REPRESENTATION TYPE OF COMMUTATIVE NOETHERIAN RINGS I: WILDNESS

LEE KLINGLER AND LAWRENCE S. LEVY

Abstract. This is the first 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. This first paper identifies the wild rings, in the complete local case, and proves that they are wild. The second paper proves that all remaining complete local rings (with the possible exception noted above) are tame, by describing their finitely generated modules explicitly. The third paper deals with the non-complete, non-local, situation and the local-versus-global behavior of modules over tame rings.

The second and third papers of this series also establish tameness of a small number of noncommutative rings.

1. Introduction

The goal of this project is to describe all finitely generated modules — including the indecomposables and direct-sum behavior — over as wide as possible a class of commutative noetherian rings, thus extending Steinitz’s well-known 1911 theorem on modules over Dedekind domains [St]. Steinitz’s theorem has been particularly resistant to generalization. In fact, the only noetherian integral domains (other than Dedekind domains) for which such a module structure theorem was known prior to completion of this project seem to be the Dedekind-like domains studied in [L2]. However, for rings that are not domains, some other results exist [L1, L2, NR, NRSB].

In all cases for which we cannot obtain such a description (with a possible exception involving characteristic 2), the obstruction is wild representation type. Informally, wild representation type can be described as follows. Let k⟨X, Y ⟩ be the free (noncommutative) algebra on two symbols X, Y over a field k, and let fdmod-k⟨X, Y ⟩ denote the category of all finite dimensional left k⟨X, Y ⟩-modules. Any k-algebra W such that fdmod-W contains a suitable copy of the category fdmod-k⟨X, Y ⟩ is said to have “wild representation type”. The reason for this name is that, by a well-known simple but clever trick [B, Theorem 3], one can show that fdmod-k⟨X, Y ⟩ contains a copy of fdmod-A, for every finite dimensional k-algebra A. Thus, any classification of all isomorphism classes in fdmod-k⟨X, Y ⟩ would contain a classification of all isomorphism classes of finite dimensional modules over every finite dimensional k-algebra. This seeming impossibility is what lies behind the term “wild.”

Date: May 16, 1999.

1991 Mathematics Subject Classification. 13E05, 16G60.

Key words and phrases. Wild representation type.

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

One of the surprising facts about finite dimensional algebras is that most kalgebras are wild [Gabriel]. The well-known tame-wild theorem of Drozd and Crawley-Boevey [CB, Theorem B] states that, if k is algebraically closed, then every finite dimesional k-algebra is either tame or wild. As applied to finite dimensional algebras, the term “tame” can be made precise, but the spirit is that, at least in principle, the k-algebra A is tame if the isomorphism classes (and sometimes morphisms) in fdmod-A can be explicitly described. One of the aims of the current project is to establish an analogous tame-wild theorem for commutative noetherian rings. This situation provides an interesting link between noetherian ring theory and representation theory of finite dimensional algebras.

We show that all commutative noetherian rings of Krull dimension greater than one are wild (Theorem 5.13). Thus, the rings with which we are really dealing are either artinian or, like Dedekind domains, have Krull dimension one.

Remark 1.1 (Results of Drozd). If Λ is a finitely generated commutative algebra over an algebraically closed field and the Λ-modules involved have finite length, the tame-wild dichotomy we are studying was obtained by Drozd [D], and the solution improved by Ringel [R]. The actual module structure in the tame case had been given by earlier results of others. This collection of results was the source of our interest in the subject.

In Drozd’s and Ringel’s situation — Λ-modules of finite length — one can assume, without loss of generality, that Λ is a complete local ring. Since our rings do not have to be algebras over fields and do not have to be local rings, our results apply to rings of algebraic integers, the rings that originally interested Steinitz. Our results also appear to be new in the situation that Λ is an artinian ring that is not an algebra over a field. Moreover, our tameness results deal with arbitrary finitely generated modules, not just modules of finite length.

In a later paper [D2], Drozd states a tameness-wildness dichotomy over a class of rings that can be noncommutative and need not be algebras over a field. However, most of the statements and all of the proofs of theorems in that paper assume that the ring is an algebra over an algebraically closed field. Thus, in the commutative case, the results proved in that paper do not go beyond those in his earlier paper. Moreover, in the commutative case, his statement of which rings are tame does not take the simple, explicit form that is given our definition of split and unsplit Dedekind-like rings and Klein rings.

For readability, this project is divided into a series of three papers.

Paper I: Wildness. We focus on complete local (commutative noetherian) rings Λ, proving that a large class of them have wild type. Section 2 contains the precise definition that we use for wildness, together with the definitions of the types of ring that lie on the border between tameness and wildness.

Following the terminology of Ringel [R], we define a class of rings called “minimal wild” and a class called “maximal tame”. The minimal wild rings consist of two types of rings that we call artinian triads and Drozd rings. The maximal tame rings consist of three types, that we call split and unsplit Dedekind-like rings, and Klein rings.

Section 3 consists of the statement and proof of our Dichotomy Theorem 3.1. This theorem shows that every complete local ring either maps onto a minimal wild ring or is a homomorphic image of a maximal tame ring. The proof is a piece of pure commutative algebra, and makes no use of the actual meanings of “tame” or “wild”.

Section 4 contains the proof that every Drozd ring is finite-length wild, that is, its category of modules of finite length is indeed wild. Moreover, a theorem of Warfield [GLW, Lemma 3] states that every artinian triad is finite-length wild. It is clear from our definition of finite-length wildness that any ring that maps onto a finite-length wild ring is again finite-length wild. Thus, as a corollary to our Dichotomy Theorem 3.1 (and the results of section 4), we obtain (Corollary 4.9) that every complete local ring is finite-length wild or is a homomorphic image of a maximal tame ring. We postpone, to the second paper of this series, the proof that the maximal tame rings and their homomorphic images are indeed finitely generated tame (with a possible exception involving charcteristic 2). Once that is done, it is clear that our Dichotomy Theorem is indeed a result about the tame-wild dichotomy of module categories.

Section 5, “Examples and Miscellaneous Results,” completes the present paper.

Paper II: Tameness (Complete Local Case) [KL2]. We give a detailed description of all finitely generated modules over Klein rings and over complete split Dedekind-like rings, and — with a possible exception involving characteristic 2 — over unsplit Dedekind-like rings. This justifies the phrase “finitely generated tame.” We also show how to extend our description of finitely generated modules over Dedekind-like rings to the description of finitely generated modules over their homomorphic images. This completes our tame-wild dichotomy for complete local commutative noetherian rings Λ: we either describe all finitely generated Λ-modules or show that Λ is finite-length wild.

The nature of the “possible exception” is as follows. Let Λ be complete, unsplit Dedekind-like (see Definition 2.3 below), with maximal ideal 𝔪 and residue field k, and let Γ be the normalization of Λ. By definition of “unsplit Dedekind-like”, 𝔪 is also the unique maximal ideal of Γ, and Γ/𝔪 is a quadratic field extension of k. When the quadratic field extension Γ/𝔪 of k is inseparable, our theory breaks down, and we do not know whether Λ is tame or wild or neither. Note, however, that this exception cannot occur for rings of algebraic integers since finite extensions of finite fields are always separable. The exception also cannot occur for rings of geometric origin since algebraically closed fields never have finite dimensional extension fields.

Paper III: Global Tameness [KL3]. We drop the assumptions that our noetherian ring Λ be local and complete. If some completion of Λ at a maximal ideal is finite-length wild, it is clear that Λ has this same property. Thus we focus our attention on the situation that every completion of Λ at a maximal ideal is either a Klein ring or a homomorphic image of a complete split or unsplit (and non-exceptional!) Dedekind-like ring. We then describe the local-versus global structure of finitely generated Λ-modules.

Two interesting applications are: (i) every ring of algebraic integers of the form ℤ[\sqrt{n}], with n a square-free integer, is among the tame rings whose modules we describe. (ii) The noncommutative ring ℤ[i, j, k] of integer quaternions is “commutative enough” for our results apply to it, too.

Notation 1.2. Throughout this paper, Λ denotes a commutative noetherian ring. We say that (Λ, 𝔪, k) is a local ring if Λ is a local ring with maximal ideal 𝔪 and residue field k. When we say that Λ is complete, we mean that it is 𝔪-adically complete.

We consistently write functions on the left except when they represent matrix multiplication. It is typographically simpler to display a row of a matrix than a column. Therefore our matrices normally act via right multiplication, and we write the corresponding functions as right operators. This occurs in all three papers of this series. In the present paper it occurs in Section 4, and we include a reminder when it happens.

For Λ-module M, we let μ_{Λ}(M) denote the minimal number of generators required by M as a Λ-module.

2. Definitions

Definition 2.1 (Wildness). Let 𝔪 be a maximal ideal of Λ and k = Λ/𝔪 the corresponding residue field. We say that Λ is finite length 𝔪-wild if there is a full subcategory W of the category of Λ-modules of finite length and an additive functor Φ : W → fdmod-k⟨X, Y ⟩ such that Φ is a representation equivalence; that is, Φ is: dense (onto all isomorphism classes), faithful (Φ(M) ≅ Φ(N) ⟺ M ≅ N), and full (a surjection on homomorphism groups). Thus, encoded in W we find not only all of the isomorphism classes in fdmod-k⟨X, Y ⟩, but (after reducing the homomorphism groups in W modulo suitable kernels) all of the hom-structure of fdmod-k⟨X, Y ⟩ as well.

When Λ is local it suffices to say that Λ is finite-length wild; that is, we do not need to explicitly mention 𝔪. However, when Λ is not local it can be finite length 𝔪-wild for infinitely many 𝔪, a situation we call “superwild” (Example 5.2).

The subcategory W that we use in proving Λ to be 𝔪-wild will always be 𝔪primary; that is, every element in W is annihilated by some power of 𝔪 (depending on the module). Modules of this type remain unchanged by the process of 𝔪localization and 𝔪-adic completion. This has the advantage that, in order to prove that Λ is finite-length 𝔪-wild, we can assume that (Λ, 𝔪, k) is a complete local ring.

Definition 2.2 (Minimal wild rings). We call the local ring (Λ, 𝔪, k) an artinian triad if μ_{Λ}(𝔪) = 3 and 𝔪^{2} = 0. (Since 𝔪^{2} = 0, every artinian triad is indeed an artinian ring.)

In the case of finite dimensional algebras over the field k = Λ/𝔪, this ring is Λ = k[X, Y, Z]/(X, Y, Z)^{2}, where X, Y, and Z are indeterminates.

We call the local ring (Λ, 𝔪, k) a Drozd ring if μ_{Λ}(𝔪) = μ_{Λ}(𝔪^{2}) = 2, 𝔪^{3} = 0, and there is an element x ∈ 𝔪 − 𝔪^{2} such that x^{2} = 0. (Since 𝔪^{3} = 0, Drozd rings are also artinian.)

We note: (2.2.1) The composition length of every Drozd ring is 5. To see this note that the dimensions of Λ/𝔪, 𝔪/𝔪^{2}, and 𝔪^{2} as k-vector spaces are 1, 2, 2 respectively.

In the case of finite dimensional algebras over the field k = Λ/𝔪, this ring is the 5-dimensional k-algebra with k-basis 1, x, y, xy, y^{2} and all other monomials equal to zero. An example of a Drozd ring that is not a algebra over a field is the ring A_{p} in Example 5.2.

The above, abstract definition definition of Drozd rings is tailored to the needs of our dichotomy results, but leaves one wondering what these rings really look like. It turns out that every Drozd ring is a subring of an artinian principal ideal ring, and we use this to give a general, explicit construction of Drozd rings in Theorem 5.11.

Our minimal wild rings have the following simple minimality property. (2.2.2) Let ϕ: Λ → → Λ′ be a surjective ring homomorphism, where Λ is an artinian

triad or a Drozd ring, and the same is true of Λ′. Then ϕ is an isomorphism. If Λ and Λ′ are both artinian triads or both Drozd rings, the statement is clear since both rings have the same length. The remaining case is that exactly one of the two rings is a Drozd ring, in which case length considerations show that Λ is a Drozd ring and Λ′ an artinian triad. But in this case we are trying to map a local ring whose maximal ideal requires 2 generators onto a local ring whose maximal ideal requires 3 generators, a clear impossibility.

Definition 2.3 (Maximal tame rings). Let (Λ, 𝔪, k) be a local ring, and suppose that Λ is reduced (no nilpotent elements). Let Γ denote the normalization of Λ (integral closure of Λ in its total quotient ring), and suppose that Γ has Krull dimension one, that μ_{Λ}(Γ) = 2, and that 𝔪 is an ideal of Γ. We call Λ split Dedekind-like if Γ/𝔪 ≅ k × k (as rings), and we call Λ unsplit Dedekind-like if Γ/𝔪 is a field (necessarily 2-dimensional over k).

