Conversely, if p and q are non-reduced pairs with I′ = I and J′ = J — or with I′ = μ(J) and J′ = μ(I)) — then reversing the above argument easily yields p ≅ q.

Types (ii) and (iii): bottom-reduced and top-reduced, respectively. In both types f(p) is a single deleted cycle 4-tuple, with label sequences {I, μ(J)} and {J, μ(I)}. A similar statement holds for f(q), I′ and J′.

Suppose p ≅ q, and hence f(p) ≅ f(q). Since isomorphic deleted cycle 4-tuples have the same pairs of label sequences, we have {I, μ(J)} = {I′, μ(J′)}, which is equivalent to I = I′ and J = J′, as desired.

Conversely, since f(p) and f(q) are each a single deleted cycle 4-tuple, if I′ = I and J′ = J, then f(p) ≅ f(q) and hence p ≅ q.

We now turn to the types involving block cycle 4-tuples. Let F and F′ be such 4-tuples, where F has label sequences H and K, and F′ has label sequences H′ and K′. We recall [Theorem 5.3] that F ≅ F′ if and only both 4-tuples have similar blocking matrices and, for some t,

ν^{t}(H) = H′ and ν^{t}(K) = K′ (6.18.3)

Type (vii): cycle pairs. Here, f(p) ≅ g ⊕ ḡ, where g is the block cycle 4-tuple with label sequences I and J and blocking matrix U. Therefore ḡ is isomorphic to the block cycle 4-tuple with label sequences μ(J) and μ(I) and blocking matrix Ū^{−1} [Lemma 6.12]. Similarly, f(q) ≅ h ⊕ h̄ where h has label sequences I′ and J′ and blocking matrix V; and h̄ has label sequences J′ and I′ and blocking matrix V̄^{−1}

As in the the analysis of type (i) above, we have either h ≅ g or h ≅ ḡ. In the situation h ≅ g, we have I′ = ν^{t}(I) and J′ = ν^{t}(J) (for some t) and V is similar to U [see (6.18.3)]. Similarly, in the situation h ≅ ḡ we have I′ = ν^{t}μ(J) and J′ = ν^{t}μ(I) (for some t) and V is similar to Ū^{−1}. As in type (i), the converse follows easily by reversing the argument.

Types (iv)–(vi): bottom-bottom-reduced, bottom-top-reduced, and top-top-reduced, respectively. Here f(p) is isomorphic to a single block cycle 4-tuple. Its label sequences are

{I, μ(J)} and {J, μ(I)}  (type (iv) or (v))
{J, μ(I)} and ν({I, μ(J)})  (type (vi))

(6.18.4)

and the blocking matrix is UŪ^{−1} in all three types. Similar remarks apply to f(q).

Suppose that p ≅ q. Then f(p) ≅ f(q) and hence their blocking matrices V V̄^{−1} and UŪ^{−1} are similar, as desired. Let f(p) have label sequences H and K; and let f(q) have label sequences H′ and K′. Then (6.18.3) holds. Let d be the number of terms in the sequence I. We now consider each of these three types separately.

Type (v): bottom-top reduced. Here, the number of terms in ({I, μ(J)} is 2d−1, because j_{d} is missing in J. In view of (6.18.4), relations (6.18.3) become

ν^{t}({I, μ(J)}) = {I′, μ(J′)} and ν^{t}({J, μ(I)}) = {J′, μ(I′)} (6.18.5)

The second equation in (6.18.5) can be rewritten as ν^{t}μ({I, μ(J)}) = μ({I′, μ(J′)}). By (6.6.1), we can substitute ν^{t}μ = μν^{−t}, so after canceling μ, we obtain the equation ν^{−t}({I, μ(J)}) = {I′, μ(J′)}. Combining this with the first equation in (6.18.5) then yields ν^{−t}({I, μ(J)}) = ν^{t}({I, μ(J)}), and hence ν^{2t}({I, μ(J)}) = {I, μ(J)}. By the “repetition-free” condition (6.10.2)(ii), the permutation ν^{2t} must act as the identity on the sequence {I, μ(J)}, from which it follows that 2d − 1 (the length of this sequence) divides 2t. But then 2d − 1 divides t, so that ν^{t} is the identity permutation in (6.18.5). This completes the proof that q satisfies the conditions for type (v).

Type (iv): bottom-bottom reduced. Here I and J have the same length d, and therefore the concatenated sequence {I, μ(J)} has length 2d. The same argument as for type (v) above still applies, with one twist: When we reach the relation ν^{2t}({I, μ(J)}) = {I, μ(J)}, the “repetition-free” condition implies that 2d (rather than 2d − 1) divides 2t, and hence d divides t, say t = xd. This yields the two possibilities for type (iv) as follows. If x is even, then 2d divides t, in which case ν^{t} is the identity permutation in (6.18.5), and therefore I = I′ and J = J′.

Otherwise x is odd, say x = 2y + 1 and hence ν^{t} = ν^{(2y+1)d} = ν^{d}. Then the second equation of (6.18.5) yields μ(I) = J′ and J = μ(I′). The second of these is equivalent to I′ = μ(J). This completes the proof that q satisfies the conditions for type (iv).

Type (vi): top-top reduced. This is similar to type (iv). I and J have the same length d, and therefore the two concatenated sequences in the second row of (6.18.4) have length 2d. Substituting these concatenated sequences into (6.18.3), and then canceling ν from the second of equation, again yields the pair of equations (6.18.5). As in type (v) this implies ν^{2t}({I, μ(J)}) = {I, μ(J)}. Since the concatenated sequences have length 2d we finish exactly as in type (iv), reaching the same conclusion as in type (iv), as desired.

This completes the proof that q satisfies the conditions for type (vi), and hence for all types.

The converse parts of (iv)–(vi) are easily verified by using the fact that isomorphism of block cycle 4-tuples is implied by the equations in (6.18.3), for some t, together with similarity of blocking matrices.

We conclude this section by noting the structure of those matrix pairs that fail the “non-symmetry conditions” in (6.7.2)(ii) or (6.10.5)(i) or (ii).

Proposition 6.19. Let p be a matrix pair. Suppose that either:

(i) p would be a nonreduced pair except that its pair of label sequences has the form I and μ(I); or

(ii) p would be a cycle pair except that its pair of label sequences has the form I

and ν^{t}μ(I) for some cycle ν^{t}, and its blocking matrix L is similar to L̄^{−1}. Then p ≅ p′ ⊕ p′ for some indecomposable matrix pair p′.

Proof. (ii) By (6.14.5) we have f(p) ≅ g ⊕ ḡ where g is the block cycle pair whose blocking matrix is L and whose label sequences are the same as the pair used to define p, that is, I and ν^{t}μ(I). By Lemma 6.12, g is self-conjugate, and therefore f(p) ≅ g ⊕ g. Since g ≅ ḡ, the 4-tuple g is a package [Theorem 6.15]; say g ≅ f(p′). Then f(p) ≅ f(p′ ⊕ p′) and therefore p ≅ p′ ⊕ p′ [Theorem 6.3]. Moreover, g ≅ f(p′) is indecomposable since it is a block cycle 4-tuple. Therefore p′ is also indecomposable.

(i) This is essentially the same as the proof of (ii), except that one uses the deleted cycle analogues, (6.9.6) and Lemma 6.8, of the corresponding block cycle results in that proof.

7. Indecomposable Modules, Unsplit Case

In this section we describe how to construct all indecomposable finitely generated Λ-modules from indecomposable (necessarily uniserial) finitely generated Γ-modules, in the unsplit case. We also describe, very briefly, how homomorphisms of Λ-modules arise from homomorphisms of Γ-modules. The statements of our main results (but not of their proofs) can be read independently of the sections between here and the introduction to this paper. To enhance the continuity of this section, as a continuation of the Introduction, the proofs of the main results do not begin until subsection 7.13.

Notation 7.1. In this section (Λ, 𝔪, k) is unsplit Dedekind-like with normalization (Γ, 𝔪, F), and π denotes the distinguished Γ-generator of 𝔪, as in Notation 1.1. Moreover, we assume that the quadratic extension field F of k is separable over k (so that the results of §6 apply).

Recall that ρ denotes the natural surjection Γ → → Γ/𝔪 = F, and let ρ̄: Γ → → F be the composition of ρ with the non-identity k = Λ/𝔪-automorphism of F. Similarly, for α ∈ F the notation ᾱ (the conjugate of α) denotes the image of α under this automorphism, and for a matrix A over F, Ā denotes the matrix whose entries are the conjugates of those of A.

Let t be a positive integer or ∞. For each such t let ρ: Γ/𝔪^{t} → → F denote the map induced by the map called ρ in the preceeding paragraph. We set 𝔪^{∞} = 0. Thus Γ/𝔪^{t} is always a uniserial Γ-module, and has infinite length when t = ∞.

When t is finite, note that, once π has been chosen, there is a natural F -linear isomorphism σ: F ≅ 𝔪^{t−1}/𝔪^{t} defined by σ(γ +𝔪) = γπ^{t−1} +𝔪^{t}. We usually regard this map as an identification, in which case we have (Γ/𝔪^{t})π^{t−1} = 𝔪^{t−1}/𝔪^{t} = F, the distinguished copy of F in Γ/𝔪^{t}. This copy of F is the Γ-socle of Γ/𝔪^{t}. The following simple fact will be used many times.

(7.1.1) When t ≠ ∞, ≠ 1, the distinguished copy of F in Γ/𝔪^{t} satisfies ρ(F) = 0.

Notation 7.2 (Diagrams). Our constructions of indecomposable finitely generated Λ-modules M(D) begin with a nonempty direct sum of nonzero uniserial Γ-modules:

X = (Γ/𝔪^{i_{1}})^{(m)} ⊕ Γ/(𝔪^{j_{1}})^{(m)} ⊕ … ⊕ (Γ/𝔪^{i_{d}})^{(m)} [⊕ (Γ/𝔪^{j_{d}})^{(m)}] (7.2.1)

and a diagram D. The superscript (m) denotes “direct sum of m copies of”. We call m the block size of D and of X. The brackets at the end indicate that, in some cases, the final term is not present. For examples of the diagrams that we are referring to, see (7.4.1) and (7.5.2). We now give the set of rules for forming and interpreting diagrams.

Let X be given, as in (7.2.1). Each block of summands (Γ/𝔪^{t})^{(m)} is represented, in our diagrams, by a vertical bar with the length label t written over it, as shown in the diagram (7.2.2). If D_{0} denotes this diagram, we define M(D_{0}) = X.

X i_{1} j_{1} … i_{d} [j_{d}] (7.2.2)

We define D to be any diagram that can be formed from D_{0} by a finite number of applications of the following four operations, each of which attaches an “edge” to these vertical bars.

Top-glue. Choose some pair of vertical bars, neither of whose tops has an edge attached to it and connect the tops of these bars by an edge, as shown in the two “top-glue” diagrams in (7.2.3). Then label the left or right end of this edge with an invertible m × m matrix U over F, as shown. We view U as being attached to the top of the corresponding vertical bar, as well as to the gluing edge.

top-glue i(U) j  top-glue i (U)j  ^{top-reduce i (U)}

(7.2.3)

In the case of the first top-glue diagram in (7.2.3), replace the (external) direct sum (Γ/𝔪^{i})^{(m)} ⊕ (Γ/𝔪^{j})^{(m)} by the Λ-submodule given by the following pullback.

{(x, y) ∈ (Γ/𝔪^{i})^{(m)} ⊕ (Γ/𝔪^{j})^{(m)} | ρ(x) = ρ̄(y) · U} (7.2.4)

(One applies ρ to a tuple by applying ρ to each entry.) The reason for considering U to be attached to the i-labeled (rather than j-labeled) vertical bar is that, since U is invertible, the set of ordered pairs (ρ(x), ρ(y)) that arise from (7.2.4) is the set of all pairs

{(αU, ᾱ) ∈ F^{m} ⊕ F^{m} | α ∈ F^{m}} (7.2.5)

This will be consistent with our definition of bottom-gluing, below, and with the matrix pairs that will appear when we prove the main theorems of this section.

In the case of the second top-glue diagram, replace the condition ρ(x) = ρ̄(y) · U in (7.2.4) by ρ̄(x) · U = ρ(y). The effect of this is to move U to the “j” side of the equal sign in (7.2.5).

If U = I we usually do not explicitly display it. If we do display it, we get the same module whether we attach it to the i-side or the j-side of the top-gluing edge.

To explain our view of what has happened in this operation, first note that the pullback in (7.2.4) contains (𝔪/𝔪^{i})^{(m)} ⊕ (𝔪/𝔪^{j})^{(m)}; that is, it contains all but the topmost part of (Γ/𝔪^{i})^{(m)} ⊕ (Γ/𝔪^{j})^{(m)}. We think of the residue module

\frac{(Γ/𝔪^{i})^{(m)} ⊕ (Γ/𝔪^{j})^{(m)}}{(𝔪/𝔪^{i})^{(m)} ⊕ (𝔪/𝔪^{j})^{(m)}} = F^{(m)} ⊕ F^{(m)} (7.2.6)

(canonical isomorphism via ρ) as the “top” of (Γ/𝔪^{i})^{(m)} ⊕ (Γ/𝔪^{j})^{(m)}.

