PACKAGE DEAL THEOREMS AND SPLITTING ORDERS IN DIMENSION 1

LAWRENCE S. LEVY AND CHARLES J. ODENTHAL

March, 1995

Abstract. Let Λ be a module-finite algebra over a commutative noetherian ring R of Krull dimension 1. We determine when a collection of finitely generated modules over the localizations Λ_{m}, at maximal ideals of R, is the family of all localizations M_{m} of a finitely generated Λ-module M. When R is semilocal we also determine which finitely generated modules over the J(R)-adic completion of Λ are completions of finitely generated Λ-modules.

If Λ is an R-order in a semisimple artinian ring, but not contained in a maximal such order, several of the basic tools of integral representation theory behave differently than in the classical situation. The theme of this paper is to develop ways of dealing with this, as in the case of localizations and completions mentioned above. In addition, we introduce a type of order called a “splitting order” of Λ that can replace maximal orders in many situations in which maximal orders do not exist.

1. Introduction

The original motivation for this paper comes from the companion paper [LO] in which the authors determine which R-orders Λ in semisimple artinian rings satisfy the Krull-Schmidt theorem for f.g. (finitely generated) modules; and which such Λ satisfy the Krull-Schmidt theorem for f.g. torsionfree modules. Since we do not assume that Λ is contained in a maximal order, it is necessary to rebuild a number of the basic tools of integral representation theory, finding answers to the questions mentioned in the abstract of this paper. We hope that these tools have some interest in their own right, and we therefore go beyond the requirements of [LO] in a number of places. In any event, clarity of exposition seems to require separating the general tools studied here from those directly related to the Krull-Schmidt problem.

Notation 1.1. The following notation will remain in effect throughout this paper. R denotes a commutative noetherian ring of Krull dimension 1, and Λ denotes a module-finite R-algebra. Q = Q(R) denotes the (finite) set of minimal prime ideals of R; and R_{Q} and Λ_{Q} denote the localizations obtained by inverting the elements of R − ⋃ Q.

1991 Mathematics Subject Classification. 16P40, 16P50,16W60,13E05, 13B30, 13J10.

Key words and phrases. localization, completion, normalization, splitting order, package deal.

Levy’s research was partially supported by NSF and NSA grants.

Since R_{Q} is an artinian ring, so is Λ_{Q}. When Λ has no artinian ring direct summands (e.g. in integral representation theory) we think of Λ_{Q} as the natural artinian quotient ring of Λ, because Λ_{Q} = Λ_{Q(R)} is then independent of the particular ring R over which Λ is a module-finite algebra. See Lemma 1.5 and Remark 1.2. Note that the natural map Λ → Λ_{Q} is not necessarily one-to-one. The purpose of Λ_{Q} is the same as that of the quotient field of an integral domain: to establish a link, via fraction-formation, between rings of dimension 0 and dimension 1. The fact that the natural map Λ → Λ_{Q} can fail to be one-to-one is no disadvantage in this regard.

The Λ-modules that really interest us, in this paper, are always f.g. But we have to look at non-f.g. modules often enough that (unlike in [LO]) we say f.g. when we mean it.

We say that an R-module M is Q-torsionfree if the elements of R − ⋃ Q act as non-zerodivisors on M, that is, if the natural map M → M_{Q} is one-to-one. Some care is required in dealing with this notion, since R itself can fail to be Q-torsionfree. This may seem disconcerting at first. But we show that, for any f.g. R-module M, M_{m} is Q-torsionfree for almost all maximal ideals m of R, a simple fact that turns out to be very useful (see Section 2).

When R has no nilpotent elements, R itself becomes a Q-torsionfree R-module. In this situation we relax the terminology, and use the more common term “torsionfree module”. Whenever we refer to a torsionfree or Q-torsionfree Λ-module M, we always mean that M is Q-torsionfree as an R-module. When Λ has no artinian ring direct summands this property of M is independent of the ring R over which Λ is a module-finite algebra, because R_{Q} ⊗_{R} M ≅ Λ_{Q} ⊗_{Λ} M and Λ_{Q} is independent of R. (1.1.1) We call Λ an R-order in the semisimple artinian ring Λ_{Q} if (in addition to

the properties of R and Λ mentioned in the first paragraph of Notation 1.1) R has no nilpotent elements ≠ 0, the canonical map Λ → Λ_{Q} is a monomorphism and the ring Λ_{Q} is semisimple artinian.