It is often more informative to view Λ in terms of its conductor square. Consider the commutative diagram

\begin{matrix}Λ & ⟶^{ι_{1}} & Γ \\ \begin{matrix}↓ & ↓ \\ ρ_{1} \end{matrix} & & \begin{matrix}↓ & ↓ \\ ρ_{2} \end{matrix} \\ k & ⟶^{ι_{2}} & Γ/𝔪 \end{matrix} (2.3.1)

where ρ_{1} and ρ_{2} are reduction modulo 𝔪, and ι_{1} and ι_{2} are inclusion maps. Since both Λ and Γ contain 𝔪 we have: Λ is the pullback of its conductor square (2.3.1); that is, Λ = ρ_{2}^{−1}ι_{2}(k). This way of viewing Dedekind-like rings makes it easy to construct them, as we now proceed to show.

Split Dedekind-like rings. Let (V_{1}, 𝔪_{1}, k) and (V_{2}, 𝔪_{2}, k) be DVRs (“discrete valuation rings,” that is, local principal ideal domains) with a common residue field k, and for each i let f_{i}: V_{i} → → k be a surjective ring homomorphism. Then let

Λ = {[v_{1}, v_{2}] ∈ V_{1} × V_{2} | f_{1}(v_{1}) = f_{2}(v_{2})}. (2.3.2)

We claim: (2.3.3) (Λ, (𝔪_{1}×𝔪_{2}), k) is a split Dedekind-like ring with normalization Γ = V_{1}×V_{2}.

Moreover, every complete split Dedekind-like ring is isomorphic to such a ring Λ.

To verify the first assertion, build conductor square (2.3.1) as follows. Write k × k in place of Γ/𝔪 and let ρ_{2} = f_{1} × f_{2}. Let ι_{1} be the inclusion map, ι_{2} the diagonal map: k → k×k, and ρ_{1} be the restriction of ρ_{2} to Λ. Then the diagram commutates (because of definition (2.3.2) of Λ) and the kernel of both ρ_{1} and ρ_{2} is 𝔪 = 𝔪_{1} ×𝔪_{2}. Therefore Λ is the pullback of the square just formed and we can identify k × k with Γ/𝔪. Moreover, the kernel 𝔪 of ρ_{1} and ρ_{2} is the (Jacobson) radical of both rings Λ and Γ. The ring Γ is generated by two elements as a Λ-module because the same is true of k × k as a k-module. Therefore Γ is integral over Λ and hence is the normalization of Λ. Λ is noetherian since Γ is noetherian and Γ/Λ is a finitely generated Λ-module [Eakin]. The rest of the proof is easy.

The proof of the second statement is easy once one notes that the two nontrivial idempotents of k × k lift to Γ because we are in the complete case, and all maximal ideals of Γ contain 𝔪 since Γ is integral over Λ.

In the case of complete local algebras over the field k = Λ/𝔪, the only possiblity for split Dedekind-like rings is k[[X, Y]]/(XY) where [[…]] denotes formal power series. To verify this, note first that, by the Structure Theorem of Complete Local Rings [N, Theorem 31.1], we must have V_{1} ≅ k[[X]] and V_{2} ≅ k[[Y]] as kalgebras. It is then easily checked that the ring Λ defined in (2.3.2) is isomorphic to k[[X, Y]]/(XY). Finally, apply (2.3.3).

Unsplit Dedekind-like rings. Consider the conductor square (2.3.1) for the unsplit Dedekind-like ring (Λ, 𝔪, k) with normalization Γ. We claim:

(2.3.4) Γ is a DVR with maximal ideal 𝔪.

To see this, note that since 𝔪 = rad(Λ) and 𝔪 is an ideal of Γ, then 𝔪 ⊆ rad(Γ). Because Γ/𝔪 is a field by hypothesis, we see that 𝔪 is the Jacobson radical of Γ. Since Γ/𝔪 is a field, 𝔪 is the unique maximal ideal of Γ. Thus Γ is a normal local domain of Krull dimension one, and therefore a DVR.

It is now easy to construct a general unsplit Dedekind-like ring Λ (complete or not). Let Γ be any DVR whose residue field Γ/𝔪 is a 2-dimensional extension of a subfield k, and define Λ to be the pullback of diagram (2.3.1).

The easiest example of this type is the ring Λ = ℝ + Xℂ[[X]] of power series whose constant terms are real numbers and whose other coefficients are complex numbers. For an example that is not an algebra over a field, let Γ′ be the ring of all integers in any number field such that some residue field F of Γ′ is a 2-dimensional extension of some subfield k. Then Let Γ be the localization or completion of Γ′ with respect to the kernel of Γ → F, and finish the definition by letting Λ be the pullback of diagram (2.3.1).

Consistency with terminology of [L2]. All localizations and completions at maximal ideals of the rings called “Dedekind-like” in [L2] are split Dedekind-like, in our present terminology. But in the present series of papers we use the name “Dedekind-like” only for local rings.

We also need one “sometimes maximal” tame ring.

A local ring (Λ, 𝔪, k) is a Klein ring if μ_{Λ}(𝔪) = 2, μ_{Λ}(𝔪^{2}) = 1, 𝔪^{3} = 0, and x^{2} = 0 (∀x ∈ 𝔪). (Since 𝔪^{3} = 0, Klein rings are artinian.) We claim:

(2.3.5) k has characteristic 2, and Λ has characteristic 2 or 4.

Let 𝔪 = (x, y). Then 0 = (x + y)^{2} = x^{2} + 2xy + y^{2} = 2xy. Since μ_{Λ}(𝔪^{2}) = 1 and x^{2} = y^{2} = 0, we cannot have xy = 0. Therefore 2xy = 0 implies that 2 ∈ 𝔪. Since Λ is a Klein ring, this implies that 4 = 2^{2} = 0 in Λ, completing the proof of (2.3.5)

The group algebra of the Klein 4-group over a field of characteristic 2 is an example of a Klein ring. The phrase “sometimes maximal” (tame) refers to the fact that some Klein rings are homomorphic images of Dedekind-like rings (in which case they are not “truly maximal” tame), while some Klein rings are truly maximal tame. (See Examples 5.6 and 5.7.)

Lemma 2.4. Every DVR and every artinian local principal ideal ring is a homomorphic image of a split Dedekind-like ring.

Proof. Let (V, 𝔫, k) be a DVR, and f: V → k the natural homomorphism. Then let

Λ = {[x, y] ∈ V × V | f(x) = f(y)}

The ring Λ is split Dedekind-like, by (2.3.3), and maps onto V.

Let A be an artinian local principal ideal ring. A theorem of Hungerford [HU] states that some complete discrete valuation ring V maps onto A. Since Λ maps onto V, the proof is complete.

3. Dichotomy Theorem.

The purpose of this section is to prove the following result.

Theorem 3.1 (Dichotomy). Let (Λ, 𝔪, k) be a complete, local ring. Then exactly one of the following two possibilities holds.

(i) Λ has an artinian triad or a Drozd ring as homomorphic image. (ii) Λ is a Klein ring or a homomorphic image of a complete split or unsplit

Dedekind-like ring.

Remarks 3.2. Note that DVRs and artinian local principal ideal rings do not occur explicitly in Theorem 3.1. The reason is that such rings are homomorphic images of split Dedekind-like rings (Lemma 2.4). Of course, one would never use this fact to prove anything about the (well-known, simple) module theory of principal ideal rings.

The remainder of this section is devoted to the proof of this theorem, which we accomplish by means of a series of propositions.

We begin by noting that, if μ_{Λ}(𝔪) ≥ 3, then clearly Λ maps onto an artinian triad. On the other hand, if μ_{Λ}(𝔪) ≤ 1, then by a theorem of Hungerford [HU], Λ is a homomorphic image of a complete discrete valuation ring (Γ, 𝔪, k), and hence a homomorphic image of a complete split Dedekind-like ring (Lemma 2.4). Therefore, for the remainder of this section we can assume that μ_{Λ}(𝔪) = 2.

Thus, we can write 𝔪 = (x, y), so that 𝔪^{2} = (x^{2}, xy, y^{2}), and hence μ_{Λ}(𝔪^{2}) ≤ 3. If in fact μ_{Λ}(𝔪^{2}) = 3, then Λ/((x^{2}) + 𝔪^{3}) is a Drozd ring. Therefore, for the remainder of this section, we can also assume that μ_{Λ}(𝔪^{2}) ≤ 2.

The argument continues as we attempt to use generators x and y of 𝔪 and the relations satisfied by x^{2}, xy, and y^{2} to construct a split or unsplit Dedekind-like ring that maps onto Λ.

Lemma 3.3. Let (Λ, 𝔪, k) be a complete, local ring, and suppose that μ_{Λ}(𝔪) = 2 and μ_{Λ}(𝔪^{2}) ≤ 2. If x and y are generators of the maximal ideal 𝔪, and if xy ∈ 𝔪^{3}, then we can find generators u and v of 𝔪 such that uv = 0.

Proof. Note that for each integer n ≥ 2 we have 𝔪^{n} = (x^{n}, x^{n−1}y, ...xy^{n−1}, y^{n}). But xy ∈ 𝔪^{3}, so it follows that x^{n−1}y, ..., xy^{n−1} ∈ 𝔪^{n+1}, and hence by Nakayama’s Lemma we have 𝔪^{n} = (x^{n}, y^{n}).

The strategy is to construct sequences u_{1}, u_{2}, u_{3}, ... and v_{1}, v_{2}, v_{3}, ... of elements of Λ, such that

𝔪 = (u_{n}, v_{n}), u_{n}v_{n} ∈ 𝔪^{n+2}, and u_{n} − u_{n+1}, v_{n} − v_{n+1} ∈ 𝔪^{n+1} (3.3.1)

for each index n. It then follows immediately that the sequences are Cauchy sequences and, if we let u = lim_{n→∞}(u_{n}) and v = lim_{n→∞}(v_{n}), that 𝔪 = (u, v) and uv = 0.

The construction of the sequences is by induction on n. For n = 1, set u_{1} = x and v_{1} = y. Now suppose, inductively, that we have selected elements u_{1}, ..., u_{n} and v_{1}, ..., v_{n} of Λ satisfying (3.3.1).

As noted above, we have 𝔪^{n+2} = (u_{n}^{n+2}, v_{n}^{n+2}). Thus we can write

u_{n}v_{n} = au_{n}^{n+2} + bv_{n}^{n+2} (3.3.2)

for some a, b ∈ Λ. Rearranging (3.3.2) yields

u_{n}(v_{n} − au_{n}^{n+1}) = bv_{n}^{n+2} (3.3.3)

Therefore setting v_{n+1} = v_{n} − au_{n}^{n+1} gives a new pair of generators 𝔪 = (u_{n}, v_{n+1}) such that v_{n} − v_{n+1} ∈ 𝔪^{n+1}. Now (3.3.3) can be rewritten as

u_{n}v_{n+1} = b(v_{n+1} + au_{n}^{n+1})^{n+2}

Expanding the right-hand side and rearranging yields

(u_{n} − bv_{n+1}^{n+1})v_{n+1} ∈ 𝔪^{n+3}

Therefore setting u_{n+1} = u_{n} − bv_{n+1}^{n+1} completes the induction, and hence the proof of the lemma.

Under the conditions of this technical lemma, we are ready to construct a complete split Dedekind-like ring that maps onto Λ.

Proposition 3.4. Let (Λ, 𝔪, k) be a complete local ring, such that μ_{Λ}(𝔪) = 2, and suppose that x and y are a pair of generators of 𝔪 such that xy ∈ 𝔪^{3}. Then Λ is a homomorphic image of a complete split Dedekind-like ring.

Proof. By Lemma 3.3, we can assume that xy = 0.

By the “Structure Theorem of Complete Local Rings” [N, Theorem 31.1 and preceding paragraph], there is a surjective homomorphism ϕ : V [[X, Y]] → Λ such that ϕ(X) = x and ϕ(Y) = y, where V [[X, Y]] is a formal power series ring, and either V = k or V is a complete DVR of characteristic 0 with residue field k. Moreover, in this latter case, k has characteristic p ≠ 0, and the maximal ideal of V is generated by p.

If V is a field, let us set R = V [[X, Y]], a complete, two-dimensional, regular local ring.

If V is not a field, we note that V [[X, Y]] is a complete, local, three-dimensional domain with maximal ideal (p, X, Y), so that V [[X, Y]] is a regular local ring (by [AM, Theorem 11.22]) and therefore a unique factorization domain. Now ϕ(p) ∈ 𝔪 since k has characteristic p, and hence there are elements A, B ∈ V [[X, Y]] such that ϕ(p − (AX + BY)) = 0. Since p is irreducible in V, clearly p − (AX + BY) is irreducible in V [[X, Y]], and hence (p − (AX + BY)) is a prime ideal. We set R = V [[X, Y]]/(p − (AX + BY)), a complete, local, two-dimensional domain by [AM, Corollary 11.18]. Moreover, the maximal ideal of R is now generated by two elements (the images of X and Y), so that R is in fact a regular local ring in this case also (again by [AM, Theorem 11.22]). Changing notation, we let ϕ denote the induced map from R onto Λ, and we denote by X, Y the cosets of the original elements X, Y, respectively, in R.