If U = I_{m}, then forming the pullback keeps only half of F^{(m)} ⊕ F^{(m)} in (7.2.6), namely the set of ordered pairs of the form (α, ᾱ). A general invertible matrix U “twists” the half of F^{(m)} ⊕ F^{(m)} that we are keeping.

Top-reduce. Choose some vertical bar whose top has no attached edge. Say its label is i. Attach an edge to this bar — displayed in diagram (7.2.3) by a short thick horizontal line segment — and label the edge with an invertible m×m matrix U over F, as shown. Then replace the corresponding summand (Γ/𝔪^{i})^{(m)} of X by its Λ-submodule

{x ∈ (Γ/𝔪^{i})^{(m)} | ρ(x) ∈ k^{(m)} U} (7.2.7)

As before, we usually do not display U if U = I, and we consider U to be attached to the vertical bar as well as to the reduction edge.

To understand what has happened here, note that the module in (7.2.7) contains the submodule (𝔪/𝔪^{t})^{(m)} of (Γ/𝔪^{t})^{(m)} — all but the top, as before. When U = I_{m} the replacement also keeps the half k^{(m)} of F^{(m)} = (Γ/𝔪^{t})^{(m)}/(𝔪/𝔪^{t})^{(m)}, the top of (Γ/𝔪^{t})^{(m)}. A general invertible matrix U again “twists” half we are keeping.

Let S(D) be the Λ-submodule of X that results from whatever top-gluing and top-reductions that have been done. (The various matrix labels U do not need to be the same.) Our discussion of which parts of the original summands of X remain intact by top-gluing and top-reduction shows:

(7.2.8) S(D) contains the distinguished copy of F^{(m)} in every summand (Γ/𝔪^{t})^{(m)} of X for which t ≠ ∞, ≠ 1.

Our remaining two operations on diagrams are shown symbolically below. As before we do not usually display U if U = I; and we regard U as being attached to the appropriate vertical bar as well as to the gluing or reduction edge.

bottom-glue j(U) i  bottom-glue j (U)i  bottom-reduce i (U)

(7.2.9)

Bottom-glue. Choose some pair of vertical bars whose bottoms have no attached edges and whose length labels j, i are ≠ ∞ and ≠ 1. Connect the bottoms of these bars with an edge, in the form of an elongated equal sign (as shown), and label the left or right end of this edge with an invertible m×m matrix U over F, as shown in (7.2.9). Since j ≠ ∞ and i ≠ ∞, the module (Γ/𝔪^{j})^{(m)} ⊕ (Γ/𝔪^{i})^{(m)} contains our distinguished copy of F^{(m)} ⊕ F^{(m)}. In the case of the first diagram in (7.2.9), form the Λ-submodule (7.2.5) of F^{(m)} ⊕ F^{(m)}, and call this the bottom-gluing module associated with the diagram. In the case of the second diagram in (7.2.9), move U to the “i”side of the comma in (7.2.5), and again call the resulting collection of ordered pairs the bottom-gluing module associated with the diagram.

Bottom-reduce. Choose some vertical bar whose bottom has no attached edge and whose label is ≠ ∞ and ≠ 1. Attach an edge to the bottom of this bar, displayed as a short thick horizontal line segment, and label this edge with an m×m invertible matrix U over F, as shown. Then form the following Λ-submodule of F^{(m)} = (𝔪^{i−1}/𝔪^{i})^{(m)}

k^{(m)} · U (7.2.10)

and call it the bottom-reduction module associated with the diagram.

Let D be the diagram that results from any such bottom and top operations, and let K(D) be the (necessarily direct) sum of the direct sum of bottom-gluing and the bottom-reduction modules thus formed. Then let

M(D) = S(D)/K(D) (7.2.11)

For (7.2.11) to make sense we need to prove that K(D) ⊆ S(D), but this follows from (7.2.8).

In order to see what passing modulo K(D) does, first consider the case that every U equals I_{m}. Then every bottom-gluing amalgamates one distinguished copy of F^{(m)} with another; but α in one summand is identified with −ᾱ in the other. Every bottom-reduction, on the other hand, reduces S(D) modulo the copy of k^{(m)} in the distinguished copy of F^{(m)}. When U ≠ I_{m}, basically the same thing happens (since U is invertible), but it is twisted by U.

The Λ-modules M(D) thus formed are not always indecomposable. Connectivity of diagram D is clearly a necessary condition for indecomposability, but is not sufficient.

Reversals. We close these introductory definitions by explicitly noting the effect of moving a matrix label U from one end of a (bottom or top) gluing edge to the other. Top-gluing and bottom-gluing — with U attached to the left-hand end of the edge — involve submodules of F^{(m)} ⊕ F^{(m)} consisting of all elements of the form (αU, ᾱ), which can be rewritten in the form (αU, (ᾱŪ)Ū^{−1}) = (β, β̄Ū^{−1}). We shall use this in the following form:

(7.2.12) Let U be the matrix label attached to one end of a gluing edge in D. Then moving U to the opposite end of that edge and then replacing U by Ū^{−1} leaves M(D) (not just its isomorphism class!) unchanged.

Notation 7.3 (Sequence manipulation). As in Notation 6.6, let μ be the mirror image permutation, which reverses the order of a sequence, so that μ{i_{1}, i_{2}, ..., i_{d}} = {i_{d}, ..., i_{2}, i_{1}}. Let ν be the unit forward rotation defined by ν{i_{1}, i_{2}, ..., i_{d}} = {i_{2}, i_{3}, ..., i_{d}, i_{1}}. A cycle is any cyclic permutation of the form ν^{t}, where t is an integer. If I and J are finite sequences, we let {I, J} denote the concatenation of I and J, that is, the sequence consisting of the terms of I followed by those of J. Thus {I, μ(J)} denotes I followed by the mirror image of J, and we have μ({I, μ(J)}) = {J, μ(I)}.

The definitions, below, of our seven “standard diagrams” all make use of the pair of label sequences in decomposition (7.2.1):

I = {i_{1}, i_{2}, …, i_{d}} J = {j_{1}, j_{2}, …, j_{d−1}, [j_{d}]} (7.3.1)

Thus every i_{ν} and j_{ν} is a positive integer or ∞, and the brackets around j_{d} indicates that j_{d} sometimes does not occur.

We now define some connected diagrams (and their associated modules, which turn out to be indecomposable) that we call standard diagrams.

Definitions 7.4 (Block size 1). The three standard diagrams in this series occur only with block size 1. The matrix labels U are all one-by-one identity matrices, and are therefore not displayed. (Thus, we must learn from context that the block size is 1.)

D_{Nrd}: i_{1} j_{1} i_{2} j_{2} … i_{d} j_{d}

D_{Brd} : i_{1} j_{1} i_{2} j_{2} … i_{d} j_{d} D_{Trd} : i_{1} j_{1} i_{2} j_{2} … i_{d} (7.4.1)

We now complete these definitions.

Nonreduced diagrams, D_{Nrd}. Here the following conditions must be satisfied.

(7.4.2) (i) The block size is m = 1.

(ii) Only length labels i_{1} and j_{d} can equal ∞ or 1.

(iii) The pair of label sequences I and J must be unsymmetrical in the sense that J ≠ μ(I).

Since the one-by-one identity matrix is its own conjugate inverse, the following is an immediate consequence of (7.2.12):

(7.4.3) Replacing the pair of label sequences I and J in diagram D_{Nrd} by μ(J) and μ(I), respectively — that is, drawing the left-right mirror image of the diagram — does not change the isomorphism class of M(D_{Nrd}).

In the extreme case d = 1, no bottom-gluing occurs. When condition (7.4.2)(iii) fails, M(D_{Nrd}) becomes the direct sum of two indecomposable modules [Proposition 7.16].

Bottom-reduced and top-reduced diagrams, D_{Brd} and D_{Trd}. The restrictions here are:

(7.4.4)

(i) The block size is m = 1.

(ii) Only length label i_{1} can equal ∞ or 1.

(Note that j_{d} does not occur in the top-reduced diagram D_{Trd}.) In the extreme case d = 1, D_{Trd} consists of a single top-reduced vertical bar. If in addition i_{1} = 1, this becomes the standard diagram for k, the residue field of Λ:

k : 1 (7.4.5)

Before proceeding, we note that there are other possibilites for diagrams with block size 1. Some of these [see diagrams (7.5.2)] yield indecomposable modules in all block sizes, as we shall see, while others are not standard diagrams [see Remark 7.10], even if they are connected.

Definitions 7.5 (Arbitrary block size). The remaining four standard diagrams occur with arbitrary block size. Before stating details, we mention some important differences between these and the standard diagrams of block size 1: (i) all length labels are ≠ ∞, ≠ 1, and therefore the associated modules M(D) all have finite length. (ii) Both the top and bottom of every vertical bar is either glued or reduced.

(iii) The additional conditions that guarantee indecomposability are more complicated. (iv) Each diagram explicitly displays one invertible m × m matrix label, U or U^{−1} over F, that we call the blocking matrix of the diagram. The reason for sometimes using U and sometimes U^{−1} is to achieve uniformity in the uniqueness formulas in Theorem 7.7.

According to our convention, the remaining matrix labels are identity matrices, and are not displayed explicitly. The placement of the blocking matrix in the diagram is somewhat arbitrary. (See Proposition 7.8 about moving U.)

We require the following definition.

(7.5.1) A finite sequence W is repetition-free if there is no strictly shorter sequence W′ such that W consists of repetitions of W′.

The following are the standard diagrams associated with arbitrary block size.

D_{BBrd} i_{1} j_{1} i_{2} j_{2} … i_{d} j_{d} (U^{−1})

D_{BTrd}  i_{1}  (U) j_{1} i_{2} j_{2} … i_{d}

D_{TTrd} ^{i_{1} j_{1} i_{2} j_{2} … i_{d} j_{d} (U)}

D_{Cy} i_{1}(U) j_{1} i_{2} j_{2} … i_{d} j_{d}

(7.5.2)

Bottom-bottom-reduced diagrams, D_{BBrd}. We require the following conditions:

(7.5.3) (i) UŪ^{−1} is indecomposable under similarity.

(ii) All length labels are ≠ ∞ and ≠ 1.

(iii) The concatenated sequence {I, μ(J)} is repetition-free [see (7.5.1)].

Although it is not obvious, there exist matrices U of all sizes satisfying condition (i). (See Remark 7.12.) Note that, when d = 1, no bottom-gluing occurs.

Bottom-top-reduced diagrams, D_{BTrd}. We require conditions (7.5.3). Note that j_{d} does not occur in this diagram. When d = 1 the diagram becomes a single vertical bar, reduced both at the bottom and the top.

Top-top-reduced diagrams, D_{TTrd}. We require conditions (7.5.3) here, too. When d = 1 no pullback occurs.

Cycle diagrams, D_{Cy}. We require:

(7.5.4) (i) The blocking matrix U is indecomposable under similarity.

(ii) Either the blocking matrix U is not similar to Ū^{−1} or, for all cycles ν^{t},

J ≠ ν^{t}μ(I). (iii) All length labels are ≠ ∞ and ≠ 1. (iv) The sequence of pairs (i_{1}, j_{1}), …, (i_{d}, j_{d}) is repetition-free.

When condition (ii) is not satisfied, the resulting module becomes the direct sum of two indecomposable modules [Proposition 7.16]. Unlike the previous types, there is nothing exceptional about the case d = 1.

For square matrices U, V over F we write U ∼ V for “U is similar to V ”.

Theorem 7.6. Every indecomposable finitely generated Λ-module is isomorphic to M(D) where D is one of the standard diagrams in (7.4.1) or (7.5.2); and all such modules are indecomposable. Moreover, every indecomposable finitely generated Λmodule of infinite length is isomorphic M(D) for one of the standard diagrams D in (7.4.1).

Theorem 7.7. Let D and D′ be standard diagrams, with pairs of label sequences I and J, and I′ and J′ respectively. If these diagrams have blocking matrices, call them U or U^{−1}, and V or V^{−1} respectively, and assume that the blocking matrices are located as shown in diagram (7.5.2).

Then M(D) ≅ M(D′) if and only if D and D′ are of the same type (i)–(vii) below, and the conditions listed for their type hold.

(i) (Nonreduced) Either I′ = I and J′ = J, or I′ = μ(J) and J′ = μ(I). (ii) (Bottom reduced) I′ = I and J′ = J. (iii) (Top reduced) I′ = I and J′ = J. (iv) (Bottom-bottom reduced) Either I′ = I and J′ = J, or I′ = μ(J) and

J′ = μ(I); and V V̄^{−1} ∼ UŪ^{−1}.