Thus, in either case, we have a surjective homomorphism ϕ : R → Λ such that ϕ(X) = x and ϕ(Y) = y, where R is a complete, two-dimensional, regular local ring, (X, Y) is the maximal ideal of R, and R/(X, Y) ≅ k.

Now ϕ(XY) = xy = 0, so that ϕ induces a surjective homomorphism from R/(XY) onto Λ. But the regular local ring R is a unique factorization domain, and X and Y are non-associate primes (being generators of the two-generated maximal ideal of R). Therefore, (XY) = (X) ∩ (Y) as ideals of R. We set V_{1} = R/(X) and V_{2} = R/(Y). Each of V_{1} and V_{2} is a complete, local domain of Krull dimension one (since X and Y are irreducible), and each has a principal maximal ideal (since we mod out by the other generator), so that both V_{1} and V_{2} are complete DVRs.

Finally, by [CR1, 2.12], there is a pullback diagram

\begin{matrix}R/(XY) = R/((X) ∩ (Y)) & ⟶ & V_{2} = R/(Y) \\ ↓ & & ^{f_{2}}↓ \\ V_{1} = R/(X) & ⟶^{f_{1}} & k = R/((X) + (Y)) \end{matrix} (3.4.1)

That is, R/(XY) is isomorphic to the pullback

Ω = {[s, t] ∈ V_{1} × V_{2} | f_{1}(s) = f_{2}(t)}

of the pair of (natural) maps f_{1} and f_{2}. Since V_{1} and V_{2} are DVRs and k is a field, the ring R/(XY) ≅ Ω is split Dedekind-like by (2.3.3). Since ϕ : R → Λ is a surjective ring homomorphism whose kernel contains (XY), our proof is complete.

Given the result of Proposition 3.4, we can assume that, for every pair of generators x and y of the maximal ideal 𝔪 of Λ, xy ∉ 𝔪^{3}. We also recall that we assume that μ_{Λ}(𝔪^{2}) ≤ 2. The next step is to reduce to the case where μ_{Λ}(𝔪^{2}) = 2.

Proposition 3.5. Let (Λ, 𝔪, k) be a complete, local ring, such that μ_{Λ}(𝔪) = 2, and suppose that xy ∉ 𝔪^{3} for every pair of generators x and y of 𝔪. Then either μ_{Λ}(𝔪^{2}) ≥ 2, or Λ is a Klein ring.

Proof. Since we assume that xy ∉ 𝔪^{3} for every pair of generators x and y of 𝔪, clearly μ_{Λ}(𝔪^{2}) ≥ 1. Thus, we can assume that μ_{Λ}(𝔪^{2}) = 1, and it suffices to show that Λ is a Klein ring.

Fix a pair of generators x and y of 𝔪. By assumption, μ_{Λ}(𝔪^{2}) = 1 and xy ∉ 𝔪^{3}, so that in fact 𝔪^{2} = (xy). Thus, x^{2} ∈ (xy), say x^{2} = axy for some element a ∈ Λ. But then x(x − ay) = 0, and if a were a unit in Λ, then we could replace y by x − ay to get a pair of generators of 𝔪 whose product is 0, contrary to assumption. Therefore, a ∈ 𝔪, and hence x^{2} ∈ 𝔪^{3}. Similarly, y^{2} ∈ 𝔪^{3}. But now 𝔪^{3} = (x^{3}, x^{2}y, xy^{2}, y^{3}) ⊆ 𝔪^{4} implies, by Nakayama’s Lemma, that 𝔪^{3} = 0. In particular, x^{2} = y^{2} = 0.

To complete the proof that Λ is a Klein ring, we must show that z^{2} = 0 for every element z ∈ 𝔪. If we write z = ax+by for some elements a, b ∈ Λ, with x and y the fixed generators of 𝔪 from above, then z^{2} = a^{2}x^{2} + 2abxy + b^{2}y^{2} = 2abxy, so that it suffices to show that the residue field of Λ has characteristic 2. But if the residue field of Λ were not of characteristic 2, then x + y and x − y would be generators of 𝔪, where (x + y)(x − y) = x^{2} − y^{2} = 0, contrary to assumption.

Given the result of Proposition 3.5, we can assume that μ_{Λ}(𝔪) = μ_{Λ}(𝔪^{2}) = 2 and that xy ∉ 𝔪^{3}, for every pair of generators x and y of 𝔪. If, for some element x ∈ 𝔪 − 𝔪^{2}, we had x^{2} ∈ 𝔪^{3}, then by definition Λ/𝔪^{3} would be a Drozd ring. Therefore, we can also assume that x^{2} ∉ 𝔪^{3} for every element x ∈ 𝔪 − 𝔪^{2}. We are now ready to map an unsplit Dedekind-like ring onto Λ.

Proposition 3.6. Let (Λ, 𝔪, k) be a complete local ring such that μ_{Λ}(𝔪) = μ_{Λ}(𝔪^{2}) = 2. Suppose that xy ∉ 𝔪^{3} for every pair of generators x and y of 𝔪, and x^{2} ∉ 𝔪^{3} for every element x ∈ 𝔪 − 𝔪^{2}. Then Λ is a homomorphic image of a complete unsplit Dedekind-like ring.

Proof. Fix generators x and y of 𝔪. By assumption, μ_{Λ}(𝔪^{2}) = 2, and xy ∉ 𝔪^{3}. Thus, we can choose xy as one of the two generators of 𝔪^{2}. Without loss of generality we can assume that 𝔪^{2} = (x^{2}, xy). Hence we can write

y^{2} + sxy + tx^{2} = 0 (3.6.1)

for some elements s, t ∈ Λ. Let s̄ and t̄ denote the images of s and t, respectively, in k = Λ/𝔪.

We claim that the polynomial Z^{2} + s̄Z + t̄ is irreducible in the polynomial ring k[Z]. Suppose it were reducible, say Z^{2} + s̄Z + t̄ = (Z + ū)(Z + v̄), where ū and v̄ are the images in k of the elements u, v ∈ Λ, respectively. This would mean that y^{2} + sxy + tx^{2} − (y + ux)(y + vx) ∈ 𝔪^{3}. Then by (3.6.1) we would have (y + ux)(y + vx) ∈ 𝔪^{3}. On the one hand, if u − v ∈ Λ − 𝔪, then y + ux and y +vx would be generators of 𝔪 whose product is in 𝔪^{3}, contrary to the hypotheses of the proposition. On the other hand, if u − v ∈ 𝔪, then it would follow that (y + ux)^{2} ∈ 𝔪^{3}, where y + ux ∈ 𝔪 − 𝔪^{2}, again contrary to the hypotheses of the proposition. Therefore the claim holds.

We continue as in the proof of Proposition 3.4. Using the Structure Theorem of Complete Local Rings [N, Theorem 31.1], there is a surjective homomorphism ϕ : V [[X, Y]] → Λ such that ϕ(X) = x and ϕ(Y) = y, where V [[X, Y]] is a formal power series ring, and either V = k or V is a complete DVR of characteristic 0 with residue field k, and where, in this latter case, k has characteristic p ≠ 0 and the maximal ideal of V is generated by p.

If V is a field, we set R = V [[X, Y]]. If V is not a field, we set R = V [[X, Y]]/(p− (AX +BY)), where A, B ∈ V [[X, Y]] are elements such that ϕ(p−(AX +BY)) = 0. In the latter case we change notation: denote again by ϕ the induced map from R onto Λ, and by X and Y the cosets in R of X and Y, respectively. Then, exactly as in the proof of Proposition 3.4, ϕ : R → Λ is a surjective homomorphism such that ϕ(X) = x and ϕ(Y) = y, where R is a complete two-dimensional regular local ring and 𝔪_{R} = (X, Y) is the maximal ideal of R, and of course ϕ induces an isomorphism between the residue fields of R and Λ. Note that (3.6.1) still holds since Λ has not changed.

Now select elements S, T ∈ R such that ϕ(S) = s and ϕ(T) = t. We claim that Y^{2} + SXY + T X^{2} is an irreducible element in the unique factorization domain R. Suppose, by way of contradiction, that Y^{2} + SXY + T X^{2} = CD in R, where neither C nor D is a unit. We know that μ_{R}(𝔪_{R}^{2}) = 3 (by [AM, Theorem 11.22]), and 𝔪_{R}^{2} = (X^{2}, XY, Y^{2}), so that CD = Y^{2} + SXY + T X^{2} ∈ 𝔪_{R}^{2} − 𝔪_{R}^{3}. Note that ϕ(C)ϕ(D) = ϕ(Y^{2} + SXY + T X^{2}) = 0 by (3.6.1). But ϕ(𝔪_{R}) = 𝔪, and both maximal ideals require two generators, from which we get a contradiction as follows. If C and D are linearly independent modulo 𝔪_{R}^{2}, then they generate 𝔪_{R}, so that ϕ(C) and ϕ(D) are generators of 𝔪 such that ϕ(C)ϕ(D) = 0, contrary to the hypotheses of the proposition. If C and D are linearly dependent modulo 𝔪_{R}^{2}, then they are associates, from which it follows that ϕ(C)^{2} = 0, again contrary to the hypotheses of the proposition. Therefore the claim holds.

We set Ω = R/(Y^{2} + SXY + T X^{2}). Since ϕ(Y^{2} + SXY + T X^{2}) = 0, the map ϕ factors through Ω to yield a surjective homomorphism from Ω onto Λ; moreover, this homomorphism induces an isomorphism of the residue fields of Ω and Λ. To complete the proof of the proposition, we need only show that Ω is an unsplit Dedekind-like ring.

First we recall that R is a unique factorization domain, and Y^{2} + SXY + T X^{2} is irreducible, so that (Y^{2} + SXY + T X^{2}) is prime, and therefore Ω is a (complete local noetherian one-dimensional) domain. Since we need only concern ourselves with the ring Ω for the remainder of this proposition, let us denote by x, y, s, and t the images in Ω of the elements X, Y, S, and T, respectively, of R. Let us also denote by 𝔪 the maximal ideal of Ω, and note that 𝔪 = (x, y). By the definition of Ω, we have y^{2} + sxy + tx^{2} = 0; that is, (3.6.1) still holds. Therefore (y/x)^{2} + s(y/x) + t = 0 in the quotient field of Ω; that is, y/x is integral over Ω in its quotient field. Let us set Γ = Ω[y/x] = Ω + Ω · y/x (using the fact that (y/x)^{2} = −s(y/x) − t).

Now Γ is a finite, integral extension of Ω in its quotient field, so that Γ is a semilocal, (noetherian) one-dimensional domain, and Γ𝔪 is contained in the Jacobson radical of Γ. On the other hand, Γ𝔪 = Γ(x, y) ⊆ 𝔪 (since (y/x)x = y ∈ 𝔪 and, by (3.6.1), (y/x)y = −(sxy + tx^{2})/x = −sy − tx ∈ 𝔪), from which it follows that Γ𝔪 = 𝔪, and F = Γ/𝔪 is an algebra over Ω/𝔪 = k, of vector space dimension 2. But the coset of y/x in F = Γ/𝔪 satisfies the irreducible polynomial Z^{2} + s̄Z + t̄ over k, and so F is in fact a quadratic field extension of k. Therefore, 𝔪 is the Jacobson radical of Γ, and hence Γ is also a local ring with maximal ideal 𝔪. Moreover, 𝔪 = Γ(x, y) = Γx (since y = (y/x)x ∈ Γx), so that 𝔪 is principal as a Γ-ideal, and hence Γ is a DVR. Thus Γ is integrally closed and therefore must be the normalization of Ω. Finally, we note that μ_{Ω}(Γ) = 2, so that, by Definition 2.3, Ω is unsplit Dedekind-like.

We remark that, since Γ is finitely generated as an Ω-module, Γ is complete in the 𝔪-adic topology as an Ω-module. But since 𝔪 is also an ideal of Γ, this is the same as the 𝔪-adic topology on Γ as a ring. Therefore, (Γ, 𝔪, F) is a complete DVR.

Propositions 3.4, 3.5, and 3.6 together show that, for a given complete, local ring (Λ, 𝔪, k), at least one of the conditions (i) or (ii) of Theorem 3.1 must hold. It remains to show that these two conditions are mutually exclusive, which we show in the final lemma and proposition of this section.

Lemma 3.7. Let (Λ, 𝔪, k) be a complete Dedekind-like ring.

(i) If w ∈ 𝔪 − 𝔪^{2} then w^{2} ∉ 𝔪^{3}. (ii) The (composition) length of the ring Λ/𝔪^{3} is 5.

Proof. Let Γ be the normalization of Λ.

(i) The important idea here is that 𝔪 is an ideal of Γ, and both the hypothesis w ∈ 𝔪 − 𝔪^{2} and the conclusion w^{2} ∉ 𝔪^{3} say the same thing whether we consider 𝔪 to be an ideal of Λ or of Γ. So we work in the ring Γ instead of Λ.

Case 1: Λ unsplit Dedekind-like. Then Γ is a DVR and 𝔪 is its maximal ideal (see (2.3.4)). The hypothesis w ∈ 𝔪 − 𝔪^{2} implies that w generates the maximal ideal of Γ (as a Γ-module). Since Γ is a DVR, w^{2} generates 𝔪^{2}, and hence is not in 𝔪^{3}.

Case 2: Λ split Dedekind-like. Here 𝔪 is an ideal of Γ, but not a maximal ideal of Γ. We have that Γ = V_{1} × V_{2}, the direct product of two DVRs and 𝔪 = 𝔪_{1} × 𝔪_{2} where each 𝔪_{i} is the maximal ideal of V_{i} (see (2.3.3)). Thus we can write the given element w as an ordered pair w = [w_{1}, w_{2}] with each w_{i} ∈ 𝔪_{i}. The hypothesis w ∈ 𝔪 − 𝔪^{2} implies that, for at least one index i, w_{i} generates 𝔪_{i}, say w_{1} generates 𝔪_{1}. Since V_{1} is a DVR, it now follows that w_{1}^{2} ∉ 𝔪_{1}^{3}, and therefore w^{2} ∉ 𝔪^{3}.

(ii) The bottom row of conductor square (2.3.1) shows that Γ/Λ is a simple Λ-module. Therefore so is (Γ/𝔪^{3})/(Λ/𝔪^{3}). So it suffices to show that Γ/𝔪^{3} has length 6 as a Λ-module.

Case 1: Λ unsplit Dedekind-like. Here Γ is a DVR and its unique simple module is Γ/𝔪, which has dimension 2 over k. In different terminology: the simple Γ-module has length 2 as a Λ-module. Γ/𝔪^{3} has length 3 as a Γ-module, and therefore has length 6 as a Λ-module.

Case 2: Λ split Dedekind-like. Here Γ = V_{1} × V_{2} is the direct product of two DVRs and 𝔪 = 𝔪_{1} × 𝔪_{2}, where each 𝔪_{i} is the maximal ideal of V_{i} (see (2.3.3)). Therefore Γ/𝔪^{3} has length 6 as a Γ-module. Γ has two simple modules, namely, the two simple modules over the ring Γ/𝔪 = k × k in the lower right corner of the conductor square for Γ. Each of these simple Γ-modules is a copy of k, and therefore remains simple as a Λ-module. Therefore, in this case the Λ-length of Γ/𝔪^{3} is the same as its Γ-length, namely 6.

To show that conditions (i) and (ii) of Theorem 3.1 are mutually exclusive, clearly it suffices to show that Klein rings and complete split or unsplit Dedekind-like rings cannot be mapped onto either an artinian triad or a Drozd ring. This we show in the final proposition of this section, which completes the proof of the theorem.

Proposition 3.8. Let (Λ, 𝔪, k) be either a Klein ring or a complete split or unsplit Dedekind-like ring. Then Λ does not have either an artinian triad or a Drozd ring as homomorphic image.

Proof. Since μ_{Λ}(𝔪) = 2, clearly Λ cannot map onto an artinian triad. Similarly, if Λ is a Klein ring, then μ_{Λ}(𝔪^{2}) = 1, and hence Λ cannot map onto a Drozd ring.

Therefore, we can suppose that Λ is a complete split or unsplit Dedekind-like ring, and we must show that Λ cannot map onto a Drozd ring. Suppose, by way of contradiction, that ϕ : Λ ⟶ A were a homomorphism from Λ onto the Drozd ring A. Then ϕ(𝔪^{3}) = ϕ(𝔪)^{3} = 0 implies that ϕ would induce a map from Λ/𝔪^{3} onto A. Since both Λ/𝔪^{3} and A have composition length five (by Lemma 3.7 and (2.2.1)), we would obtain the isomorphism Λ/𝔪^{3} ≅ A. That is, Λ/𝔪^{3} would be a Drozd ring.

But then Λ would contain an element w whose image w̄ in Λ/𝔪^{3} satisfies w̄ ∈ 𝔪̄ − 𝔪̄^{2} and w̄^{2} = 0. Therefore w ∈ 𝔪 − 𝔪^{2} and w^{2} ∈ 𝔪^{3}, contrary to Lemma 3.7.

4. Wildness of Drozd Rings

Caution. When functions represent matrix multiplication, we write the functions as right-hand operators. This occurs many times in this section.

Notation 4.1. Recall (see Definition 2.2) that a Drozd ring is an artinian local commutative ring (Λ, 𝔪, k) (maximal ideal 𝔪, residue field k) such that μ_{Λ}(𝔪) = μ_{Λ}(𝔪^{2}) = 2, 𝔪^{3} = 0, and (∃x ∈ 𝔪 − 𝔪^{2}) x^{2} = 0. Since μ_{Λ}(𝔪) = 2 we have 𝔪 = (x, y) for some y ∈ 𝔪 − 𝔪^{2}. Fix such elements x, y for the rest of this section. Then 𝔪^{2} = (xy, y^{2}), and the only nonzero monomials in x and y are x, y, xy, y^{2}.

In Drozd’s original paper [D], Λ is a 5-dimensional k-algebra, and 1, x, y, xy, x^{2} form a basis of Λ. In our more general setting we partially repair the lack of a basis by establishing the following standard form for elements of 𝔪:

(4.1.1) (∀c ∈ 𝔪) c = u_{1}x + u_{2}y + u_{3}xy + u_{4}y^{2} (with each u_{i} a unit or 0.)

Note that we do not claim uniqueness of the coefficients u_{i}.

Proof. There is an expression c = ax + by. We can write a = u_{1} + a′x + b′y where u_{1} is a unit or 0. (If u_{1} is a unit, we can take a′ = 0 = b′, but this is not important.) Similarly, b = u_{2} + a′′x + b′′y. Then c = u_{1}x + u_{2}y + (b′ + a′′)xy + b′′y^{2}. Making similar substitutions for b′ + a′′ and b′′ completes the proof.

Viewing Λ/(y) and Λ/(xy) as Λ-modules, for λ ∈ Λ we denote by λ_{y} (respectively λ_{xy}) the coset of λ in Λ/(y) (respectively Λ/(xy)). Thus, Λ/(y) = Λ1_{y} has submodule Λx_{y}, etc. The following lemma establishes some basic facts about Λ and its homomorphic images Λ1_{y} and Λ1_{xy}.

Lemma 4.2. (i) All of the monomials x, xy, x_{xy}, … in diagram (4.2.1) below

are nonzero. (ii) soc Λ1_{y} = Λx_{y}, soc Λ1_{xy} = Λx_{xy} ⊕ Λy_{xy}^{2}, and soc Λ = Λxy ⊕ Λy^{2} = 𝔪^{2}. (iii) Λx ∩ Λy = Λxy and Λx_{xy} ∩ Λy_{xy} = 0. (iv) Λx ≅ Λy_{xy} as Λ-modules via the multiplication map cx → cy_{xy}.

Λ1_{y}  Λ1_{xy}  Λ
[figure]
Λx_{y} Λx_{xy} Λy_{xy} ^{(≅)} \overline{}_{(mult)} Λx Λy
[figure]
Λy_{xy}^{2}  Λxy  Λy^{2}

(4.2.1)

Proof. Λ. Since 𝔪^{2} = (xy, y^{2}) and μ_{Λ}(𝔪^{2}) = 2 the monomials displayed in Λ are nonzero. To prove the nontrivial inclusion in Λx ∩ Λy = Λxy, assume that cx = dy. The element c cannot be a unit, since this would imply μ_{Λ}(x, y) = 1. Therefore we can write c in the standard form (4.1.1), getting cx = u_{2}xy, as desired.

Next we verify the description of the socle of Λ. Note that, for any Λ-module M, we have 𝔪M = 0 if and only if M is (a k-module hence) a semisimple Λ-module. Therefore 𝔪^{3} = 0 implies that 𝔪^{2} ⊆ soc Λ. Conversely, take c ∈ soc Λ and write c in the standard form (4.1.1). Then 0 = cx = u_{2}xy shows that u_{2} is not a unit, and hence u_{2} = 0. But then 0 = cy = u_{1}xy, and so u_{1} = 0, completing the proof that 𝔪^{2} = soc Λ = Λxy + Λy^{2}. The sum Λxy + Λy^{2} must be direct since μ_{Λ}(𝔪^{2}) = 2. This completes the proof of the assertions about Λ.

Λ1_{xy}. The second intersection in (iii) holds since Λx∩Λy = Λxy. The monomial x_{xy} is nonzero since otherwise x ∈ Λxy contradicting μ_{Λ}(x, y) = 2. Similarly y_{xy} ≠ 0; and y_{xy}^{2} ≠ 0 because μ_{Λ}(xy, y^{2}) = 2.

We have soc Λ1_{xy} ⊇ Λx_{xy} + Λy_{xy}^{2} because the right-hand side is annihilated by x and y. For the opposite inclusion take c ∈ Λ such that c_{xy} ∈ soc Λ1_{xy}. Since c_{xy}𝔪 = 0 we have c𝔪 ⊆ Λxy. Now write c in the standard form (4.1.1). Then cy = u_{1}xy + u_{2}y^{2} ∈ Λxy. Since cy ∈ Λxy, directness of the sum Λxy ⊕ Λy^{2} shows that u_{2} cannot be a unit, hence equals zero. Therefore c = u_{1}x + u_{3}xy + u_{4}y^{2}, showing that c_{xy} ∈ Λx_{xy} + Λy_{xy}^{2}. It remains to prove directness of this last sum; and this follows from Λx_{xy} ∩ Λy_{xy} = 0, completing the proof of all assertions about Λ1_{xy}.

Λ1_{y}. All assertions here are obvious.

(iv). It suffices to prove that cy_{xy} = 0 ⟺ cx = 0. Since c is not a unit, we can write it in the standard form (4.1.1). Then cy_{xy} = 0 is equivalent to u_{1}xy + u_{2}y^{2} ∈ Λxy. Directness of the sum in our expression for soc Λ shows that u_{2} cannot be a unit, hence equals 0. Therefore cy_{xy} = 0 implies that c = u_{1}x + u_{3}xy + u_{4}y^{2}, and therefore cx = 0. The converse is proved similarly.

Definition 4.3 (S(1\frac{1}{2}), S(2)). We define an object of the category S(1\frac{1}{2}), or more completely, k-S(1\frac{1}{2}) (“one and one-half k-similarity”) to be an ordered triple (m, n, ϕ) where m ≤ n are positive integers and ϕ is an n × n matrix over k. We define a morphism of S(1\frac{1}{2}) to be a pair of k-linear maps (σ, τ) (equivalently, matrices of the appropriate sizes) such that the following diagram commutes.

S(1\frac{1}{2}) : \begin{matrix}k^{(m)} & ⟶^{i = [I 0]} & k^{(n)} & ⟶^{ϕ} & k^{(n)} \\ ^{σ}↓ & & ^{τ}↓ & & ^{τ}↓ \\ k^{(m′)} & ⟶^{i′ = [I 0]} & k^{(n′)} & ⟶^{ϕ′} & k^{(n′)} \end{matrix} (4.3.1)

Here k^{(m)} denotes the direct sum of m copies of k, written as row vectors, and i = i_{m,n} denotes the inclusion map b → (b, 0), where b denotes an arbitrary vector in k^{(m)} and 0 denotes the zero vector in k^{(n−m)}. The map i′ is defined analogously.

Similarly, we define the the category S(2) (more completely, k-S(2)) to be “simultaneous similarity” of a pair of matrices over the field k. That is, the objects of S(2) are ordered triples (n, A, B) where n is a positive integer and A and B are n×n matrices over k. We define a morphism of S(2) to be a k-linear map (equivalently, matrix of the appropriate size) τ such that the following diagram commutes.

S(2): \begin{matrix}k^{(n)} & ⟶^{A} & k^{(n)} & ⟶^{B} & k^{(n)} \\ ^{τ}↓ & & ^{τ}↓ & & ^{τ}↓ \\ k^{(n′)} & ⟶^{A′} & k^{(n′)} & ⟶^{B′} & k^{(n′)} \end{matrix} (4.3.2)

The category S(2) is easily seen to be equivalent to the category fdmod-k⟨X, Y ⟩ of finite dimensional k⟨X, Y ⟩-modules by sending the object (n, A, B) to the k⟨X, Y ⟩- module consisting of the vector space k^{(n)}, on which X and Y act via multiplication by the matrices A and B, respectively.