(v) (Bottom-top reduced) I′ = I and J′ = J, and V V̄^{−1} ∼ UŪ^{−1}. (vi) (Top-top reduced) Either I′ = I and J′ = J, or I′ = μ(J) and J′ = μ(I);

and V V̄^{−1} ∼ UŪ^{−1}. (vii) (Cycle) Either I′ = ν^{t}(I) and J′ = ν^{t}(J) (for some cyclic permutation ν^{t})

and V ∼ U; or I′ = ν^{t}μ(J) and J′ = ν^{t}μ(I) (for some cyclic permutation ν^{t}) and V ∼ Ū^{−1}.

We note that, in every case where the pair of label sequences is not uniquely determined by the isomorphism class of M(D), the alternative pair corresponds to the geometric left-right or rotational symmetry of the diagram. However (according to the statement of the theorem), when applying this symmetry to the bars, edges, and length-labels, one leaves the location of the blocking matrix unchanged, in the position shown in standard diagrams (7.5.2).

As mentioned earlier, the placement of the blocking matrix in standard diagrams (7.5.2) is somewhat arbitrary. The precise rules for moving the blocking matrix — and appropriately modifying it — are the subject of our next result.

Proposition 7.8 (Moving U). Let D be any standard diagram in the “arbitrary block size” family (7.5.2), except that the blocking matrix U or U^{−1} is located at an arbitrary position in the diagram.

Then each of the following operations on U leaves the isomorphism class of M(D) unchanged.

(i) Move U from one end (left or right) of any gluing edge to the other end of that edge, replacing it with Ū^{−1}.

(ii) Move U from one end (top or bottom) of any vertical bar to the other end,

replacing it with U^{−1}. In addition — if D is one of the standard diagrams with a pair of reduction edges — U can be moved unchanged from the top of any vertical bar to the top of any other vertical bar (or from bottom to bottom) without changing the isomorphism class of M(D).

For example, in the first of the diagrams in (7.5.2) — diagram D_{BBrd} — we can move U^{−1} to the top of its attached vertical bar, replacing it with U. This is consistent with the other standard diagrams in (7.5.2). However, it introduces the inconsistency that the blocking matrix is no longer attached to a reduction edge. Moreover, this will cause some awkwardness when we relate these diagrams to their associated matrices, in the proofs that occur later in this section.

Our survey of module structure is completed by the following result, already proved in Lemma 1.3.

Theorem 7.9. The Krull-Schmidt Theorem holds for finitely generated Λ-modules.

Remark 7.10 (Efficient vs. standard representations). Our standard diagrams handle some indecomposable Λ-modules inefficiently. This includes all of the indecomposable finitely generated Γ-modules. One might expect the standard diagram for Γ/𝔪^{j} (1 ≤ j ≤ ∞) to be the second diagram in (7.10.1), but it is the first diagram that is standard.

(7.10.1)  Γ/𝔪^{j} :  ^{1 j} (standard)  j (not standard)

To see that the second diagram is not standard, it suffices to note that it does not appear among the three standard “block size one” diagrams in (7.4.1). (Set d = 1, the smallest value of d that makes sense in each of these diagrams.) The first diagram in (7.10.1) is a nonreduced diagram; and to see that it actually describes Γ/𝔪^{j} is a simple verification using the definition (7.2.4) of top-gluing.

To see the general principle involved, first note that the only places that the label 1 can occur in a standard diagram are: the left-hand end of the three diagrams in (7.4.1) [the “block size 1 only” diagrams], and the right-hand end of the first of these. It is then easy to see:

(7.10.2) If a vertical bar with length label 1 occurs in a standard diagram D in (7.4.1) with an attached top-gluing edge, then the diagram D′ that results from deleting the vertical bar and attached edge satisfies M(D′) ≅ M(D). But D′ is not a standard diagram.

Making these deletions gives what we call the efficient representation of the module involved, as opposed to the “standard” representation of that module.

If the only purpose of this paper were to describe finitely generated Λ-modules, we would use the efficient representation. A minor disadvantage of this would be that we would have to increase the number of types of standard diagrams by including, for example, diagrams whose left-most gluing edge is a bottom-gluing edge. But this minor disadvantage would be more than offset by the fact that not having an irrelevant vertical bar makes M(D′) easier to visualize.

The real advantage of keeping the “irrelevant” edge has to do with our Theorem 7.11 about morphisms in mod-Λ: We do not know whether the theorem applies when standard representations are replaced by efficient ones.

Note that if D is the disjoint union of diagrams D_{1}, …, D_{m} then M(D) = M(D_{1})⊕…⊕M(D_{m}). Therefore (by the theorems stated so far in this section) every finitely generated Λ-module is isomorphic to M(D) where D is a disjoint union of standard diagrams. We can now state our main theorem on morphisms in mod-Λ, which shows that all such morphisms arise from homomorphisms of modules over the DVR Γ.

Theorem 7.11. Let D′ and D be disjoint unions of standard diagrams, and let f: M(D′) → M(D) be a Λ-homomorphism. Then there is a Γ-homomorphism f^{∗∗}: X(D′) → X(D) whose restriction f^{∗} to S(D′) is a Λ-homomorphism: S(D′) → S(D) that takes K(D′) → K(D) and induces f. Moreover, if f is one-to-one or onto, then so is any such f^{∗}.

As we shall see in the proof of this theorem, the natural surjection S(D) → → M(D) is a separated cover of M(D), that is, a “best possible approximation” to M(D) by a Λ-submodule of some Γ-module. The basic idea upon which this paper is based is that separated covers are unique and every homomorphism of Λ-modules can be lifted to a homomorphism of their separated covers. Section 2 is devoted to properties of separated covers.

Remark 7.12 (Indecomposability of UŪ^{−1}). Condition (7.5.3)(i) on the indecomposability of UŪ^{−1} under similarity, and condition (7.5.4)(ii) that U not be similar to Ū^{−1}, fit in with Theorem 7.7 in a particularly interesting way.

Let U be an invertible m × m matrix over F, and suppose that U is indecomposable under similarity. Thus U can be the companion matrix of any power of any irreducible polynomial ≠ x in F [x].

By (7.5.4)(ii), U can be the blocking matrix of a cycle diagram D_{Cy} in (7.5.2) except if U ∼ Ū^{−1} and the label sequences {i_{ν}} and {j_{ν}} fail to satisfy a certain non-symmetry condition. When these restrictions are satisfied, it is the similarity class of U that is an isomorphism invariant of M(D_{Cy}). However it is not obvious what similarity classes are excluded here.

A “Hilbert Theorem 90” for matrices, due to Ballantine [see Lemma 6.11], states that a matrix satisfies the condition U ∼ Ū^{−1} if and only if there is a matrix V such that V V̄^{−1} = U. According to condition (7.5.3)(i), such matrices V are precisely the matrices that are allowed to be blocking matrices for the remaining three diagrams that have blocking matrices [see (7.5.2)]. Moreover, according to our structure Theorem 7.7, it is the similarity class of V V̄^{−1} = U that is an isomorphism invariant of M(D) in these cases — precisely the similarity classes that are excluded from M(D_{Cy})!

To see precisely what similarity classes of matrices we are talking about, let U = C(f) be the companion matrix of a polynomial f ∈ F [x], where f = x^{m} + a^{m−1}x^{m−1} + ⋯ + a_{0}. Since U is invertible in this discussion, we have a_{0} ≠ 0. Let g be the monic polynomial g = (x^{m}/a_{0})f(1/x). Then it follows easily, with the help of the Cayley-Hamilton Theorem, that U^{−1} ∼ C(g). Thus

U ∼ Ū^{−1} ⟺ f = (x^{m}/ā_{0})f̄(1/x) (7.12.1)

It is easy, using this, to see that there exist invertible indecomposable matrices U of all sizes that satisfy conditions (7.12.1). For example, use f(x) = (x + 1)^{m}.

7.13. Connection with matrices. The proofs in this section require definitions and theorems that are scattered throughout the earlier sections of this paper. In this subsection we give a brief introduction and directory to the basic definitions and ideas.

By Theorem 3.9 every finitely generated Λ-module (when Λ is unsplit) is isomorphic to a module of the form M = M(A, B) where A, B are suitable matrices over F constructed from a matrix setup X. Associated with X is a direct sum X of uniserial Γ-modules Γ/𝔪^{t}, t a positive integer or ∞ [see (3.6.2)]. We will soon rearrange and group these summands — when M is indecomposable — into the form of the Γ-module X displayed in (7.2.1). Let n be the number of uniserial summands of X, and e_{∞} the number of uniserial summands of X of infinite length. Then A is an n × n invertible matrix and B is q × (n − e_{∞}) for some q. We have

M = M(A, B) = S(A)/K(B) where
S(A) = {x ∈ X | ρ(x) ∈ k^{(n)} · A} and
K(B) = k^{(q)} · B

(7.13.1)

To interpret this, note that ρ is the map X → F^{(n)} defined, coordinate by coordinate, by the maps ρ: Γ/𝔪^{t} → F defined at the beginning of this section.

We review the identifications by which K(B) is considered to be a submodule of S(A). Since the number n of uniserial summands of X is the same as the number of columns of A, we can speak of the summand of X corresponding to each column of A. The matrix B has one column for each uniserial summand of X that has finite length. If we disregard the uniserial summands of X of infinite length — and the corresponding columns of A — then the new A has the same number of columns as B. Each column of B thus has a corresponding column of A and a corresponding uniserial summand (of finite length) of X. We view B as being placed under A, with each column of B written under the corresponding column of A (and no column of B under each column of A whose corresponding summand of X has infinite length). Having done this, we attach a length label to each column of (A, B), namely the length of the corresponding uniserial summand of X. (See (5.1.1) for an example of all this.)

Since the uniserial summand of X corresponding to each column of B has finite length, it contains a distinguished copy of F [Notation 7.1]. Each element of K(B) has one coordinate for each column of B and therefore each element of K(B) can be considered to be an element of the direct sum of those uniserial summands of X that have finite length. The rules for constructing B then imply that K(B) ⊆ S(A), and therefore the definition of M(A, B) makes sense. (See Notation 3.7 for details.)

The matrix pair (A, B) — together with its length labels — can be put into canonical form, as described in Corollary 6.17 — a direct sum of indecomposable matrix pairs, with each indecomposable pair corresponding to a single indecomposable direct summand of the Λ-module M.

Now assume that M is indecomposable. Part of the process of reduction to canonical form involves simultaneously permuting the columns of (A, B) — together with their length labels and corresponding summands of X — so that we can still refer to uniserial summands of X and their corresponding columns of (A, B). At this point X is in the form (7.2.1), and we can now refer to the block size m of (A, B): Each m-column block of (A, B) has a length label t, corresponding to the fact that all m corresponding uniserial summands of X are isomorphic to Γ/𝔪^{t}. We can now form diagram D_{0}, consisting of one vertical bar for each m-summand block of X, labeled with the length of the uniserials in that block of summands of X, as in the diagram (7.2.2).

The detailed description of all possible canonical forms of (A, B) is given Notation 6.7 and 6.10. There is one canonical form for each of the seven standard diagrams in (7.4.1) and (7.5.2), and each canonical form has the same name as the corresponding standard diagram.

7.14. Proof of Theorems 7.6 and 7.7. We continue the notation of the previous subsection, letting the Λ-module M be indecomposable. Our first objective is to show that the Λ-module determined by each of the seven canonical forms of (A, B) is the module determined by the diagram with the same name. Recall that the element ε in the canonical forms is an element of F such that F = k[ε].

In each canonical form, each of A and B is a direct sum of zero or more basic tiles of the form

T = [\begin{matrix}U & I \\ Uε & Iε̄ \end{matrix}] ,  usually  [\begin{matrix}I & I \\ Iε & Iε̄ \end{matrix}]  (I = I_{m}) (7.14.1)

and not more than two exceptional tiles equal to the single m × m matrix U or I_{m}. If (A, B) corresponds to one of the diagrams that has a blocking matrix U, then U appears in exactly one tile in the pair (A, B). On the other hand, if (A, B) corresponds to one of the diagrams with no blocking matrix, then all identity matrices have size one-by-one (that is, the block size m equals 1).

It is possible for one of the basic tiles to be “wrapped”, with its first column block actually occurring in the last column block of (A, B) and its second column block occurring at the beginning of (A, B) [for example, see the last and first columnblocks of B in (6.10.6)].

Construct a diagram D from (A, B) as follows. Begin with D_{0}, which consists of one vertical bar for each block of uniserial summands of X, labeled by the common length of the summands in that block. Form D by attaching edges to D_{0} according to the following rules. In these rules: “Column blocks” always refer to m-column blocks of (A, B), with B underneath A and corresponding column blocks lined up. Every basic tile that occurs in A or B has two corresponding vertical bars, namely the vertical bars that correspond to the column blocks containing that tile. Similarly, every exceptional tile has one corresponding vertical bar.

(7.14.2) (i) For each basic tile that occurs in A, connect its two corresponding vertical bars with a top-gluing edge. If U occurs in this basic tile (and hence in the first of the two column blocks of the tile), label the left-hand end of the edge with U.

(ii) For each basic tile that occurs in B, connect the two corresponding vertical bars with a bottom-gluing edge. Suppose U occurs in this basic tile — this does not happen in the seven canonical forms, but will happen later in some modified canonical forms — say, in column block j. Then label the side of the gluing edge corresponding to column block j with U.

(iii) For each exceptional tile that occurs in A attach a top-reduction edge to the corresponding vertical bar. If the tile consists of the blocking matrix U, label the top-reduction edge with U.

(iv) For each exceptional tile that occurs in B, attach a bottom-reduction edge to the corresponding vertical bar. If the tile consists of the blocking matrix U, label the bottom-reduction edge with U.

It is immediate that D is the standard diagram with the same name as the canonical form (A, B).

Next we show that S(A) = S(D). Choose any basic tile T that occurs in A. This tile interacts with precisely two uniserial summand-blocks of X, namely those corresponding to the column blocks containing T, and T is the only tile contained in A that interacts with either of these summand-blocks. Let the length labels of the first and second column blocks of T be i and j respectively, and denote the corresponding direct sum of uniserial summands of X by X_{i} and X_{j} respectively (ignoring the slight abuse of notation that results when i = j and X_{i} ≠ X_{j}). Thus X_{i} = (Γ/𝔪^{i})^{(m)} and X_{j} = (Γ/𝔪^{j})^{(m)}. If x ∈ X_{i} and y ∈ X_{j} and (x, y) ∈ S(A), then formula (7.13.1) implies that ρ(x, y) ∈ (k^{(m)} ⊕k^{(m)})T. Writing this out, using the form of a basic tile given in (7.14.1) (where possibly U = I), yields

(ρ(x), ρ(y)) = ((a + bε)U, a + bε̄) (a, b ∈ k^{(m)}) (7.14.3)

Writing α = a + bε we see that the ordered pair in (7.14.3) equals (αU, ᾱ) where α is an arbitrary element of F^{(m)}. Therefore ρ(x) = ρ̄(y)U; in other words, X_{i} and X_{j} are top-glued as in (7.2.4).

On the other hand, consider any exceptional tile that occurs in A. This block equals U or I; we write U in either case. This tile interacts with the uniserial summand X_{i} of X that corresponds to the column block containing the tile, and is the only tile in A that interacts with X_{i}. Take x ∈ X_{i}. If x ∈ S(A) then (7.13.1) implies that ρ(x) ∈ k^{(m)}U. Thus X_{i} is top-reduced in the sense of (7.2.7). This completes the proof that S(A) = S(D).

K(B) = K(D). The analysis here is very similar to the analysis of S(A) except that the basic and exceptional tiles (whichever occur) result in bottom gluing and bottom reduction, respectively.

Thus we now have M(A, B) = M(D). The rest of the proofs of the two theorems under consideration consists of reciting results from §6, together with the fact that indecomposable matrix pairs correspond to indecomposable modules: Every indecomposable finitely generated Λ-module is isomorphic to M(A, B) = M(D), where D is a standard diagram with the same name as the matrix pair (A, B) [Corollary 6.17]. Since the Krull-Schmidt Theorem holds for matrix pairs [Corollary 6.17], it follows that every M(D), with D a standard diagram, is indeed indecomposable. Finally, the uniqueness assertions in Theorem 7.7 are just restatements of the uniqueness Theorem 6.18 on matrix pairs.

7.15. Proof of Proposition 7.8. Statement (i) was proved in (7.2.12). Let M = M(D), where D is a standard diagram, except that the blocking matrix U labels an arbitrary gluing or reduction edge (to be specified).

Statement (ii). We want to show that moving U to the opposite end of its attached vertical bar and then changing it to U^{−1} does not change the isomorphism class of M. If this move can be done, then it can also be done in reverse, so there is no loss of generality in assuming that we start at the top of a vertical bar. We need to consider seven situations separately [see (7.15.1)]. In each situation we wish to move U from position 1 to position 2, replacing it by U^{−1}. We usually change notation for each configuration, placing a matrix U at position 3 and then comparing the effects of moving it to position 1 and position 2, and determining how U must then be changed to preserve the isomorphism class of M. We consider the seven situations in the order in which they are displayed, doing the first in more detail than the others.

1_{2}^{3}  1 2 3  ^{1}_{2}^{3}  1 2 3  {1 \atop 2}  {1 \atop 2}  {1 \atop 2} (7.15.1)

Let D be the diagram before the blocking matrix is moved. Thus D is a standard diagram except that U is in a position that we shall specify. In the notation of subsection 7.13 and 7.14 we have M = S(A)/K(B) for the matrix pair (A, B) over F, where (A, B) is in one of our standard canonical forms — the form with the same name as D — except that the blocking matrix is located in a basic tile to be specified, instead of the standard place.

Our proof makes use of the matrix 4-tuple (A, Ā, B, B̄) corresponding to the given matrix pair. Recall that (A, B) ≅ (A′, B′) if and only if (A, Ā, B, B̄) ≅ (A′, \overline{A′}, B′, \overline{B′}) [Theorem 6.3]. We always assume that M(A, B) ≅ M(D) before the blocking matrix is moved, and then prove that the isomorphism continues to hold afterwards. Our proofs make use of the fact that sweeping-similarity operations do not change the isomorphism class of (A, Ā, B, B̄) [Theorem 4.7]. But recall that we use F -F sweeping similarity instead of k-k-similarity since our matrices have entries in F.

Top-gluing edge at left (of vertical bar) to bottom-gluing edge. This is the first situation displayed in (7.15.1). We assume that U is placed in position 3. By statement (i) we can move U to position 1, changing it to Ū^{−1}. Therefore it suffices to prove that we can move U from position 3 to position 2, changing it to Ū.

The matrix 4-tuple in (7.15.2) shows the portion of (A, Ā, B, B̄) that we wish to alter: We wish to move U from the two top tiles to the two bottom tiles and replace it with Ū.

\begin{matrix}[ & \begin{matrix}U & I \\ Uε & Iε̄ \end{matrix} & ] \\ [ & \begin{matrix}I & I \\ Iε & Iε̄ \end{matrix} & ] \end{matrix} \begin{matrix}[ & \begin{matrix}Ū & I \\ Ūε̄ & Iε \end{matrix} & ] \\ [ & \begin{matrix}I & I \\ Iε̄ & Iε \end{matrix} & ] \end{matrix} (7.15.2)

The first step is to reduce the basic tiles, using Lemma 6.4 twice. This changes them to the form shown in the 4-tuple in (7.15.3).

\begin{matrix}[ & \begin{matrix}U & 0 \\ 0 & I \end{matrix} & ] \\ [ & \begin{matrix}I & 0 \\ 0 & I \end{matrix} & ] \end{matrix} \begin{matrix}[ & \begin{matrix}0 & I \\ Ū & 0 \end{matrix} & ] \\ [ & \begin{matrix}0 & I \\ I & 0 \end{matrix} & ] \end{matrix} (7.15.3)

Next, left multiplication of the first row-block of the upper two matrices by U^{−1} and the second row-block by Ū^{−1} yields the form shown in (7.15.4).

\begin{matrix}[ & \begin{matrix}I & 0 \\ 0 & Ū^{−1} \end{matrix} & ] \\ [ & \begin{matrix}I & 0 \\ 0 & I \end{matrix} & ] \end{matrix} \begin{matrix}[ & \begin{matrix}0 & U^{−1} \\ I & 0 \end{matrix} & ] \\ [ & \begin{matrix}0 & I \\ I & 0 \end{matrix} & ] \end{matrix} (7.15.4)

Then, right multiplication of the second column-block of the first upper-lower pair by Ū and the second column-block of the second upper-lower pair by U yields the form shown in (7.15.5).

\begin{matrix}[ & \begin{matrix}I & 0 \\ 0 & I \end{matrix} & ] \\ [ & \begin{matrix}Ū & 0 \\ 0 & I \end{matrix} & ] \end{matrix} \begin{matrix}[ & \begin{matrix}0 & I \\ I & 0 \end{matrix} & ] \\ [ & \begin{matrix}0 & I \\ U & 0 \end{matrix} & ] \end{matrix} (7.15.5)

Finally, note that this matrix 4-tuple in (7.15.5) can also be obtained by starting with the 4-tuple in (7.15.6) and applying Lemma 6.4 to each of the two pairs of basic tiles.

\begin{matrix}[ & \begin{matrix}I & I \\ Iε & Iε̄ \end{matrix} & ] \\ [ & \begin{matrix}Ū & I \\ Ūε & Iε̄ \end{matrix} & ] \end{matrix} \begin{matrix}[ & \begin{matrix}I & I \\ Iε̄ & Iε \end{matrix} & ] \\ [ & \begin{matrix}U & I \\ Uε̄ & Iε \end{matrix} & ] \end{matrix} (7.15.6)

This completes the proof that we can move U from position 3 to Ū in position 2.

Top-gluing edge at right (of vertical bar) to bottom-gluing edge. This is done similarly.

Top-gluing edge at left (of vertical bar) to bottom-reduction edge. We want to change the first 4-tuple displayed in (7.15.7) to the second. The procedure is the same as the first situation we considered.

\begin{matrix}[ & \begin{matrix}U & I \\ Uε & Iε̄ \end{matrix} & ] \\ [ & I & ] \end{matrix} \begin{matrix}[ & \begin{matrix}Ū & I \\ Ūε̄ & Iε \end{matrix} & ] \\ [ & I & ] \end{matrix} \begin{matrix}[ & \begin{matrix}I & I \\ Iε & Iε̄ \end{matrix} & ] \\ [ & Ū & ] \end{matrix} \begin{matrix}[ & \begin{matrix}I & I \\ Iε̄ & Iε \end{matrix} & ] \\ [ & U & ] \end{matrix} (7.15.7)

Top-reduction edge to bottom-gluing edge at left (of vertical bar). Interchange the roles of A and B in the previous case.

Top-gluing edge at right (of vertical bar) to bottom-reduction edge. Here we ignore position 3, and move U directly from position 1 to position 2. Moreover, it is easier to ignore the 4-tuple and work directly with (A, B), transforming the first matrix pair in (7.15.8) to the second by means of the k-F sweeping-similarity operations of Theorem 3.10. In fact, we need only right-multiply the first column block of the first matrix pair by U^{−1} in order to obtain the second matrix pair.

\begin{matrix}[ & \begin{matrix}U & I \\ Uε & Iε̄ \end{matrix} & ] \\ [ & I & ] \end{matrix} \begin{matrix}[ & \begin{matrix}I & I \\ Iε & Iε̄ \end{matrix} & ] \\ [ & U^{−1} & ] \end{matrix} (7.15.8)

Top-reduction edge to bottom-gluing edge at right (of vertical bar). Reverse the roles of A and B in the previous case.

Top-reduction edge to bottom-reduction edge. This again is the same, but replace the two basic tiles (A-matrices) in (7.15.8) by the exceptional tiles U and I, respectively.

Supplementary statement. Now suppose that D is one of the standard diagrams with a pair of reduction edges. Then the similarity invariant of D is the similarity class of UŪ^{−1} rather than of U itself [Theorem 7.7]. Suppose that U is attached to some top gluing edge. Then replacing U by Ū^{−1}, and not moving it, leaves the similarity invariant of D unchanged. Now, by (i), moving Ū^{−1} to the other end of the gluing edge replaces it by U again. Thus, moving U from the top of one vertical bar to the top of an adjacent vertical bar does not change the isomorphism class of M(D). After doing this, we can move U to the bottom of that bar, replacing it by U^{−1}. If that bar has an attached gluing edge, the same reasoning as before allows us to move U^{−1} to the other end of that edge without changing the isomorphism class of M(D). Since all standard diagrams are connected diagrams, a combination of these moves enable us to move U from top to top (or, similarly, bottom to bottom) of any pair of vertical bars.

Proposition 7.16. Suppose that D = D_{Nrd}, except that condition (7.4.2)(iii) fails (that is, J = μ(I)), or D = D_{Cy}, except that condition (7.5.4)(ii) fails (that is, U ∼ Ū^{−1} and J = ν^{t}μ(I) for some cycle ν^{t}). Then M(D) is the direct sum of two isomorphic indecomposable Λ-modules.

Proof. As at the end of subsection 7.14, we have M(A, B) = M(D) for an appropriate (A, B). Therefore this proposition is a module-theoretic translation of Proposition 6.19.

7.17. Proof of Theorem 7.11. A separated Λ-module is a Λ-submodule of some Γ-module. The fundamental structure on which this paper is built is that of a separated cover, which is a closest possible approximation of an arbitrary finitely generated Λ-module by a separated Λ-module. Section 2 of this paper gives a precise definition of this, and develops its basic properties.

Now let ϕ: S(D) → → M(D) be the natural homomorphism. We claim that ϕ is a separated cover. We already observed that M(D) ≅ M(A, B) = S(A)/K(B) for a suitable matrix pair (A, B) [(7.13.1)]; and in subsection 7.14 we showed that S(A) = S(D) and K(B) = K(D). Therefore the claim follows from Theorem 3.9(ii). Similarly, ϕ′: S(D′) → → M(D′) is a separated cover.