The category S(1\frac{1}{2}) is known to be wild over any field k in the strongest sense: there is a full exact embedding S(2) → S(1\frac{1}{2}). The following lemma is due to Narazova [Nz, Lemma 1], except for a more functorial treatment here, and some changes and reversals of notation.

Lemma 4.4. Over any field k there is a full subcategory W′ of S(1\frac{1}{2}) and an additive functor F: W′ → S(2) that is a representation equivalence whose restrictions to homomorphism groups are isomorphisms.

Proof. Let W′ be the full subcategory of S(1\frac{1}{2}) consisting of objects of the form

(n, 2n, ϕ) in which ϕ = [\begin{matrix}0 & I \\ A & B \end{matrix}], where A, B are n × n matrices over k and 0, I are

the n×n zero and identity matrices, respectively. Then let F (n, 2n, ϕ) = (n, A, B) ∈ S(2).

Let (σ, τ): (n, 2n, ϕ) → (n′, 2n′, π′) be a morphism in W′. We claim that τ has

the 2×2 block-diagonal form τ = [\begin{matrix}T & 0 \\ 0 & T \end{matrix}] where the block-sizes in T are consistent

with the partitioning of ϕ and ϕ′. To see this, let τ = [\begin{matrix}T_{1} & T_{2} \\ T_{3} & T_{4} \end{matrix}].

Note. Since we are representing homomorphisms by matrix multiplication on the right, we write the homomorphisms as right operators. The relation iτ = σi′ yields the equation

[I 0] [\begin{matrix}T_{1} & T_{2} \\ T_{3} & T_{4} \end{matrix}] = [S 0]

which shows that T_{2} = 0. Then the relation τϕ′ = ϕτ yields the equation

[\begin{matrix}T_{1} & 0 \\ T_{3} & T_{4} \end{matrix}] [\begin{matrix}0 & I \\ A′ & B′ \end{matrix}] = [\begin{matrix}0 & I \\ A & B \end{matrix}] [\begin{matrix}T_{1} & 0 \\ T_{3} & T_{4} \end{matrix}]

which easily completes the proof of the claim. Moreover, the relation τϕ′ = ϕτ now implies that T A′ = AT and T B′ = BT, and hence these left and right multiplications by T form a morphism F (σ, τ) in S(2).

The block diagonal form of τ shows that, if (σ, τ) is an isomorphism, then T is an invertible matrix. The relations in the previous paragraph then show that F (σ, τ) is an isomorphism in S(2). The remaining assertions of the lemma now follow immediately.

Definition 4.5 (Ringel’s Λ-module M(m, n, ϕ)). Let (Λ, 𝔪, k) be a Drozd ring and (m, n, ϕ) ∈ S(1\frac{1}{2}), and let Q be the Λ-module Q = Λ^{(m)}1_{y} ⊕ Λ^{(n)}1_{xy} ⊕ Λ^{(n)} (notation as in (4.2.1)), where Λ^{(m)} denotes the direct sum of m copies of the Λ-module Λ, written as row vectors. We define the module M = M(m, n, ϕ) by “amalgamating” three pairs of submodules of Q. (The quotes indicate that one of these amalgamations can have a kernel, hence is not actually an amalgamation.) We first display the diagram that exhibits these amalgamations, then define them both informally and formally. (To save space in the diagram, we write Λ^{m} and Λ^{n} in place of Λ^{(m)} and Λ^{(n)}, respectively.)

Λ^{m}1_{y}  Λ^{n}1_{xy}  Λ^{n}
[figure]
Λ^{m}x_{y} Λ^{n}x_{xy} Λ^{n}y_{xy} \overline{}_{(mult)} Λ^{n}x Λ^{n}y
[figure]
Λ^{n}y_{xy}^{2} Λ^{n}xy Λ^{n}y^{2}
[figure: i; ϕ]

(4.5.1)

Comments on notation. Since Λ1_{xy} is a Λ-module, we have cx_{xy} = xc_{xy} for all c ∈ Λ. We also use this notation in vector form, writing cx_{xy} = xc_{xy} for vectors c ∈ Λ^{(n)}, hence c_{xy} for its natural image in Λ^{(n)}1_{xy}. For any c ∈ Λ we define c̄ to be the natural image of c in the residue field k of Λ. As above, we also use this notation in vector form.

Amalgamation map i : Λ^{(m)}x_{y} → Λ^{(n)}y_{xy}^{2}. Since the socle of every Λ-module is annihilated by 𝔪, we have Λ^{(m)}x_{y} = k^{(m)}; more precisely, this is a canonical Λ-module isomorphism via bx_{y} → b̄. Similarly Λ^{(n)}y_{xy}^{2} = k^{(n)} via cy_{xy}^{2} → c̄. We define i = i_{m,n} by (bx_{y})i = (b, 0)y_{xy}^{2}, the same map i as given in (4.3.1).

Amalgamation map mult : Λ^{(n)}x → Λ^{(n)}y_{xy}. This is by means of the multiplication map dx → dy_{xy}. By Lemma 4.2(iii) this is indeed an Λ-module isomorphism.

“Amalgamation” map ϕ : Λ^{(n)}x_{xy} → Λ^{(n)}y^{2}. We have Λ^{(n)}x_{xy} = k^{(n)} via cx_{xy} → c̄. Therefore we can right-multiply elements of Λ^{(n)}x_{xy} = k^{(n)} by the matrix ϕ in our given object (m, n, ϕ) of S(1\frac{1}{2}). Also, Λ^{(n)}y^{2} = k^{(n)} via cy^{2} → c̄. For c̄ ∈ Λ^{(n)} define (cx_{xy})ϕ = (c̄ϕ)y^{2}. Note that if ϕ has a kernel, this is not actually an amalgamation map.

Informally, we form M from Q by identifying elements of Q with their image under under the above amalgamation maps. Formally, let b, c, d be elements of Λ^{(m)}, Λ^{(n)}, Λ^{(n)} respectively. We define M = M(m, n, ϕ) to be Q/R where R denotes the following set of relations, that is, elements of Q.

(bx_{y}, −(bi)y_{xy}^{2}, 0) + (0, −dy_{xy}, dx) + (0, cx_{xy}, −(c̄ϕ)y^{2}) (4.5.2)

Note that we can replace b, c, d by b̄, c̄, d̄ respectively, whenever convenient, but we cannot replace c̄ by c in c̄ϕ.

For every element (m, n, ϕ) ∈ S(1\frac{1}{2}) we have a natural Λ-module surjection:

ρ: P = Λ^{(m)} ⊕ Λ^{(n)} ⊕ Λ^{(n)} ⟶^{ρ_{1}} Q = Λ^{(m)}1_{y} ⊕ Λ^{(n)}1_{xy} ⊕ Λ^{(n)} ⟶^{ρ_{2}} M(m, n, ϕ) (4.5.3)

where (b, c, d)ρ_{1} = (b_{y}, c_{xy}, d) and ρ_{2} equals reduction modulo the relations R displayed in (4.5.2). Note that ker ρ ⊆ 𝔪P, and therefore:

(4.5.4) The map ρ in (4.5.3) is a projective cover.

Historical Remark. In Drozd’s original paper [D], Λ is a k-algebra and the modules he uses in place of our M are represented by very large matrices (32n×32n, in the notation of 4.3) over k. He assumes that k is an algebraically closed field. But, in order to obtain wildness of Λ, the only property of algebraic closure that he uses is that k has at least 5 elements. (Algebraic closure is, however, used elsewhere in his paper.) Ringel [R, (3.4)] uses much smaller matrices over k, by relating M to S(1\frac{1}{2}) rather than directly to S(2), as Drozd did. The size of Ringel’s matrices is (m + 6n) × (m + 6n), in the notation of 4.3. Wildness of the category of Λ-modules that he obtains holds for any field k.

Our module M = M(m, n, ϕ) is a modification of Ringel’s construction, and follows him by relating Λ-modules to S(1\frac{1}{2}) rather than directly to S(2). However, since our ring Λ need not be a k-algebra, M need not have a matrix representation. But we do represent homomorphisms in the category of modules of the form M(m, n, ϕ) by matrices over the ring Λ. The critical fact is that any map of Λmodules lifts to a map of their projective covers; and since projective modules over commutative local rings are free, this lifted map can be represented by a matrix over Λ. Our Λ-matrices are much smaller than Ringel’s k-matrices. But this is deceptive because an element of Λ holds more information than a single element of k. There are two consequences of this: (i) The functor that we obtain, displaying wildness, is a only a representation equivalence rather than the full exact embedding that Ringel gets. (ii) Our wildness proof is probably more intricate but shorter than Ringel’s (omitted) proof.

Morphisms and Matrices. Let f: M = M(m, n, ϕ) → M′ = M(m′, n′, ϕ′) be any Λ-module homomorphism. We wish to represent f by a matrix F over Λ. Since P and P′ are projective Λ-modules, f lifts to an Λ-homomorphism such that the following diagram commutes.

\begin{matrix}P & ⟶^{F} & P′ \\ ^{↓ ↓}ρ & & ^{↓ ↓}ρ′ \\ M & ⟶^{f} & M′ \end{matrix} (4.5.5)

Any such F equals right multiplication by a unique (m+2n)×(m′+2n′) matrix over Λ, and we again call this matrix F. We always view F in 3×3 block form, where the rows and columns are partitioned according to the scheme (m+n+n)×(m′+n′+n′).

The crux of our wildness proof is the next lemma, which gives some information about the form of the matrix F.

Notation 4.6. We denote by “ent e” the set of entries of any array e. Recall that F̄_{ii} denotes the k-matrix obtained by reducing all entries of F_{ii} modulo 𝔪.

Lemma 4.7. (i) The matrix F has the block form

F = ((F_{ij})) = [\begin{matrix}F_{11} & 𝔪 & (x, y^{2}) \\ ∗ & F_{22} & 𝔪 \\ ∗ & ∗ & F_{33} \end{matrix}] (4.7.1)

That is, ent F_{12} ⊆ 𝔪, ent F_{23} ⊆ 𝔪, and ent F_{13} ⊆ (x, y^{2}). Moreover, the diagonal blocks satisfy the following three properties (viewed as right operators).

(a) F̄_{11}i_{m′,n′} = i_{m,n}F̄_{22} (See definition of i_{m,n} below (4.3.1).)

(b) F̄_{22} = F̄_{33}

(c) ϕF̄_{22} = F̄_{22}ϕ′

(We make no assertion about the blocks denoted by ∗.) (ii) If f is an isomorphism, then the diagonal blocks F_{ii} are invertible matrices.

Proof. Let b, c, d be elements of a local ring with residue field k (e.g. Λ or its homomorphic images).

(4.7.2) If bd = cd and d ≠ 0 then b̄ = c̄ (natural images in k)

because b ≡ c modulo the annihilator of d and this annihilator is contained in the maximal ideal of the ring. We use this fact many times in the long, intricate proof below.

(i) Since F lifts f we must have (ker ρ)F ⊆ ker ρ′. Recall that we factor ρ = ρ_{1}ρ_{2} as in (4.5.3). The information that we obtain about F is information that can be obtained by dealing separately with five types of elements of ker ρ. Type 1: elements (e, 0, 0) ∈ P such that (e, 0, 0)ρ_{1} = 0 ∈ Q; type 2: elements (0, e, 0) ∈ P such that (0, e, 0)ρ_{1} = 0 in Q; types 3–5: elements of P corresponding to the three types of amalgamations that were used to form M, one type for each term in (4.5.2). We label the five parts of the proof by the blocks of F about which they give information.

F_{12}, F_{13}. Note that (ey, 0, 0)ρ_{1} = 0 for every e because y_{y} = 0. Therefore (ey, 0, 0)F ρ′ = 0 and hence (ey, 0, 0)F ρ′_{1} ⊆ ker ρ′_{2}. Writing F in block form and using the definition of ρ′_{1}, it follows that

(0, eF_{12}y_{xy}, eF_{13}y) ∈ ker ρ′_{2} (4.7.3) for every e. Therefore every element of this form must have an expression of the form (4.5.2) (more precisely, the version of (4.5.2) that applies to M′, hence contains i′ and ϕ′ in place of i and ϕ).

Comparing coordinate 1 in (4.5.2) and (4.7.3) yields ent b ⊆ 𝔪. Therefore the first of the three terms in (4.5.2) equals zero, and can be ignored. Looking at coordinate 3 shows that

eF_{13}y = dx − (c̄ϕ)y^{2} (4.7.4)

Therefore ent d ⊆ 𝔪 (otherwise x ∈ Λy and μ_{Λ}(𝔪) ≤ 1). On the other hand, looking at coordinate 2 shows that eF_{12}y_{xy} = −dy_{xy} + cx_{xy}. But Λx_{xy} ∩ Λy_{xy} = 0 by (4.2.1). Therefore ent c ⊆ 𝔪. A second look at coordinate 2 therefore shows that ent(eF_{12}y_{xy}) ⊆ 𝔪^{2} for every e. Hence ent F_{12} ⊆ 𝔪 as desired.