Our main “almost functorial property” of separated covers [Theorem 2.6] states that any homomorphism f of finitely generated Λ-modules can be lifted to a Λhomomorphism f^{∗} of their separated covers, and this lifting preserves monomorphisms and surjections. Thus it now suffices to further extend f^{∗}S(D′) → S(D) to a Γ-homomorphism f^{∗∗}: X(D′) → X(D).

First note that X(D) = Γ · S(D) and X(D′) = Γ · S(D′), by the second equation in (7.13.1), because the matrices A and A′ are invertible. Now Γ·S(D) ≅ Γ⊗_{Λ}S(D) and Γ · S(D′) ≅ Γ ⊗_{Λ} S(D′), by Lemma 3.2, so that f^{∗∗} = 1 ⊗ f^{∗} is the desired extension of f^{∗}.

Remark 7.18 (on repetition-freeness). When the condition on repetition-freeness (7.5.4)(iv) fails for cycle diagrams, it is easy to see that a larger block size can be used, with a correspondingly different blocking matrix. The new, larger blocking matrix can fail to be indecomposable under similarity, in which case M(D) decomposes.

When the repetition-freeness condition (7.5.3)(iii) fails for one of the other three diagrams that occur with arbitrary block size, a similar (but slightly more complicated) thing happens. For readers familiar with the results of §6, the following more informative discussion actually applies to all four diagrams that use blocking matrices. Let p be the matrix pair for which the condition fails. Then, in the associated 4-tuple f(p), a larger block size can be used. Let its (larger) blocking matrix be L. Our desire for a canonical form then allows us to disregard the smaller block size. Changing to the larger blocking matrix L can also change the type of diagram that we are dealing with. Moreover, the new, larger blocking matrix might be decomposable under similarity, which easily shows that f(p) decomposes. If this decomposition of f(p) contains a pair of mutually conjugate summands, or a single self-conjugate summand, and contains additional terms, then f(p) is a direct sum of two packages, and hence p decomposes. In this case, M(p) decomposes as well.

8. Indecomposable Modules, Strictly Split Case

In this section we describe how to construct all indecomposable finitely generated Λ-modules from indecomposable (necessarily uniserial) finitely generated Γ-modules, in the strictly split case. This is similar to — but simpler than — the unsplit case. We also describe, very briefly, how homomorphisms of Λ-modules arise from homomorphisms of Γ-modules. This structure was determined previously, in [L3]. Therefore our focus, in this short section, is to make the terminology of this description consistent with the other terminology of the present paper. This will be especially important when we deal with the nonlocal situation, in the third paper of this series. As in the unsplit case, the statements of our main results (but not of their proofs) can be read independently of the sections between here and the introduction to this paper, and so, to enhance the continuity of this section as a continuation of the Introduction, the proofs of the main results occur at the end [Subsection 8.9].

Notation 8.1. Throughout this section (Λ, 𝔪, k) is a strictly split Dedekind-like (residue commutative) ring, as in Notation 1.1. Thus a normalization of Λ is Γ = Γ_{1} ⊕ Γ_{2} where each (Γ_{ν}, 𝔪_{ν}, k) is a (residue commtative) DVR and 𝔪 = 𝔪_{1} ⊕ 𝔪_{2}.

Following the style of Notation 3.6 we denote by ρ not only the map Γ → → Γ̄ = Γ/𝔪 = k ⊕ k in pullback diagram (1.1.1), but also the maps Γ_{ν}/𝔪_{ν}^{t} → → k induced by the original map ρ, and direct sums of such maps. Moreover, we let 𝔪_{ν}^{∞} = 0, so that Γ_{ν} = Γ_{ν}/𝔪_{ν}^{∞}.

Recall, from Notation 1.1, that we have chosen a distinguished generator π_{ν} of the maximal ideal 𝔪_{ν} = Γ_{ν}π_{ν} = π_{ν}Γ_{ν} of each Γ_{ν}.

When t is finite, there are natural k-linear isomorphisms σ: k ≅ 𝔪_{ν}^{t−1}/𝔪_{ν}^{t} defined by σ(γ_{ν} + 𝔪_{ν}) = γ_{ν}π_{ν}^{t−1} + 𝔪_{ν}^{t}. We usually regard this map as an identification, in which case we have (Γ_{ν}/𝔪_{ν}^{t})π_{ν}^{t−1} = 𝔪_{ν}^{t−1}/𝔪_{ν}^{t} = k, the distinguished copy of k in Γ_{ν}/𝔪_{ν}^{t}. This copy of k is the Γ_{ν}-socle of Γ_{ν}/𝔪_{ν}^{t}. The following simple fact will be used many times.

(8.1.1) When t ≠ ∞, ≠ 1, the distinguished copy of k in Γ_{ν}/𝔪_{ν}^{t} satisfies ρ(k) = 0.

Notation 8.2 (Diagrams). As in the unsplit case, our constructions of indecomposable finitely generated Λ-modules M(D) begin with a nonempty direct sum of nonzero uniserial Γ-modules:

X = X_{1} ⊕ X_{2} where

X_{1} = (Γ_{1}/𝔪_{1}^{i_{1}})^{(m)} ⊕ … ⊕ (Γ_{1}/𝔪_{1}^{i_{d}})^{(m)} X_{2} = (Γ_{2}/𝔪_{2}^{j_{1}})^{(m)} ⊕ … ⊕ (Γ_{2}/𝔪_{2}^{j_{d}})^{(m)} (8.2.1)

Here each i_{ν} and j_{ν} is a positive integer or ∞, and the exponent (m) denotes “direct sum of m copies of”. Since each Γ_{ν} is a DVR, every finitely generated Γ_{ν}-module is isomorphic to a module of the form X_{ν} (i = 1, 2) [J, Chap. 3, Theorem 19]. However, note that the same number md of indecomposable summands occurs in both X_{1} and X_{2} in (8.2.1). We call m the block size of D and of X. For examples of diagrams, see (8.3.1). We now give the set of rules for forming and interpreting diagrams.

Let X be given, as in (8.2.1). Each block of summands (Γ_{ν}/𝔪_{ν}^{t})^{(m)} is represented, in our diagrams, by a vertical bar with a length label t written over it and the index ν written below it, as shown in (8.2.2). If D_{0} denotes this diagram, we define M(D_{0}) = X.

X i_{1}_{1} j_{1}_{2} … i_{d}_{1} j_{d}_{2} (8.2.2)

We define D to be any diagram that can be formed from D_{0} by a finite number of applications of the following two operations, each of which attaches an “edge” to a pair of vertical bars.

top-glue i_{1}(U) j_{2}  top-glue i_{1} (U)j_{2}

(8.2.3)

Top-glue. Choose a vertical bar with a 1 below it, and a vertical bar with a 2 below it, and suppose that neither of the tops of these bars has an edge attached; then connect the tops of these bars by an edge, as shown in (8.2.3). Then label the left or right end of this edge with an invertible m × m matrix U over k, as shown. We view U as being attached to the top of the corresponding vertical bar, as well as to the gluing edge.

In the case of the first top-glue diagram in (8.2.3), replace the (external) direct sum (Γ_{1}/𝔪_{1}^{i})^{(m)} ⊕(Γ_{2}/𝔪_{2}^{j})^{(m)} by the Λ-submodule given by the following pullback.

{(x, y) ∈ (Γ_{1}/𝔪_{1}^{i})^{(m)} ⊕ (Γ_{2}/𝔪_{2}^{j})^{(m)} | ρ(x) = ρ(y) · U} (8.2.4)

(One applies ρ to a tuple by applying ρ to each entry.) The reason for considering U to be attached to the i-labeled (rather than j-labeled) vertical bar is that, since U is invertible, the set of ordered pairs (ρ(x), ρ(y)) that arise from (8.2.4) is the set of all pairs

{(αU, α) ∈ k^{(m)} ⊕ k^{(m)} | α ∈ k^{(m)}} (8.2.5)

This will be consistent with our definition of bottom-gluing, below, and with the matrix pairs that will appear when we prove the main theorems of this section.

In the case of the second top-glue diagram, replace the condition ρ(x) = ρ(y) · U in (8.2.4) by ρ(x) · U = ρ(y). The effect of this is to move U to the “j” side of the equal sign in (8.2.5).

If U = I we usually do not explicitly display it. If we do display it, we get the same module whether we attach it to the i-side or the j-side of the top-gluing edge.

To explain our view of what has happened in this operation, first note that the pullback in (8.2.4) contains (𝔪_{1}/𝔪_{1}^{i})^{(m)} ⊕ (𝔪_{2}/𝔪_{2}^{j})^{(m)}; that is, it contains all but the topmost part of (Γ_{1}/𝔪_{1}^{i})^{(m)} ⊕ (Γ_{2}/𝔪_{2}^{j})^{(m)}. We think of the residue module

\frac{(Γ_{1}/𝔪_{1}^{i})^{(m)} ⊕ (Γ_{2}/𝔪_{2}^{j})^{(m)}}{(𝔪_{1}/𝔪_{1}^{i})^{(m)} ⊕ (𝔪_{2}/𝔪_{2}^{j})^{(m)}} = k^{(m)} ⊕ k^{(m)} (8.2.6)

(canonical isomorphism via ρ) as the “top” of (Γ_{1}/𝔪_{1}^{i})^{(m)} ⊕ (Γ_{2}/𝔪_{2}^{j})^{(m)}.

If U = I_{m}, then forming the pullback keeps only half of k^{(m)} ⊕ k^{(m)} in (8.2.6), namely the set of ordered pairs of the form (α, α). A general invertible matrix U “twists” the half of k^{(m)} ⊕ k^{(m)} that we are keeping.

Let S(D) be the Λ-submodule of X that results from whatever top-gluing operations that have been done. (The various matrix labels U do not need to be the same.) Our discussion of which parts of the original summands of X remain intact by top-gluing shows:

(8.2.7) S(D) contains the distinguished copy of k^{(m)} in every summand (Γ_{ν}/𝔪_{ν}^{t})^{(m)} of X for which t ≠ ∞, ≠ 1.

Our remaining operations on diagrams are shown symbolically below. As before we do not usually display U if U = I; and we regard U as being attached to the appropriate vertical bar as well as to the bottom-gluing edge.

bottom-glue j_{2}(U) i_{1}  bottom-glue j_{2} (U)i_{1}

(8.2.8)

Bottom-glue. Choose a vertical bar with 1 below it and a vertical bar with 2 below it, both with length labels ≠ 1, ≠ ∞, and without any edge attached to the bottom of either of them. Connect the bottoms of these bars with an edge, in the form of an elongated equal sign (as shown), and label the left or right end of this edge with an invertible m × m matrix U over k, as shown in (8.2.8). Since the length labels ≠ ∞, (Γ_{2}/𝔪_{2}^{j})^{(m)} ⊕ (Γ_{1}/𝔪_{1}^{i})^{(m)} contains our distinguished copy of k^{(m)} ⊕ k^{(m)}. In the case of the first diagram in (8.2.8), form the Λ-submodule (8.2.5) of k^{(m)} ⊕ k^{(m)}, and call it the bottom-gluing module associated with the diagram. In the case of the second diagram in (8.2.8), move U to the “i” side of the comma in (8.2.5), and again call the resulting set of ordered pairs the bottom-gluing module associated with the diagram .

Let D be the diagram that results from any such bottom- and top-gluing operations, and let K(D) be the (necessarily direct) sum of the bottom-gluing modules thus formed. Then let

M(D) = S(D)/K(D) (8.2.9)

For (8.2.9) to make sense we need to prove that K(D) ⊆ S(D), but this follows from (8.2.7).

In order to see what passing modulo K(D) does, first consider the case that every U equals I_{m}. Then every bottom-gluing amalgamates one distinguished copy of k^{(m)} with another. When U ≠ I_{m}, basically the same thing happens (since U is invertible), but it is twisted by U.

The Λ-modules M(D) thus formed are not always indecomposable. Connectivity of diagram D is clearly a necessary condition for indecomposability, but is not sufficient.

We now define two connected diagrams (and their associated modules, which turn out to be indecomposable) that we call standard diagrams. These definitions make use of the pair of label sequences in decomposition (8.2.1):

I = {i_{1}, i_{2}, …, i_{d}} J = {j_{1}, j_{2}, …, j_{d}} (8.2.10)

Thus every i_{ν} and j_{ν} is a positive integer or ∞.

Definitions 8.3 (Standard diagrams). Our first standard diagram — a “deleted cycle” diagram — occurs only with block size 1. For deleted cycle diagrams, the matrix labels U are all one-by-one identity matrices, and are therefore not displayed. The second diagram — a “block cycle” diagram — can occur with arbitrary block size. Each block cycle diagram explicitly displays one invertible m×m matrix label U over k — the blocking matrix of the diagram. According to our convention, the remaining matrix labels of a block cycle diagram are identity matrices. The placement of the blocking matrix in the diagram is somewhat arbitrary. (See Proposition 8.5 about moving U.)

\begin{matrix}D_{DCy}: i_{1}_{1} j_{1}_{2} i_{2}_{1} j_{2}_{2} … i_{d}_{1} j_{d}_{2} \\ D_{BCy}: i_{1}_{1}(U) j_{1}_{2} i_{2}_{1} j_{2}_{2} … i_{d}_{1} j_{d}_{2} \end{matrix} (8.3.1)

We now complete these definitions.

Deleted cycle diagrams, D_{DCy}. Here the following conditions must be satisfied.

(8.3.2) (i) The block size is m = 1.

(ii) Only length labels i_{1} and j_{d} can equal ∞ or 1.

Block cycle diagrams, D_{BCy}. We require:

(8.3.3) (i) The blocking matrix U is indecomposable under similarity.

(ii) All length labels are ≠ ∞ and ≠ 1.

(iii) The sequence of pairs (i_{1}, j_{1}), …, (i_{d}, j_{d}) is repetition-free.

(Recall, from (6.10.2), that a finite sequence W is repetition-free if there is no strictly shorter sequence W′ such that W consists of repetitions of W′.) When condition (ii) is not satisfied, a larger block size can be used and the module M(D) might be decomposable.

Let I, J be as in (8.2.10); and let I′, J′ be another pair of sequences, each again of length d. We say that I, J and I′, J′ are equal modulo simultaneous cyclic permutations if there is an integer c such that

I′ = {i_{c}, i_{c+1}, …, i_{d}, i_{1}, …, i_{c−1}}  and  J′ = {j_{c}, j_{c+1}, …, j_{d}, j_{1}, …, j_{c−1}}

Theorem 8.4. Every indecomposable finitely generated Λ-module (Λ strictly split) is isomorphic to M(D) where D is a deleted cycle or block cycle diagram, and all such modules are indecomposable.

Let D, D′ be diagrams of these types. (If D has a blocking matrix U, assume that U is located where shown in standard diagram (8.3.1).)

Then M(D) ≅ M(D′) if and only if one of the following holds.

(i) Both D and D′ are deleted cycle diagrams, and D = D′, (that is, they have the same label sequences).

(ii) Both D and D′ are block cycle diagrams, they have the same pairs of label seqeunces modulo simultaneous cyclic permutations, and their blocking matrices are similar.

The placement of the blocking matrix in block cycle diagrams is somewhat arbitrary. The precise rules for moving the blocking matrix are the subject of our next result.

Proposition 8.5 (Moving U). Let D be any block cycle diagram, except that the blocking matrix U is located at an arbitrary position in the diagram. Then each of the following operations on D leaves the isomorphism class of M(D) unchanged.

(i) Move U from one end (left or right) of any gluing edge to the other end of that edge, replacing it with U^{−1}.

(ii) Move U from one end (top or bottom) of any vertical bar to the other end, replacing it with U^{−1}.

Our survey of module structure is completed by the following result, already proved in Lemma 1.3.

Theorem 8.6. The Krull-Schmidt Theorem holds for finitely generated Λ-modules.

Remark 8.7 (Efficient vs. standard representations). The discussion of this given in Remark 7.10 applies here, with only one very minor change: The indecomposable Γ-modules are now Γ_{ν}/𝔪_{ν}^{t}, rather than Γ/𝔪^{t} as displayed in (7.10.1).

The way that homomorphisms of Λ-modules arise from homomorphisms of Γmodules is given in our final result.

Theorem 8.8. Let D′ and D be disjoint unions of block cycle and deleted cycle diagrams, and let f: M(D′) → M(D) be a Λ-homomorphism. Then there is a Γ-homomorphism f^{∗∗}: X(D′) → X(D) whose restriction f^{∗} to S(D′) is a Λhomomorphism: S(D′) → S(D) that takes K(D′) → K(D) and induces f. Moreover, if f is one-to-one or onto, then so is any such f^{∗}.

8.9. Proofs of theorems. The proofs in this section are minor modifications of the corresponding similar — but not identical — results in the unsplit case. The purpose of the truncated proofs that follow is to give a directory to the needed earlier results in this paper.

Let M be any finitely generated Λ-module. Then M ≅ M(A_{1}, A_{2}, B_{1}, B_{2}) where (A_{1}, A_{2}, B_{1}, B_{2}) is a matrix 4-tuple (over k) associated with some matrix setup X [Notation 4.3 – Theorem 4.6]. Moreover, two such modules are isomorphic if and only if their corresponding matrix 4-tuples can be obtained from each other by means of k-k-similarity operations and display operations [Theorem 4.10].

Now suppose that M is indecomposable. Then, after suitable k-k sweepingsimilarity operations and display operations, (A_{1}, A_{2}, B_{1}, B_{2}) becomes either a deleted cycle 4-tuple or a block cycle 4-tuple [Theorem 5.4], and each 4-tuple of either of these types yields an indecomposable M [Theorem 5.3].

Proof of Theorem 8.4. The first step is to show that each deleted cycle or block cycle 4-tuple yields, respectively, a module M(D_{DCy}) or M(D_{BCy}), and this is an easy consequence of the definitions of these two types of 4-tuples [Definitions 5.2] and diagrams [Definitions 8.3]. Thus it now suffices to prove the uniqueness assertions of Theorem 8.4, and these follow from Theorem 5.3.

Proof of Proposition 8.5. This proof refers to the block cycle 4-tuple displayed in diagram (5.2.5), where the blocking matrix is called L instead of U. The matrix operations we use are among the k-k sweeping-similarity operations listed in Theorem 4.7.

We can simultaneously left multiply the first m rows of A_{1} and B_{1} by L^{−1}, thus replacing the L in the upper left corner of A_{1} by I and replacing the I in the upper left corner of A_{2} by L^{−1}. Thus we have moved L from the left side of a top-gluing edge in D to the right side of that edge, replacing L by L^{−1}, as desired.

Once L^{−1} is in this new position, we simultaneously right multiply the j_{1}-labeled columns of A_{2} and B_{2} by L, thus replacing L^{−1} by I and replacing the I below it (in B_{2}) by L. Thus we have moved L^{−1} from the top of the j_{1}-labeled bar in D to the bottom of that bar, replacing it by L, as desired. Note that L is now attached to the bottom-gluing edge attached to the j_{1}-labeled vertical bar.

Analogous simultaneous row operations in B_{1} and B_{2} now move L from one end of its attached bottom-gluing edge to the other, replacing it by L^{−1}. Continuing in the fashion around the entire cycle completes the proof of the proposition.

Proof of Theorem 8.8. We have M = S/K where S = S(D) and K = K(D), as in (8.2.9). If we can show that the natural homomorphism S → → S/K is a separated cover of S/K ≅ M, then we can just repeat the proof, in subsection 7.17, of the corresponding theorem in the unsplit case. For this, see Theorem 4.6.

9. Klein Rings and Homomorphic Images of Dedekind-like Rings

In this section we complete our commutative noetherian tame-wild theorem by describing all indecomposable finitely generated modules over Klein rings (defined below). We also show how to use our description of indecomposable modules over Dedekind-like rings [§7, §8] to describe modules over homomorphic images of these rings. The point here is that our previous description involves both Λ and its normalization Γ, and we need to describe how to deal with Γ when passing to homomorphic images of Λ. We use the following well-known result. (For the finitely generated case, which is all we need, see for example [CR1, Theorem 6.30].)

Lemma 9.1. Let A be a quasi-Frobenius (i.e. artinian self-injective) ring with left socle H. Then every left A-module is the direct sum of a projective A-module and and an A/H-module.

For any module M over a ring Ω, we let μ_{Ω}(M) denote the minimal number of generators required by M.

Notation 9.2 (Klein rings). We call the commutative artinian local ring (Ω, 𝔫, k) a Klein ring if μ_{Ω}(𝔫) = 2, μ_{Ω}(𝔫^{2}) = 1, 𝔫^{3} = 0, and every element of 𝔫 has square 0. (See [KL2, Introduction and subsections 5.3–5.6] for more about these rings.)

The next result reduces the description of modules over Klein rings to modules over homomorphic images of Dedekind-like rings.

Theorem 9.3. Suppose that (Ω, 𝔫, k) is a Klein ring; then the following hold. (i) Ω is a quasi-Frobenius ring with simple socle 𝔫^{2}, and k has characteristic 2. (ii) Ω/𝔫^{2} is a homomorphic image of a strictly split commutative Dedekind-like

ring (Λ, 𝔪, k) which is 𝔪-adically complete; in fact, Ω/𝔫^{2} ≅ Λ/𝔪^{2}. (iii) Every Ω-module is the direct sum of a free module and an Ω/𝔫^{2} ≅ Λ/𝔪^{2}-

module.

Proof. (i) Since 𝔫^{3} = 0 we have 𝔫^{2} ⊆ soc Ω. We can write 𝔫 = (x, y). From the definition, we conclude that x^{2} = y^{2} = 0, but xy ≠ 0 since μ_{Ω}(𝔫^{2}) = 1, so in fact 𝔫^{2} = (xy). Thus 𝔫^{2} is a simple submdule of soc Ω. To see that 𝔫^{2} is the entire socle of Ω suppose that (ax + by)𝔫 = 0. Then 𝔫 = (x, y) shows that axy = 0 = bxy, and hence both a and b are nonunits, that is, elements of 𝔫; and hence ax + by ∈ 𝔫^{2} as desired.

Also from the definition of Klein ring, it follows that 0 = (x + y)^{2} = x^{2} + 2xy + y^{2} = 2xy, from which we conclude that 2 ∈ 𝔫, and hence the residue field k has characteristic 2. Since Ω is artinian, local, and commutative with simple socle, Ω is quasi-Frobenius [F, Theorem 3.1].

(ii) Ω/𝔫^{2} is a homomorphic image of a Dedekind-like ring Λ of the claimed form, by [KL2, Proposition 3.4]. The fact that Ω/𝔫^{2} ≅ Λ/𝔪^{2} follows from the fact that the maximal ideal of Λ is generated by two elements that annihilate each other, but no power of either of them is zero.

(iii) This follows from the earlier statements, Lemma 9.1, and the fact that projective modules over local rings are free.

The rest of this section deals with modules over homomorphic images of (residuecommutative) Dedekind-like rings.

Lemma 9.4. Let I ≠ 0 be a left ideal of a Dedekind-like ring (Λ, 𝔪, k) with normalization Γ. Then exactly one of the following holds.

(i) μ_{Λ}(I) = 1, I contains regular elements of Λ, and _{Λ}(ΓI)/I ≅ k. (ii) μ_{Λ}(I) = 2, I contains regular elements, and ΓI = I. (iii) μ_{Λ}(I) = 1, I contains no regular elements of Λ, and ΓI = I. The module _{Λ}(Λ/I) has finite length in situations (i) and (ii), but not (iii). In situation (i), (ΓI)/I = soc_{Λ}(Λ/I), and if I is a 2-sided ideal of Λ, then (ΓI)/I also equals soc(Λ/I)_{Λ}.

Proof. To begin, we recall that the definitions of unsplit and split Dedekind-like rings [Notation 1.1] are left-right symmetric. In particular, Λ is both left and right noetherian; and once we prove statements (i)–(iii) for left ideals I, we will also know that their analogues for right ideals hold. We claim that the following additional symmetry holds, for every left ideal I of Λ.

ΓI = IΓ  and  𝔪I = I𝔪 (9.4.1)

Recall that Γ is either a (residue-commutative) DVR or the direct sum of two such rings. If Γ is a DVR, the assertion ΓI = IΓ follows from the fact that all proper 1-sided ideals of Γ have the form Γπ^{e} = π^{e}Γ, where π is our distinguished Γ-generator of 𝔪. If Λ is the direct sum of two DVRs, we proceed similarly, but work one coordinate at a time. The assertion 𝔪I = I𝔪 follows similarly, after writing 𝔪 = Γπ = πΓ in the DVR case, and working analogously, one coordinate at a time, if Γ is the direct sum of two DVRs.

Recall that Γ/𝔪 is a 2-dimensional k-vector space in both the unsplit and strictly split cases [Notation 1.1]. It follows from Nakayama’s lemma that μ_{Λ}(_{Λ}Γ) = 2. Again by Nakayama’s lemma, we have

0 ≠ I/(𝔪I) ⊆ (ΓI)/(𝔪I) (9.4.2)

Note that the Λ-modules on both sides of the inclusion in (9.4.2) are annihilated by 𝔪 and are therefore k-vector spaces.

Consider the case that I contains regular elements of Λ. These are also regular elements of the principal ideal ring Γ, and therefore (ΓI)/(𝔪I) ≅ Γ/𝔪 as Λ-modules, hence has dimension 2 as a k-vector space. Suppose ΓI ≠ I. Then the inclusion in (9.4.2) is strict. Since the right-hand side has k-dimension 2, I/(𝔪I) therefore has k-dimension 1, and therefore, by Nakayama’s lemma, μ_{Λ}(I) = 1. Moreover, these dimensions show that (ΓI)/I has k-dimension 1 as claimed. Thus all of the assertions in situation (i) hold in the case. On the other hand, suppose that ΓI = I. Then the inclusion in (9.4.2) becomes equality, showing that I/(𝔪I) has k-dimension 2, and hence μ_{Λ}(I) = 2. Thus situation (ii) holds.