Since ent c ⊆ 𝔪, we have c̄ = 0. But ent d ⊆ 𝔪 also, so (4.7.4) now yields

ent eF_{13}y ⊆ ent dx ⊆ 𝔪x = Λxy

for every e. Let c be any element of ent F_{13}. Then cy ∈ Λxy, so that c ∈ 𝔪. Write c in the standard form (4.1.1). Then

cy = u_{1}xy + u_{2}y^{2} ∈ Λxy

Directness of the sum Λxy+Λy^{2} then shows that u_{2} is not a unit, and hence u_{2} = 0. Therefore c ∈ (x, y^{2}) and hence ent F_{13} ⊆ (x, y^{2}) as desired.

F_{23}. Note that (0, exy, 0)ρ_{1} = 0 for every e because (xy)_{xy} = 0. Therefore, as before, (0, exy, 0)F ρ′_{1} ⊆ ker ρ′_{2} and hence

(0, 0, eF_{23}xy) ∈ ker ρ′_{2} (4.7.5)

for every e. As before, this triple must have the form (4.5.2). Looking at coordinate 1 in both triples shows that ent b ⊆ 𝔪, hence term 1 of (4.5.2) equals zero and we can ignore it. Coordinate 2 then yields −dy_{xy} + cx_{xy} = 0. Since Λx_{xy} ∩ Λy_{xy} = 0 we have

cx_{xy} = 0  and  dy_{xy} = 0 (4.7.6)

In particular, ent c ⊆ 𝔪 and ent d ∈ 𝔪. Write an arbitrary entry d of d in standard form d = u_{1}x+u_{2}y +u_{3}xy +u_{4}y^{2}. Then (4.7.6) yields 0 = dy_{xy} = u_{2}y_{xy}^{2}. Therefore u_{2} is not a unit, and hence u_{2} = 0. We conclude that d ∈ (x, y^{2}). That is, ent d ⊆ (x, y^{2}), from which it follows that dx = 0. Since ent c ⊆ 𝔪 we have c̄ = 0 and hence coordinate 3 of (4.5.2) equals 0. Therefore (4.7.5) yields eF_{23}xy = 0 for every e. This, in turn, implies that ent F_{23} ⊆ 𝔪, as desired.

F̄_{11}i_{m′,n′} = i_{m,n}F̄_{22}. The first amalgamation term in (4.5.2) is carried by ρ to zero: (ex, −(ei_{m,n})y^{2}, 0)ρ = 0 for every e. Therefore (ex, −(ei_{m,n})y^{2}, 0)F ρ′_{1} ∈ ker ρ′_{2}. Writing F in block form and using the definition of ρ′_{2} then shows

(eF_{11}x_{y} − (ei_{m,n})F_{21}y_{y}^{2}, eF_{12}x_{xy} − (ei_{m,n})F_{22}y_{xy}^{2}, eF_{13}x − (ei_{m,n})F_{23}y^{2}) ∈ ker ρ′_{2}

Since y_{y} = 0, F_{12}x_{xy} ∈ 𝔪x_{xy} = 0, ent F_{13} ⊆ (x, y^{2}), and ent F_{23} ⊆ 𝔪 this simplifies to

(eF_{11}x_{y}, −(ei_{m,n})F_{22}y_{xy}^{2}, 0) ∈ ker ρ′_{2} (4.7.7)

We have Λx ∩ Λy^{2} = Λx ∩ Λy ∩ Λy^{2} = Λxy ∩ Λy^{2} = 0 by (4.2.1). Comparing coordinate 3 of (4.7.7) with coordinate 3 of (4.5.2) yields dx−(c̄ϕ)y^{2} = 0; therefore dx = 0, and hence ent d ⊆ 𝔪. Then writing a typical entry of d in standard form (4.1.1), together with the relation dx = 0, shows that ent d ⊆ (x, y^{2}). Therefore term 2 in (4.5.2) equals zero.

Coordinate 2 of (4.7.7) and (4.5.2) now yields

−(ei_{m,n})F_{22}y_{xy}^{2} = −(bi_{m′,n′})y_{xy}^{2} + cx_{xy}

Then Λx_{xy} ∩ Λy_{xy}^{2} = 0 yields

(ei_{m,n})F_{22}y_{xy}^{2} = (bi_{m′,n′})y_{xy}^{2} (4.7.8)

Coordinate 1 yields eF_{11}x_{y} = bx_{y}. Then (4.7.2) yields ēF̄_{11} = b̄ and hence ēF̄_{11}i_{m′,n′} = b̄i_{m′,n′}. But (4.7.8) yields (ēi_{m,n})F̄_{22} = b̄i_{m′,n′} (also by (4.7.2)). We conclude that ēF̄_{11}i_{m′,n′} = (ēi_{m,n})F̄_{22} for every e, and therefore F̄_{11}i_{m′,n′} = i_{m,n}F̄_{22}, proving (a).

F̄_{22} = F̄_{33}. We use the second amalgamation term in (4.5.2): (0, −ey, ex)ρ = 0 and therefore (0, −ey, ex)F ρ′_{1} ⊆ ker ρ′_{2} for every e. Writing F in block form and applying the definition of ρ′_{1} yields

(…, −eF_{22}y_{xy} + eF_{32}x_{xy}, −eF_{23}y + eF_{33}x) ∈ ker ρ′_{2} (4.7.9)

Comparing coordinate 2 in this and (4.5.2) and multiplying by y yields eF_{22}y_{xy}^{2} = dy_{xy}^{2}. Then (4.7.2) yields ēF̄_{22} = d̄.

Comparing coordinate 3 and multiplying by y yields −eF_{23}y^{2} + eF_{33}xy = dxy. Since Λxy ∩ Λy^{2} = 0, this yields ēF̄_{33} = d̄ (by (4.7.2)). Therefore ēF̄_{22} = ēF̄_{33} for every e, and hence F̄_{22} = F̄_{33}, proving (b).

ϕF̄_{22} = F̄_{22}ϕ′. Use the third amalgamation term in (4.5.2): (0, ex, −(ēϕ)y^{2})ρ = 0 and hence (0, ex, −(ēϕ)y^{2})F ρ′_{1} ⊆ ker ρ′_{2}. As before this yields

(…, eF_{22}x_{xy} − (ēϕ)F_{32}y_{xy}^{2}, eF_{23}x − (ēϕ)F_{33}y^{2}) ⊆ ker ρ′_{2} (4.7.10)

and this therefore has the form (4.5.2) for all e. Comparing coordinate 2 in these expressions yields

eF_{22}x_{xy} − (ēϕ)F_{32}y_{xy}^{2} = −(bi′)y_{xy}^{2} − dy_{xy} + cx_{xy} (4.7.11)

Since Λx_{xy} ∩ Λy_{xy} = 0, comparing the x_{xy}-terms on both sides yields eF_{22}x_{xy} = cx_{xy}, and therefore ēF̄_{22} = c̄ (by (4.7.2)). We conclude that

ēF̄_{22}ϕ′ = c̄ϕ′ (4.7.12)

for all e.

Comparing coordinate 3 in (4.7.10) and (4.5.2) yields

eF_{23}x − (ēϕ)F_{33}y^{2} = dx − (c̄ϕ′)y^{2} (4.7.13)

As already observed, Λx ∩ Λy^{2} = 0. Therefore (ēϕ)F_{33}y^{2} = (c̄ϕ′)y^{2}, so that by (4.7.2) we get (ēϕ)F̄_{33} = c̄ϕ′. Comparing this with (4.7.12) yields (ēϕ)F̄_{33} = ēF̄_{22}ϕ′ for every e, and hence ϕF̄_{33} = F̄_{22}ϕ′. Since F̄_{33} = F̄_{22}, statement (c) follows, and the proof of part (i) of the lemma is now complete.

(ii) We assume that f is an isomorphism. Since ρ and ρ′ are projective covers (see (4.5.4)), the map F is again an isomorphism, and therefore the matrix F is invertible. Hence F̄ is invertible. But F̄ is a block-triangular matrix over the field k, by part (i) of this proof, and hence each diagonal block F̄_{ii} is invertible. Therefore F_{ii} is also invertible.

Theorem 4.8 (Wildness of Drozd Rings). Let (Λ, 𝔪, k) be a Drozd ring and W the full subcategory of mod Λ consisting of all modules of the form M(m, n, ϕ). Then there is an addtive functor E: W → S(1\frac{1}{2}) that is a representation equivalence.

Proof. Let M = M(m, n, ϕ) and M′ = M(m′, n′, ϕ′) be objects in W. With notation as in Definition 4.3, set E(M) = (m, n, ϕ) ∈ S(1\frac{1}{2}). Let f: M → M′ be a morphism in W, that is, an Λ-module homomorphism. We wish to define the morphism E(f) = (σ, τ); that is, we wish to define σ and τ in diagram (4.3.1).

Take projective covers ρ: P → M and ρ′ : P′ → M′. Then f lifts to a homomorphism F: P → P′ such that diagram (4.5.5) commutes. The matrix of F (again called F) has a 3 × 3 block upper triangular form, as displayed in (4.7.1). Then we define

E(f) = (F̄_{11}, F̄_{22}) (4.8.2)

We proceed to prove that E has the required properties.

First we claim that E(f) is well-defined; that is, choosing a different lifting F of f does not change E(f). It suffices to show that if f = 0 then F̄_{11} and F̄_{22} equal 0. But f = 0 implies that F (P) ⊆ ker ρ′ ⊆ 𝔪P′, and hence all entries of the matrix F are elements of 𝔪. Hence the entire matrix F̄ equals 0, proving the claim.

Next, we claim that E(f) is a morphism in S(1\frac{1}{2}), that is, that diagram (4.3.1) commutes. For that, we must show that, for every b ∈ Λ^{(m)} and c ∈ Λ^{(n)}, both b̄iF̄_{22} = b̄F̄_{11}i′ and c̄ϕF̄_{22} = c̄F̄_{22}ϕ′. But these two equations follow immediately from parts (i)(a) and (i)(c) of Lemma 4.7.

Since E is well-defined on morphisms, it is now easy to complete the proof that E is an additive functor. For example the identity map on a module can be lifted to the identity map on its projective cover and then again becomes an identity map modulo 𝔪, showing that E(1) = 1. Similarly E preserves compositions and sums of morphisms.

We now check the three defining properties of a representation equivalence. Dense, that is, E maps onto all isomorphism classes. This is clear from the definition of W as the full subcategory of mod Λ consisting

of all modules of the form M(m, n, ϕ).

Full, that is, E is a surjection on hom groups.

Let (σ, τ) be a morphism in S(1\frac{1}{2}), as displayed in (4.3.1). Each of σ, τ equals right multiplication by a k-matrix that we again call σ, τ respectively. Let F be a block diagonal matrix over Λ such that F̄_{11} = σ, F̄_{22} = τ, F_{33} = F_{22}, and all other entries of F are zero. Right multiplication by F is a homomorphism of free Λ-modules P → P′. To show that F induces an Λ-homomorphism M → M′ it suffices to check that (ker ρ)F ⊆ ker ρ′. The block diagonal form of F shows that F induces an Λ-module homomorphism F′: Q = P ρ_{1} → Q′ = P′ρ′_{1}. Thus it now suffices to check that the induced map F′ induces a homomorphism M → M′. For this, it suffices to check that F′ takes each of the three terms of (4.5.2) to terms of the same form with respect to M′. First we have

(bx_{y}, −(bi)y_{xy}, 0)(F_{11} ⊕ F_{22} ⊕ F_{33}) = (bF_{11}x_{y}, −(bi)F_{22}y_{xy}, 0)

Since we are working in socles of Λ-modules, we can replace F_{11} and F_{22} by their images mod 𝔪. Moreover, since (F̄_{11}, F̄_{22}) = (σ, τ) is a morphism in S(1\frac{1}{2}), we have iF̄_{22} = F̄_{11}i′, and this shows that the form of (bx_{y}, −(bi)y_{xy}, 0) is preserved.

The form of the second term in (4.5.2) is preserved because F_{22} = F_{33}. For the third term we have

(0, cx_{xy}, −(c̄ϕ)y^{2})(F_{11} ⊕ F_{22} ⊕ F_{33}) = (0, cF_{22}x_{xy}, −(c̄ϕ)F_{33}y^{2})

Since x_{xy} and y^{2} are in the socles of their respective modules, we can replace each of F_{22} and F_{33}, on the right-hand side, by their respective images (both τ) modulo 𝔪. Since ϕτ = τϕ′ it is clear that the form of (0, cx_{xy}, −(c̄ϕ)y^{2}) has been preserved.

Faithful, that is, E(M) ≅ E(M′) if and only if M ≅ M′.

Since E is a functor, clearly E(M) ≅ E(M′) if M ≅ M′.