This leaves the case that I consists of zero divisors, and hence the Dedekind-like ring Λ is strictly split. It follows that I is contained in one of the two coordinate rings of Γ. In this situation situation (iii) is easily seen to hold, and Λ/I does not have finite length.

Now we claim that, in situation (i),

soc_{Λ}(Λ/I) = (ΓI)/I (9.4.3)

We already have the inclusion (⊇) by (i). Let soc(Λ/I) = Y/I. Since 𝔪 annihilates the simple Λ-module, we have 𝔪Y ⊆ I. Note that the left-hand side is a Γ-module while the right-hand side is not [since ΓI ≠ I in situation (i)]. Therefore strict inequality holds. Moreover 𝔪I is the unique maximal Λ-submodule of I, since I is principal and generated by a regular element. It follows from these two observations that 𝔪Y ⊆ 𝔪I. Substituting 𝔪 = πΓ, and canceling the regular element π then shows that Y ⊆ ΓI, completing the proof of the claim.

To complete the assertions about the socle of Λ/I, suppose that we are in situation (i) and I is a 2-sided ideal. By (i) we have ΓI ≠ I, and therefore (9.4.1) shows that IΓ ≠ I. Since I contains regular elements, the right-hand version of statement (i) or (ii) must hold; and since IΓ ≠ I, we see that the right-hand version of statement (i) holds — with IΓ replacing ΓI. Therefore the right-hand analog of (9.4.3) shows that the socle of the right Λ-module Λ/I equals IΓ/I; and this equals ΓI/I by (9.4.1), as desired.

If Λ is commutative, the finite length assertion about situations (i) and (ii) follows from the facts that Λ has Krull dimension 1 and I contains regular elements. In our slightly noncommutative situation we first note that, in each situation (i) and (ii), it suffices to show that _{Λ}(Γ/I) has finite length. Since, in both situations (i) and (ii), _{Λ}(ΓI/I) has finite length, it suffices to show that _{Λ}(Γ/ΓI) has finite length. Since the simple Γ-module(s) has length ≤ 2 as a Λ-module, it suffices to show that Γ(Γ/ΓI) has finite length. This, in turn, follows from the fact that Γ is the direct sum of at most 2 DVRs.

Definition 9.5. A (residue-commutative) AVR — artinian valuation ring — is either a division ring or a local ring (V, 𝔪, k) containing an element π ≠ 0 such that each proper one-sided ideal has the form V π^{e} = π^{e}V for some unique positive integer e, and π^{n} = 0 for some positive integer n. In particular, all 1-sided ideals are 2-sided and V is artinian.

Let (Λ, 𝔪, k) be an unsplit or strictly split Dedekind-like ring with normalization Γ, and let I be a proper ideal of Γ such that I ⊂ 𝔪. Then Γ/I is an AVR or the direct sum of two AVRs or the direct sum of one DVR and one AVR. In any case, Λ/I is the pullback of the following commutative square analogous to (1.1.1).

Λ/I ⊆ Γ/I
↓ ↓ρ  ^{↓ ↓}ρ (ker = 𝔪/I = rad Γ/I)
k ⊆ Γ̄

(9.5.1)

We discuss the strictly split case in more detail. Here Γ = Γ_{1} ⊕ Γ_{2} where each Γ_{ν} is a DVR. We also have I = I_{1} ⊕ I_{2} where each I_{ν} ⊂ 𝔪_{ν}. If both ideals I_{ν} are nonzero, then each ring Γ_{ν}/I_{ν} is an AVR. Otherwise one of the ideals I_{ν}, say I_{1}, equals zero. In this case Γ_{1}/I_{1} = Γ_{1} is a DVR and Γ_{2}/I_{2} is an AVR.

One special case needs explicit mention: (say) I_{2} = 𝔪_{2}, the maximal ideal of Γ_{2}. In this case Λ/I ≅ Γ_{1}/I_{1}. Then Λ/I is an AVR if I_{1} ≠ 0, and a DVR otherwise.

The next theorem reduces the description of modules over homomorphic images of Dedekind-like rings to the situation in diagram (9.5.1). We ignore the extreme case I = 𝔪, since then Λ/I = k.

Theorem 9.6. Let 0 ≠ I ⊂ 𝔪 be an ideal of the Dedekind-like ring (Λ, 𝔪, k). Then one of the following holds.

(i) I ≠ ΓI and Λ/I is a quasi-Frobenius ring with simple left and right socle (ΓI)/I; or

(ii) I = ΓI and Λ/I is the pullback of commutative square (9.5.1). In situation (i), every Λ-module is the direct sum of a free module and a Λ/(ΓI)- module [and Λ/ΓI is a pullback as in (9.5.1)].

Proof. First consider the case that I ≠ ΓI. Then, by Lemma 9.4, the ring Λ/I is an artinian ring with simple left and right socle (ΓI)/I and (obviously) has no nontrivial idempotents. Therefore Λ/I is quasi-Frobenius [F, Theorem 3.1].

Next suppose that I = ΓI. Since Γ is the direct sum of one or two DVRs, ΓI is a 2-sided ideal of Γ, and we are in the situation of diagram (9.5.1) — with ΓI playing the role of I in that diagram.

The supplementary statement follows from Lemma 9.1.

It now suffices to describe all indecomposable finitely generated Λ/I-modules in the situation of diagram (9.5.1), where 0 ≠ I ⊂ 𝔪 and I is an ideal of Γ.

Theorem 9.7. Let (Λ, 𝔪, k) be a Dedekind-like ring with normalization Γ, and let I be an ideal of Γ such that 0 ≠ I ⊂ 𝔪. Let M = M(D) be any finitely generated indecomposable Λ-module constructed from a diagram D and a Γ-module X, as in sections 7 and 8. Then IM = 0 (i.e. M is a Λ/I-module) if and only if IX = 0 (i.e. X is a Γ/I-module). More precisely, IM = 0 if and only if the one of the following holds:

(i) If Λ is unsplit, then all length-labels in D are less than or equal to the length of the AVR Γ/I (as a Γ/I-module).

(ii) If Λ is strictly split (so that Γ = Γ_{1} ⊕Γ_{2}, and I = I_{1} ⊕I_{2}, with each I_{ν} ⊂ Γ_{ν}), then all Γ_{ν}-length labels (ν = 1, 2) in D are less than or equal to the length of the DVR or AVR Γ_{ν}/I_{ν}.

Proof. We have M = S(D)/K(D) by (7.2.11) or (8.2.9) respectively, according to whether Λ is unsplit or strictly split. Moreover, as shown in 7.17 and Theorem 4.6, the natural surjection ϕ: S = S(D) → → M is a separated cover of M in both cases; and X = ΓS [in the unsplit case, by (7.13.1) and the fact that A is invertible; in the strictly split case, by (4.4.3) and the fact that A_{1} and A_{2} are invertible].

Now consider any separated cover ϕ: S → → M of a Λ-module, and let X = ΓS be any Γ-module generated by S. We want to prove that IM = 0 if and only if IX = 0. Since I is a 2-sided ideal of Γ, we have IX = 0 if and only if IS = 0. If IS = 0 then the fact that ϕ is a surjection shows that IM = 0.

Suppose, conversely, that IM = 0. Then IS ⊆ ker ϕ. Since I is an ideal of Γ and S is a Λ-submodule of some Γ-module (!), it follows that IS is a Γ-submodule of ker ϕ. Since ϕ is a separated cover, it follows that IS = 0 [Lemma 2.3] as desired.

Remark 9.8. It is possible to know that a commutative complete local ring (Υ, 𝔭, k) is a homomorphic image of a complete Dedekind-like ring (Λ, 𝔪, k) without knowing Λ and its normalization Γ in advance. For example, Υ might be equal Ω/𝔫^{2} where Ω is a Klein ring with radical 𝔫 [Theorem 9.3].

Assume that Υ is known to be a homomorphic image of a commutative, complete Dedekind-like (local) ring. We may assume that the socle of Υ has k-dimension 2 (for if it equals 1, we apply the reduction in the supplementary statement of Theorem 9.6). Thus we are now in the situation shown in (9.5.1): I is an ideal of Γ contained in 𝔪. In this situation, one must know something about Γ in order to obtain the structure of indecomposable Λ-modules from Theorem 9.7. A careful reading of the results in sections 7 and 8 to which Theorem 9.7 refers shows that it suffices to know Γ/I, rather than Γ itself.

We remark that the proofs of [KL2, Propositions 3.4 and 3.6] give constructions of Λ and Γ in the strictly split and unsplit cases, respectively, and this can presumably be used to find the structure of Γ/I.

10. Examples

Both geometric and arithmetic examples of split Dedekind-like rings abound. In some earlier papers, the term “Dedekind-like” applied to commutative noetherian rings Λ, not necessarily local, whose completions at maximal ideals are — in the language of the present paper — either strictly split Dedekind-like or DVRs, and whose is normalization is a finitely generated (hence 2-generated) Λ-module.

(10.0.1) Examples of such Dedekind-like rings are:

(i) The integral group ring ℤG of any finite cyclic group of square-free order. [L2]

(ii) (In the language of algebraic geometry:) The coordinate ring, over any algebraically closed field, of any affine curve whose singularities are only simple nodes (index 2).

(iii) Every subring Λ of square-free index in the direct product Γ = ℤ×ℤ×⋯×ℤ of copies of the integers; that is, such that the additive group Γ/Λ has squarefree order. [Theorem 10.3 and Example 10.4 below.]

(iv) ℤ[\sqrt{17} ]. [L3, p. 2]

Now return to our present terminology, where “Dedekind-like” applies only to (residue-commutative) local rings (Λ, 𝔪, k). Since the residue field k of an unsplit Dedekind-like ring cannot be algebraically closed, unsplit Dedekind-like rings do not occur in classical algebraic geometry, nor do homomorphic images of unsplit Dedekind-like rings occur among classical Artin algebras.

In this section, we show that many unsplit and split Dedekind-like rings occur naturally among completions of arithmetic orders, and we prove (10.0.1)(iii).

Example 10.1 (Quadratic orders). Consider the quadratic order Λ = ℤ[\sqrt{n}] for some square-free integer n. We show that the p-adic completion Λ̂_{p} of Λ is a DVR or Dedekind-like for every rational prime p.

It is well-known ([We, Theorem 6-1-1], for example) that Λ is a Dedekind domain if and only if n is congruent to 2 or 3 modulo 4. Thus, we can suppose that n is congruent to 1 modulo 4. In this situation it is well-known that Γ = ℤ[(1 + \sqrt{n})/2 ] is the normalization of Λ. (See, for example, [We, Theorem 6-1-1].) If p is an odd prime, the p-adic completions of Λ and Γ are obviously the same; so we only need to consider on the 2-adic completion Λ̂_{2}.

Let C = 2Γ; clearly C ⊆ Λ, so that C is contained in the conductor of Γ into Λ. By [We, Theorem 6-2-1], the ideal C is maximal in Γ if n is congruent to 5 modulo 8, while C is the product of two distinct maximal ideals in Γ if n is congruent to 1 modulo 8. We consider these two cases individually.

Consider the case n ≡ 5 (mod 8). Here, the residue ring Γ/C is a field with exactly four elements, and Λ/C is a proper subring of Γ/C, so that k = Λ/C is a field with exactly two elements. Therefore, C is also a maximal ideal of Λ and so is the conductor from Γ into Λ. Moreover, the rings Λ and Γ fit into a conductor square as in (1.1.1) (except for the kernels of the vertical maps being contained in the radical), in which F = Γ/C is a (separable) quadratic field extension of k. Since tensoring a pullback diagram with a flat module again yields a pullback diagram — see, e.g. the method of proof of [L3, 1.5(1)] — tensoring this diagram with ℤ̂_{2} shows that Λ̂_{2} is unsplit Dedekind-like. In fact, the localization Λ_{2} is also unsplit Dedekind-like.

Consider the case n ≡ 1 (mod 8). Here the residue ring Γ/C is the product of two fields each with exactly two elements, and Λ/C is a proper subring of Γ/C. Again it follows that k = Λ/C is a field with exactly two elements, so that C is a maximal ideal of Λ and is the conductor from Γ into Λ. Moreover, the rings Λ and Γ fit into a conductor square as in (1.1.1) (except for the vertical maps being contained in the radical), in which the inclusion in the bottom row can be taken to be the diagonal inclusion of k into Γ/C ≅ k ×k. As before, tensoring with ℤ̂_{2} shows that Λ̂_{2} is strictly split Dedekind-like. Here the localization Λ_{2} is never strictly split Dedekind-like, because it is an integral domain. (However it is split Dedekind-like, as defined in [KL2, Definitions 2.2].)

Example 10.2 (Integer quaternions). Let Λ = ℤ[i, j, k], the ring of integer quaternions. Thus Λ is a free ℤ-module with basis 1, i, j, and k; and multiplication is given by i^{2} = j^{2} = k^{2} = −1 and ij = k. We prove that the 2-adic completion Λ̂ is unsplit Dedekind-like. (The completions at odd primes are all maximal orders.)