For the converse, suppose that (σ, τ) is an isomorphism from E(M) to E(M′) in the category S(1\frac{1}{2}). Let F be a block diagonal matrix over Λ such that F̄_{11} = σ, F̄_{22} = τ, F_{33} = F_{22}, and all other entries of F are zero; as in the proof of “full” above, right multiplication by F induces a Λ-homomophism f : M → M′ such that E(f) = (σ, τ). But (σ, τ) is an isomorphism in the category S(1\frac{1}{2}), and so σ and τ are invertible matrices over the field k. Then the matrix F is invertible, too. The same argument shows that the (block diagonal) matrix F^{−1} induces a Λ-homomorphism g : M → M′ such that E(g) = (σ^{−1}, τ^{−1}) = (σ, τ)^{−1}. Moreover, since F F^{−1} is the identity map on P (the projective cover of M), it follows that the induced map fg is the identity map on M. Similarly, gf is the identity map on M′, and hence f is an isomorphism between M and M′.

This completes the proof that Drozd rings are finite-length wild. Combined with the fact that artinian triads are finite-length wild [GLW, Lemma 3], together with the fact that any ring mapping onto a finite-length wild ring is also finite-length wild, we refine our Dichotomy Theorem 3.1 as follows.

Corollary 4.9. If (Λ, 𝔪, k) is a complete, local ring, then Λ is either finite-length wild, or a Klein ring or a homomorphic image of a complete split or unsplit Dedekindlike ring.

In [KL2] we complete the picture by showing that Klein rings and homomorphic images of complete Dedekind-like rings are finitely generated tame (with a possible exception involving characteristic 2, as noted in the introduction).

5. Examples and Miscellaneous Results

Among the results in this sections are examples of tame and wild rings, especially examples that are not algebras over fields. We also give our promised construction of Drozd rings as subrings of artinian valuation rings that serve as a kind of “artinian normalization” [Theorem 5.11]. Other results relate the rings of this paper to each other and to ramified orders, Krull dimension, and infinite lattice type.

For further examples of tame rings, see the last section of [KL2]. Comments about actual tameness anticipate results from later papers in this series.

Example 5.1. The simplest example of a Drozd ring that is not an algebra over a field, is the ring A_{p} = ℤ[X]/(X^{2}, p^{3}, p^{2}X), where p denotes any prime number.

Example 5.2 (Superwild). The innocuous-looking ring Λ = ℤ[X]/(X^{2}) is superwild in the sense that, for every prime number p, Λ has a maximal ideal 𝔪 such that Λ is finite length 𝔪-wild and Λ/𝔪 has characteristic p. This follows because, for each prime number p, the ring A_{p} in Example 5.1 is a Drozd ring with residue field ℤ̄_{p} = ℤ/(p), and A_{p} is a homomorphic image of Λ.

Example 5.3 (Split Dedekind-like). Let Λ = ℤ̂_{p}[[X]]/(pX) where p is any prime number, ℤ̂_{p} denotes the p-adic integers, and [[…]] denotes formal power series. Then Λ is a complete split Dedekind-like ring.

To see this, let V_{1} = ℤ̂_{p}[[X]]/(X) ≅ ℤ̂_{p} and V_{2} = ℤ̂_{p}[[X]]/(p) ≅ ℤ̄_{p}[[X]], both of which are complete DVRs. The natural map ρ: Λ → V_{1} ⊕ V_{2} is clearly one-to-one, and it is not difficult to verify that ρ maps Λ onto the split Dedekind-like ring described in (2.3.2) and (2.3.3), with k = ℤ̄_{p} = ℤ/(p) and f_{1} and f_{2} the natural homomorphisms.

This example is, in a sense, as far as possible from being an algebra over a field, since V_{1} and V_{2} have characteristics 0 and p, respectively. Displaying it in the present form — as opposed to the form (2.3.2) — shows it to be the numbertheoretic analog of the split Dedekind-like ring k[[X, Y]]/(XY).

Example 5.4 (Klein versus split Dedekind-like). For each prime number p, let Λ_{p} = ℤ[X]/(X^{2}, p^{2}), a local artinian ring with composition length four and maximal ideal 𝔪_{p} = (x, p̄), where x and p̄ denote the images of X and p respectively, in Λ_{p}.

Clearly Λ_{p} is a proper homomorphic image of the Drozd ring A_{p} constructed in Example 5.1. By the Dichotomy Theorem 3.1, Λ_{p} must be either a Klein ring or a homomorphic image of a split or unsplit Dedekind-like ring. In fact, both possibilities occur for suitable values of p.

First we claim that Λ_{2} is a Klein ring of characteristic 4. We check that the square of every element of 𝔪_{2} is zero (and omit the remaining easy details). For any w = ax + ¯2b ∈ 𝔪_{2} we have w^{2} = (ax + ¯2b)^{2} = a^{2}x^{2} + ¯4abx + ¯4b^{2} = 0 in Λ_{2} = ℤ[x]/(x^{2}, 4). Note that this fails when p ≠ 2.

Next we claim that, for p ≠ 2, Λ_{p} is a homomorphic image of a split Dedekindlike ring. (Our proof must fail for p = 2, by Proposition 5.5 below.) Let Ω be the split Dedekind-like ring defined by the conductor square

\begin{matrix}Ω & ⟶ & ℤ_{p} × ℤ_{p} \\ ^{↓ ↓} & & ^{↓ ↓}ρ \\ ℤ̄_{p} & ⟶^{ι} & ℤ̄_{p} × ℤ̄_{p} \end{matrix} (5.4.1)

where ℤ_{p} denotes the p-localization of the integers ℤ, ρ is the natural map and ι is the diagonal embedding.

Since Ω is a free ℤ_{p}-module of rank two, with basis [1, 1], [p, 0], we can define a ℤ_{p}- module homomorphism ϕ : Ω ⟶ Λ_{p} induced by ϕ[1, 1] = ¯1 and ϕ[p, 0] = \frac{1}{2}(p̄− x). Clearly ϕ is surjective; we claim that it is in fact a ring homomorphism. For this, it suffices to check that ϕ is multplicative on the above basis. It is enough to notice that (ϕ[p, 0])^{2} = (\frac{1}{2}(p̄− x))^{2} = \frac{1}{4}p̄^{2} − \frac{1}{2} p̄x + \frac{1}{4}x^{2} = −\frac{1}{2} p̄x and ϕ([p, 0]^{2}) = pϕ[p, 0] = \frac{1}{2}p̄^{2} − \frac{1}{2} p̄x = −\frac{1}{2} p̄x, using the fact that p̄^{2} = 0 = x^{2} in Λ_{p}.

Note that we can map a complete Dedekind-like ring onto Λ_{p} by taking 𝔪_{Ω}-adic completions in the map ϕ : Ω ⟶ Λ_{p} (since artinian local rings are unaffected by the completion process).

Our next few results are related to the assertion, in Definition 2.3, that Klein rings are sometimes maximal tame rings, that is (in addition to being tame, as shown in the next paper of this series), they are sometimes homomorphic images of Dedekind-like rings and sometimes not.

Proposition 5.5. A Klein ring is never a homomorphic image of a split Dedekindlike ring.

Proof. Let (Ω, 𝔪, k) be a split Dedekind-like ring and Λ a Klein ring, and suppose that there is a surjective ring homomorphism ϕ : Ω → → Λ. Then there would be a surjective homomorphism of the completion Ω̂ onto Λ, since artinian rings are already complete. Therefore we may assume that Ω is complete.

Since Ω is complete can write 𝔪 = (x, y), where (see (2.3.3)) xy = 0. It follows that the maximal ideal of Λ is ϕ(𝔪) = (ϕ(x), ϕ(y)). But then ϕ(𝔪)^{2} = (ϕ(x), ϕ(y))^{2} = (ϕ(x)^{2}, ϕ(x)ϕ(y), ϕ(y)^{2}) = 0, since xy = 0, and all of the elements of the maximal ideal of any Klein ring square to zero. This contradicts the fact that the square of the maximal ideal of any Klein ring is non-zero.

Example 5.6 (Klein, not homomorphic image of Dedekind-like). Let k be an algebraically closed field of characteristic 2, and G the Klein four-group. It is wellknown (and easy to verify) that the group algebra kG is a Klein ring. By Proposition 5.5, kG is not a homomorphic image of a complete split Dedekind-like ring. Since the field k is assumed to be algebraically closed, there is no complete unsplit Dedekind-like ring with residue field k, and hence kG is not a homomorphic image of a complete unsplit Dedekind-like ring, either.

Example 5.7 (Klein, homomorphic image of Dedekind-like). Let k be an imperfect field of characteristic 2 with inseparable quadratic field extension F = k(ϵ) (so that ϵ^{2} ∈ k), and let Ω be the subring of F [[X]] consisting of those power series in F with constant term in k. The ring Ω is complete unsplit Dedekind-like, defined by the conductor square

\begin{matrix}Ω & ⟶ & F [[X]] \\ ^{↓ ↓} & & ^{↓ ↓}ρ \\ k & ⟶^{ι} & F \end{matrix} (5.7.1)

where ι is the inclusion map and ρ is the natural map sending a power series to its constant term. The maximal ideal of Ω is 𝔪 = (X, ϵX). We compute that 𝔪^{2} = (X^{2}, ϵX^{2}) and 𝔪^{3} = (X^{3}, ϵX^{3}), using the fact that ϵ^{2} ∈ k. Set Λ = Ω/(𝔪^{3} + (X^{2})), a local artinian ring with maximal ideal 𝔪/(𝔪^{3} + (X^{2})). Since 𝔪^{3} ⊂ 𝔪^{3} + (X^{2}) ⊂ 𝔪^{2} are proper inclusions, it follows that Λ = Ω/(𝔪^{3} + (X^{2})) has composition length four. Moreover, for any element w = aX + bϵX ∈ 𝔪, we get w^{2} = a^{2}X^{2} + b^{2}ϵ^{2}X^{2} ∈ (X^{2}), (since ϵ^{2} ∈ k ⊆ Ω, and using the fact that the field F has characteristic 2). Thus, Λ is a Klein ring.

Remark 5.8 (Maximal tame?). We do not know whether the unsplit Dedekindlike ring Ω in the previous example is tame or wild or neither, because the field extension in the bottom row of its pullback square is an inseparable field extension of dimension 2. On the other hand, we know that all Klein rings are tame. Thus the (tame) Klein ring Λ in the previous example might be “maximal tame” even though it is a homomorphic image of an unsplit Dedekind-like ring.

Remark 5.9 (Ramifiation). Three basic concepts in algebraic number theory — relating an order to its normalization — are those of splitting of a maximal ideal, residue-field growth, and ramification. The first two of these concepts occur in the definitions of split and unsplit Dedekind-like rings. But ramification has not occurred thus far in the present paper. In fact: if (Λ, 𝔪, k) is any complete local order that ramifies in its normalization Γ, in the sense that 𝔪 is contained in the square of some maximal ideal of Γ, then Λ is finite-length wild.

To prove this, note that our Dichotomy Theorem shows that Λ maps onto an artinian triad or Drozd ring, and hence is finite-length wild.

In particular, if μ_{Λ}(𝔪) = 2, then Λ cannot map onto a triad, and therefore maps onto a Drozd ring. It is interesting that all Drozd rings arise in this way, as our next result shows.

Proposition 5.10 (Ramified onto Drozd). Let (Λ, 𝔫, k) be a Drozd ring. Then Λ is a homomorphic image of a completely ramified integral domain Ω. In more detail, there is a commutative diagram of ring homomorphisms (5.10.1) in which:

(i) (Ω, 𝔪, k) is a complete local domain whose normalization is a DVR (Γ, (z), k) such that 𝔪 = Γz^{2};

(ii) The vertical maps are surjections; and the left-hand vertical maps are the

restrictions of ρ′ and ρ′′; (iii) The horizontal maps are inclusions.

Ω ⟶  Γ  (𝔪 = Γz^{2})
↓ ↓  ^{↓ ↓}ρ′ (ker = 𝔪^{3} = Γz^{6})
Λ ⟶ Γ/(z^{6}) (𝔫 = Γz^{2}/Γz^{6})
↓ ↓  \begin{matrix}↓ & ↓ \\ ρ′′ \end{matrix} (ker = 𝔫)
k ⟶ k + kz̄ (z̄^{2} = 0, z̄ ≠ 0)

(5.10.1)

Proof. The approach is similar to that in the proofs of Propositions 3.4 and 3.6. We can write the maximal ideal of the Drozd ring Λ as 𝔫 = (x, y), where x^{2} = y^{3} = 0 and xy ≠ 0. As in the proofs of Propositions 3.4 and 3.6, by the “Structure Theorem of Complete Local Rings” [N, Theorem 31.1], there is a surjective homomorphism ϕ : V [[X, Y]] → Λ such that ϕ(X) = x and ϕ(Y) = y, where V [[X, Y]] is a power series ring and either V = k or V is a complete DVR of characteristic 0 with residue field k. Moreover, in this latter case, k has characteristic p ≠ 0 and the maximal ideal of V is generated by p.

If V is a field, we set R = V [[X, Y]], a complete, two-dimensional, regular local ring, with maximal ideal (X, Y). If V is a DVR, then again exactly as in the proofs of Propositions 3.4 and 3.6, we note that there are elements A, B ∈ V [[X, Y]] such that ϕ(p − (AX + BY)) = 0, and we set R = V [[X, Y]]/(p − (AX + BY)), a regular local ring by [AM, Theorem 11.22]. In this latter case, we change notation and let ϕ denote the induced map from R onto Λ, and we let X and Y denote the cosets of X and Y, respectively, in R. Thus, in either case we have a surjective homomorphism ϕ : R → Λ such that ϕ(X) = x and ϕ(Y) = y, where R is a complete two-dimensional regular local ring and (X, Y) is the maximal ideal of R.

Now ϕ(X^{2} − Y^{3}) = x^{2} − y^{3} = 0, and so ϕ induces a surjective homomorphism from Ω = R/(X^{2} − Y^{3}) onto Λ. We claim that X^{2} − Y^{3} is irreducible in R, so that Ω is an integral domain.

Suppose, by way of contradiction, that X^{2} − Y^{3} were reducible, say

X^{2} − Y^{3} = (aX + bY)(cX + dY) = acX^{2} + (ad + bc)XY + bdY^{2} (5.10.2)

for some elements aX+bY and cX+dY of the maximal ideal (X, Y), with a, b, c, d ∈ R. Note that (X, Y)^{2} = (X^{2}, XY, Y^{2}) and, since R is a regular local ring of dimension 2, the elements X^{2}, XY, and Y^{2} form a basis of (X, Y)^{2}/(X, Y)^{3}. Thus reading (5.10.2) modulo (X, Y)^{3}, we see that ac is a unit in R, and hence a and c are units. Moreover, by this same reasoning, 1 − ac ∈ (X, Y).

Similarly, looking at the coefficient of Y^{2} and reading (5.10.2) modulo (X, Y)^{3} shows that bd ∈ (X, Y), and hence at least one of b and d is in (X, Y). But then both b and d must be in (X, Y), because otherwise, (using the fact that a and c are both units) the coefficient of XY would be a unit on the right-hand side of (5.10.2) but zero on the left-hand side.

We have (X, Y)^{3} = (X^{3}, X^{2}Y, XY^{2}, Y^{3}) and the elements X^{3}, X^{2}Y, XY^{2}, and Y^{3} form a basis of (X, Y)^{3}/(X, Y)^{4} (again because R is a regular local ring of dimension 2). Rearranging (5.10.2), we get that

(1 − ac)X^{2} − Y^{3} − (ad + bc)XY ≡ 0 (mod(X, Y)^{4}) (5.10.3)

since b and d are in (X, Y), so that bdY^{2} ∈ (X, Y)^{4}. Since 1 − ac, b, and d are all in (X, Y) we can substitute expressions of the form 1 − ac = eX + fY and ad + bc = gX + hY into (5.10.3). Since the coefficient of Y^{3} is a unit, this gives a nontrivial linear relation among the basis elements of (X, Y)^{3}/(X, Y)^{4}, and this contradiction proves our claim that X^{2} − Y^{3} is irreducible in R. Therefore Ω is an integral domain.

Let s and t denote the cosets of X and Y, respectively, in Ω = R/(X^{2} −Y^{3}), and recall that 𝔪 denotes the maximal ideal of Ω, so that 𝔪 = (s, t). Since s^{2} = t^{3}, we have (s/t)^{2} = t. It follows that z = s/t is integral over Ω, and hence Γ = Ω[z] is an integral extension of Ω. Moreover, since z·t = s ∈ 𝔪 and z·s = s^{2}/t = t^{3}/t = t^{2} ∈ 𝔪, we have that 𝔪 is an ideal of Γ also. Since 𝔪 is the maximal ideal of Ω, it follows that 𝔪 is the conductor from Γ to Ω. Also, 𝔪 is the radical of Ω, and Γ is finitely generated as an Ω-module, so 𝔪 is contained in the radical of Γ. We claim:

μ_{Ω}(𝔪) = 2,  z^{2} ∈ 𝔪,  z ∉ 𝔪. (5.10.4)

We already have that 𝔪 = (s, t) as an ideal of Ω. The fact that two generators are required follows from the fact that 𝔪 maps onto the maximal ideal of the Drozd ring Λ. Next, note that z^{2} = (s/t)^{2} = t ∈ 𝔪. Finally, suppose z ∈ 𝔪. Then s/t = z implies that s = tz ∈ 𝔪^{2}, which yields the contradiction that the ideal 𝔪 = (s, t) of Ω is principal.

Since Γ = Ω[z], (5.10.4) yields

(5.10.5) Γ/𝔪 = k + kz̄ with z̄^{2} = 0 and z̄ ≠ 0

This yields the first equality in

rad(Γ) = 𝔪 + Γz = Γz (5.10.6)

The second equality holds since s = z · t ∈ Γz and t = z^{2} ∈ Γz.

The fact that rad(Γ/𝔪) = (z̄) is the maximal ideal of Γ/𝔪 now shows that rad(Γ) = Γz is the maximal ideal of Γ; that is, Γ is a local domain with principal maximal ideal Γz. Therefore Γ is a DVR and hence is the normalization of Ω.

Next we prove that 𝔪 = Γz^{2}. This holds by (5.10.4) and (5.10.6), together with the fact that Γ is a DVR.

We have now completed the proof of the top line in diagram (5.10.1). We also have established the existence of a surjective ring homomorphism ρ′: Ω → → Λ. The map ρ′ takes 𝔪 = 𝔪_{Ω} onto 𝔫 = 𝔪_{Λ}; and 𝔫^{3} = 0 since Λ is a Drozd ring. We conclude that 𝔪^{3} ⊆ ker ρ′. We claim that equality holds.

It suffices to prove that the composition length of Ω/𝔪^{3} equals the length of Λ, namely 5. The length of Ω/𝔪^{3} is ≥ 5 since Ω maps onto Λ. Therefore it suffices to prove that the length of the Ω-module Ω/𝔪^{3} is < 6.

Since Ω is properly contained in Γ, it suffices to show that the length of the Ω-module Γ/𝔪^{3} equals 6. Now, the simple Γ-module k remains simple as an Ωmodule by (5.10.5) and the fact that Ω/𝔪 = k. Therefore it suffices to show that the length of the Γ-module Γ/𝔪^{3} equals 6; and this follows from the fact that Γ is a DVR whose maximal ideal Γz satisfies Γz^{2} = 𝔪.

We have now established the existence of the top commutative square in (5.10.1). The existence of the bottom commutative square now follows immediately from (5.10.5), completing the proof of the theorem.

By an AVR (artinian valuation ring) of length n we mean a local principal ideal ring (V, (z), k) whose lattice of ideals is V ⊃ V z ⊃ V z^{2} ⊃ … ⊃ V z^{n} = 0. We say that V contains its residue field if V contains a subfield that maps isomorphically onto k via the natural homomorphism V → → k.

Theorem 5.11 (Construction of Drozd rings from AVRs). Let (V, (z), k) be an AVR of length 6 such that the ring V/(V z^{2}) contains its residue field (and hence V/(z^{2}) = k + kz̄ with z̄^{2} = 0 and z̄ ≠ 0). Then:

(i) The pullback Λ of diagram (5.11.1) is a Drozd ring; and every Drozd ring arises in this way.

(ii) V = Λ + Λz and 𝔪_{Λ} = Λx + Λy where x = z^{3}, y = z^{2}, x ∈ 𝔪_{V} − 𝔪_{V}^{2} , and x^{2} = 0.

\begin{matrix}Λ & ⟶ & V \\ ^{↓ ↓} & & ^{↓ ↓}ρ & (ker = V z^{2}) \\ k & ⟶ & k + kz̄ \end{matrix} (5.11.1)

Proof. The fact that every Drozd ring arises in this way is expressed in the bottom square of diagram (5.10.1). Suppose, conversely, that Λ is the pullback of diagram (5.11.1). We want to prove that Λ is a Drozd ring satisfying conditions (ii).

Since the image of the left-hand vertical map in (5.11.1) is the field k and the kernel of this map is nilpotent, we see that Λ is a local ring with maximal ideal 𝔪_{Λ} = V z^{2} and residue field k. Moreover, V = Λ + Λz holds because it obviously holds modulo V z^{2}. Multiplying by z^{2} then shows

𝔪_{Λ} = Λz^{2} + Λz^{3} (5.11.2)

To see that 𝔪_{Λ} requires two generators (as a Λ-module), note that 𝔪_{Λ}/𝔪_{Λ}^{2} = (V z^{2})/(V z^{4}). Since V is an AVR, this is V -isomorphic to V/(V z^{2}), a k-vector space of dimension 2 by (5.11.1). Similarly, we check that 𝔪_{Λ}^{2}/𝔪_{Λ}^{3} = V z^{4} has k-dimension 2, and hence 𝔪_{Λ}^{2} requires two generators as a Λ-module.

Thus, to complete the proof that Λ is a Drozd ring, it suffices to note that x = z^{3} is an element of 𝔪 − 𝔪^{2} = V z^{2} − V z^{4} (obvious, since V is an AVR), x^{2} = 0, and 𝔪_{Λ}^{3} = V z^{6} = 0.

Examples 5.12 (Drozd vs. AVR). We display the two Drozd rings explicitly discussed in this paper, as subrings of their AVR “normalization”, V.

(i) Λ = k +kx+ky +kxy +ky^{2}, the Drozd k-algebra originally studied by Drozd. Here k is any field, and all monomials in x and y other than the displayed ones equal 0. Choose V = k[z] where z satisfies the defining relation z^{6} = 0. Then Λ is the k-subalgebra with basis 1, z^{2} = y, z^{3} = x, z^{4} = y^{2}, z^{5} = xy.

(ii) Λ = ℤ[X]/(X^{2}, p^{3}, p^{2}X), the Drozd ring in Example 5.1.

Choose V = ℤ̄_{p^{3}}[W]/(W^{6}, W^{2} − p̄) where ℤ̄_{p^{3}} denotes the integers modulo p^{3} and W is an indeterminate. Letting w be the natural image of W in V, we see that V is an AVR of length 6 with composition series

V ⊃ V w ⊃ V w^{2} = V p̄ ⊃ V w^{3} = V pw ⊃ V w^{4} = V p̄^{2} ⊃ V w^{5} = V p̄^{2}w ⊃ 0 (5.12.1)

and V/V w^{2} contains its residue field ℤ̄_{p}.

The subring of V additively spanned by the monomials 1, w^{2} = p̄, w^{3} = p̄w, w^{4} = p̄^{2}, w^{5} = p̄^{2}w is isomorphic to Λ, under the correspondence p̄ → p̄ and p̄w → x.

We conclude this section with two results that give an indication of the ubiquity of wildness.

Theorem 5.13. Every complete local ring (Λ, 𝔪, k) of Krull dimension greater than one is finite-length wild.

Proof. Dedekind-like rings have Krull dimension 1, and Klein rings are artinian, hence have Krull dimension 0. Therefore, by our Dichotomy Theorem 3.1, Λ maps onto an artinian triad or a Drozd ring, and hence is finite-length wild.

If the local ring Λ is reduced (no nilpotent elements ≠ 0), then submodules of free modules of finite rank are often called lattices. (This is equivalent to the definition of “lattice” used in integral representation theory.) Λ is said to have infinite lattice type if it has infinitely many indecomposable lattices. There exist many such Λ with infinite but tame lattice type. However:

Theorem 5.14. Let (Λ, 𝔪, k) be a complete local reduced ring. If Λ has infinite lattice type, then Λ is finite-length wild.

Proof. First note that Λ is not artinian, since then it would be field, and hence have finite lattice type.

It is easiest to prove the contrapositive of our theorem, so suppose that Λ is not finite-length wild. Then Λ does not have an artinian triad or a Drozd ring as homomorphic image. By the Dichotomy Theorem 3.1 — and the fact that Λ is not artinian — Λ must be a homomorphic image of a complete split or unsplit Dedekind-like ring. Since Λ is reduced, it must therefore be either one of these types of Dedekind-like rings or else a DVR. Since DVRs are well-known to have finite lattice type, we may suppose that Λ is Dedekind-like, with normalization Γ.

Then μ_{Λ}(Γ/Λ) = 1 and μ_{Λ}(rad(Γ/Λ)) = 0 (since 𝔪 · Γ/Λ = 0, so that Γ/Λ is a k-module, hence a simple Λ-module). Therefore, Λ satisfies the Drozd-Roiter conditions and hence, by a result of Cimen and Wiegand (extending earlier results of Drozd, Roiter, Jacobinski, Green, and Reiner on lattices over orders), must have finite lattice type [CWW].

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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