A normalization of Λ is the maximal ℤ-order

Γ = ℤ + ℤi + ℤj + ℤk + ℤγ [γ = (1 + i + j + k)/2] (10.2.1)

and therefore a normalization of Λ̂ is the ℤ̂_{2}-order Γ̂ obtained by replacing ℤ by its 2-adic completion ℤ̂_{2} in (10.2.1).

To prove that Λ̂ is unsplit Dedekind-like, it suffices to prove that Λ̂ is the pullback of the second commutative square in (10.2.2), in which ℤ̄_{2} and F_{4} are the fields of 2 and 4 elements respectively, ρ̂ is a surjective ring homomorphism such that ker ρ̂ = rad Γ̂, and Γ̂ is a (residue-commutative) DVR.

\begin{aligned}Λ & ⊂ & Γ \\ ^{↓ ↓}ρ & & ^{↓ ↓}ρ \\ Λ/C = ℤ̄_{2} & ⊂ & Γ/C = F_{4} \end{aligned} \begin{aligned}Λ̂ & ⊂ & Γ̂ \\ ^{↓ ↓}ρ̂ & & ^{↓ ↓}ρ̂ \\ ℤ̄_{2} & ⊂ & F_{4} \end{aligned} (10.2.2)

Proof. First, we define the first commutative diagram in (10.2.2) and show that Λ is its pulback. Let C = Γ(1 − i); we claim that

(10.2.3)  2, 1 ± i, 1 ± j, 1 ± k, 1 + i + j + k  are elements of C.

First, 2 = (1 + i)(1 − i) ∈ C. Then 1 + i = (1 − i) + 2i ∈ C as well. Moreover, γ(1 − i) = 1 + k ∈ C, and so 1 − k = (1 + k) − 2k ∈ C also. Similarly, γ(1 − k) = 1 + j ∈ C, and so 1 − j = (1 + j) − 2j ∈ C. Finally, 2γ = 1 + i + j + k ∈ C.

Next, we claim that C is a 2-sided ideal, for which it suffices to show that C is closed under right multiplication by i, j, k and γ. This follows easily from (10.2.3) and the (easily verified) equations (1 − i)i = 1 + i, (1 − i)j = j(1 + i), (1 − i)k = k(1 + i), and (1 − i)γ = 1 + j. Moreover, since C is a 2-sided ideal, a symmetric argument shows that

Γ(1 − i) = (1 − i)Γ (10.2.4)

Next, we claim that Γ/C = F_{4}. Since 2 ∈ C, we have ℤ̂_{2} ⊆ C. By (10.2.3), each of i, j, k becomes equal to 1 modulo C. Therefore Γ/C = ℤ̄_{2}[γ̄], where γ̄ is the natural image of γ in Γ/C. Let γ′ = (1 − i − j − k)/2, the quaternion conjugate of γ. Then γ − γ′ = i + j + k. Multiplying by γ yields γ^{2} − 1 = (i + j + k)γ. Reducing modulo C, and remembering that 1 + i + j + k ∈ C, therefore yields γ̄^{2} − γ̄ − 1 = 0, an irreducible equation over ℤ̄_{2}. This shows that Γ/C = ℤ̄_{2}[γ̄] = F_{4}, as claimed.

It follows that C is a maximal ideal of Γ. Now (10.2.4) implies that C^{2} = (2); and so, since C is prime, it follows that C is the unique maximal ideal of Γ containing 2. Moreover, since γ(1 − i) = 1 + k ∈ C, it follows that C ⊆ Λ, and hence C is an ideal of Λ as well.

In view of what we have shown, letting ρ be any ring homomorphism with kernel C allows us to construct the first commutative diagram in (10.2.2), and the fact that C ⊆ Λ shows that Λ is the pullback of this diagram. Tensoring this diagram with ℤ̂_{2} — a flat ℤ-module — yields the second commutative diagram in (10.2.2), and Λ̂ is its pullback. Moreover, the fact that C is the unique maximal ideal of Γ that contains 2 shows that ker ρ̂ = rad Γ̂.

To complete the proof we must show that Γ̂ is a DVR. Since Γ is a maximal ℤ-order in a division algebra, Γ̂ is a maximal ℤ̂_{2}-order in a semisimple algebra [Re, (11.5)]. Since ℤ̂_{2} is a complete discrete valuation ring, the maximal ℤ̂_{2}-order Γ̂ is a direct product of full matrix rings over DVRs [Re, (17.3), (13.2)]. Since ker ρ̂ = rad Γ̂ and the field F_{4} has no nontrivial idempotent elements, we conclude that Γ̂ has no nontrivial idempotent elements, and is therefore a DVR, completing the proof.

The reason for dealing only with the 2-adic completion above is that, in view of (10.2.1), Λ and Γ become equal when completed at any other prime number.

Theorem 10.3. Let Λ be a subring of square-free index (say, n) in Γ = ℤ^{s} (direct product of s copies of the integers). Then for every maximal ideal 𝔪 of Λ, the localization Λ_{𝔪} is a strictly split Dedekind-like ring or a DVR.

In fact, Λ_{𝔪} is strictly split Dedekind-like precisely for the finite number of 𝔪 that contain a prime factor of n.

Proof. We may assume that n > 1. Since |Γ/Λ| = n we have

Γn ⊂ Λ ⊂ Γ (10.3.1)

Choose a maximal ideal 𝔪 of Λ. We want to show that Λ_{𝔪} is strictly split Dedekindlike with normalization Γ_{𝔪}, or else is a DVR.

If 𝔪 ⊉ Γn, then some element of Γn becomes a unit in Λ_{𝔪}, and we have Λ_{𝔪} = Γ_{𝔪}, a local principal ideal domain, equivalently, a DVR. Thus we suppose, from now on, that 𝔪 ⊇ Γn. There are only finitely many such 𝔪 since Γ/Γn is a finite ring.

Since n ∈ 𝔪, the prime ideal 𝔪 contains some prime number p that divides n. Since localizing at 𝔪 can be done by first localizing at p, we may replace the three terms in (10.3.1) by their p-localizations. The effect of this is that, after a change of notation, we may assume that n = p, a prime number. Moreover the original aditive group Γ/Λ of square-free order n has now been replaced by its p-localization, hence is a group of order p. We now have (after our change of notation) that Λ is the pullback of the following diagram.

Λ  ⊂  Γ
↓ ↓  ^{↓ ↓}
Λ̄ = Λ/(Γp) ⊂ Γ̄ = Γ/(Γp)

(10.3.2)

Note that Λ̄ and Γ̄ are finite rings. Now localize the whole diagram at 𝔪. Then Λ_{𝔪} remains the pullback of the localized diagram. Thus, after a further change of notation, we have that Λ is a local ring. We now prove that Λ is strictly split Dedekind-like with normalization Γ.

Think of the original rings Λ and Γ as module-finite ℤ-algebras. Then our first localization — at p — converted Λ and Γ to module-finite ℤ_{p}-algebras. In particular, it converted ℤ^{s} (the original Γ) to ℤ_{p}^{s}. The effect of the further localization at 𝔪 on each coordinate ring ℤ_{p} is to either leave it unchanged or replace it with 0. Thus we now have Γ = ℤ_{p}^{t} for some t ≤ s. Therefore Γ̄ = ℤ̄_{p}^{t}, the direct product of t copies of the field of p elements.

Now consider the finite, therefore artinian ring Λ̄. Since Λ̄ ⊆ Γ̄, Λ̄ has radical zero and is therefore a direct product of fields. On the other hand Λ̄ is a local ring, because the local ring Λ maps onto it. Therefore Λ̄ is a field (of characteristic p). Since each of the coordinate rings ℤ̄_{p} of Γ̄ is a Λ̄-vector space, we see that Λ̄ = ℤ̄_{p}.

The abelian group Γ̄/Λ̄ ≅ Γ/Λ has order p. On the other hand its order is |Γ̄|/|Λ̄| = p^{t}/p = p^{t−1}. Therefore t = 2. We now have that Γ = ℤ_{p}^{2}, Γ̄ = ℤ̄_{p}^{2}, and Λ̄ = ℤ̄_{p}. Therefore the local ring (Λ, Γp, ℤ̄_{p}) is strictly split Dedekind-like, as desired.

Example 10.4. It is easy to see, by continuing the reasoning in the previous proof, that the rings Λ in Theorem 10.3 can be explicitly constructed as follows.

Choose any square-free positive integer n = p_{1}p_{2} … p_{u} > 1 and any positive integer s > 1. For each prime p_{ν} choose a pair of coordinate rings, say coordinates i_{ν} ≠ j_{ν} of ℤ^{s}. Let f_{ν}, g_{ν} be the natural map of coordinate rings i_{ν}, j_{ν} respectively, onto ℤ̄_{p_{ν}}. Then let

Λ = {(x_{1}, …, x_{s}) ∈ ℤ^{s} | (∀ν) f_{ν}(x_{i_{ν}}) = g_{ν}(x_{j_{ν}})} (10.4.1)

For some amusing module-theoretic parlor tricks over these rings — that is, many examples of non-uniqueness of direct-sum decompositions of finitely generated Λmodules — see [L4].

References

[Ba] C. S. Ballantine, “Products of complic cosquares and pseudo-involutory matrices”, Lin. and Multilin. Alg. 8 (1979), 73–78.

[Br] T. Brüstle, “On commutative tame algebras”, C. R. Acad. Sci. Paris t. 318, Série I (1994), 13–18.

[BS] M. C. R. Butler and M. Shahzamanian, “The construction of almost split sequences, III: Modules over two classes of tame local algebras”, Math. Ann. 247 (1980), 111–122.

[CR1] C. W. Curtis and I. Reiner, Methods of Representation Theory, with Applications to Finite Groups and Associative Algebras, Volume 1, Wiley-Interscience, New York, 1981.

[D1] Yu. A. Drozd, “Representations of commutative algebras” (Russian), Funkstional’nyi Analiz i Ego Prilozheniya 6 (1972), 41–43. English Translation in Functional Analysis and its Applications 6 (1972)

[D2] Yu. A. Drozd, “Finite modules over pure noetherian algebras”, Proc. Steklov Institute of Math 4 (1991), 97–108.

[F] K. Fuller, “On indecomposable injective modules over artinian rings”, Pac. J. Math. 29 (1969), 115–135.

[J] N. Jacobson, The theory of rings, American Math. Soc. (Mathematical surveys, II) (Providence, R. I., USA, 1943)

[K] L. Klingler, “Modules over the integral group ring of a nonabelian group of order pq”, Mem. Amer. Math. Soc. 59 (1986)

[KL1] L. Klingler and L. S. Levy, “Sweeping-similarity of matrices”, Lin. Alg. Appl. 75 (1986), 67–104.

[KL2] L. Klingler and L. S. Levy, “Representation type of commutative noetherian rings I: Wildness”, preprint.

[L1] L. S. Levy, “Mixed modules over ZG, G cyclic of prime order, and over related Dedekind pullbacks”, J. Alg. 71 (1981), 62–114.

[L2] L. S. Levy, “ZG_{n}-modules, G_{n} cyclic of square-free order n”, J. Alg. 93 (1985), 354–375.

[L3] L. S. Levy, “Modules over Dedekind-like rings”, J. Alg. 93 (1985), 1–116.

[L4] L. S.Levy, “Krull-Schmidt uniqueness fails dramatically over subrings of Z ⊕ ⋯ ⊕ Z”, Rocky Mountain J. Math. 13 (1983), 659–678.

[LO] L. S. Levy and C. J. Odenthal, “Krull-Schmidt theorems in dimension 1”, Trans. Amer. Math. Soc. 348 (1996) pp. 3391–3455.

[NR] L. A. Nazarova and A. V. Roiter, “Finitely generated modules over a dyad of two local rings and finite groups with an abelian normal divisor of Index p”, Zap. Nauch. Sem. Leningrad Odtel. Mat. Inst. Steklov (LOM1) Izv. Akad. Nauk SSSR Ser mat. 33 (1) (1969). English transl: USSR-Izv 3 (1) (1969)

[NRSB] L. A. Nazarova, A. V. Roiter, V. V. Sergeichuk, and V. M. Bondarenko, “Application of modules over a dyad for the classification of finite p-Groups that have an abelian subgroup of index p and of pairs of mutually annihilating operators” (Russian), Zap. Nauch. Sem. Leningrad Odtel. Mat. Inst. Steklov (LOM1) 28 (1972), 69–92. English transl: J. Soviet Math. 3 (1975), 636–653.

[Re] I. Reiner, Maximal Orders, Academic Press, New York, 1975.

[R1] K. M. Ringel, “The representation type of local algebras”, Lecture Notes in Mathematics, Springer, New York, 1975, 282–305.

[R2] K. M. Ringel, “Indecomposable representations of the dihedral 2-groups”, Math. Ann. 214 (1975), 19–34.

[We] E. Weiss, Algebraic Number Theory, Chelsea Publishing Company, New York, 1963.

Lee Klingler, Mathematics Department, Florida Atlantic University, Boca Raton, Florida 33431–0991, USA

E-mail address: klingler@fau.edu

L. S. Levy, Mathematics Department, University of Wisconsin, Madison, Wisconsin 53706–1388, USA

E-mail address: levy@math.wisc.edu