The hypothesis about no nilpotent elements is no loss of generality, because any nilpotent element of R·1_{Λ} generates a nilpotent ideal of Λ_{Q} and therefore equals zero.

Here is a summary of the problems considered in this paper.

• Section 2: Packages of Localizations. Consider a family {M(m) | m is a maximal ideal of R} where each M(m) is a f.g. Λ_{m}-module. For which such families is there a f.g. Λ-module M such that M_{m} ≅ M(m) for every m?

When Λ = R and is an integral domain the necessary and sufficient conditions are especially easy to state: (i) almost all (i.e. all but finitely many) of the M(m) must be free R_{m}-modules; and (ii) All of the R_{Q}-vector spaces R_{Q} ⊗_{R} M(m) = M(m)_{Q} must have the same dimension. [Recall that when R is an integral domain R_{Q} is the field of quotients of R.]

In order to state the general result, it is easiest to first specify the f.g. Λ_{Q}-module X that will be isomorphic to the Q-localization of the M that we wish to build. As a first approximation to M, let N be any f.g. Λ-moldule such that N_{Q} ≅ X. (For example, let N be the Λ-module generated by some finite set of Λ_{Q}-generators of X.) We prove that it is possible to change the isomorphism classes of any finite set of localizations N_{m} arbitrarily provided that we do not change (N_{m})_{Q} (hence not change X = N_{Q}). The resulting Λ-modules M yield the most general collection of isomorphism classes {M_{m}} such that M_{Q} ≅ X. The basic restriction that governs this whole section is that, although we can change any finite number of isomorphism classes of localizations arbitrarily (provided that we do not change their Q-localization), we can never change infinitely many of them.

To clarify the spirit of this complicated result, we show how it applies to the special case previously discussed: R = Λ, an integral domain. Condition (ii) is obviously necessary for the existence of M because (M_{m})_{Q} = M_{Q} in this special situation. Conversely, suppose that conditions (i) and (ii) hold. Then, by condition (ii), the Λ_{Q}-module X = M(m)_{Q} is independent of our choice of m, and provides the X needed to begin the construction in the general theorem. Let n be the dimension of this Λ_{Q}-vector space X. Then for our first approximation to the desired M we choose N = Λ^{n}. Almost every M(m) is Λ_{m}-free, by condition (i), and its rank is n by condition (ii). Thus we have N_{m} ≅ M(m) for almost all m. Our ability to arbitrarily change any finite number of localizations of N enables us to correct the fintely many remainimg localizations, and complete the proof that M exists. Moreover, the fact that we can never change infinitely many localizations (without changing M_{Q} = X) shows that condition (i) is also a necessary condition.

But what about the infinitely may isomorphism classes of localizations that we cannot change, when we are not in this special situation? We give a complete answer when Λ has no nilpotent ideals. In the commutative case R = Λ (reduced, but not necessarily a domain) almost every Λ_{m} must still be free, but the rank need not be the same for all m. For noncommutative semiprime Λ we prove that, for almost all m there is a canonical “default” projective Λ_{m}-module that is determined by its Q-localization, and this module replaces the free module in condition (i) of the special case R = Λ.

An application of this is that indecomposability is a semilocal property. That is, if a f.g. Λ-module M is indecomposable, then there is a finite collection M of maximal ideals of R such that the localization M_{M} is an indecomposable Λ_{M}-module, and remains indecomposable if M is enlarged to any larger finite set of maximal ideals. [Here M_{M} = {m/d | m ∈ M and d ∈ R − ⋃ M}.] If Λ is any of the rings studied in integral representation theory and M is a torsionfree Λ-module, this is well-known and easy to see: Λ_{m} is a maximal order for almost all m, and so we can take M to be the remaining maximal ideals. This indecomposability property, in our full generality, is one of the key steps in the reduction of the Krull-Schmidt problem to the case that R is semilocal in [LO].

• Section 3: Completions. Here R is semilocal and Λ̂ denotes the J(R)-adic completion of Λ. In order to define a f.g. Λ-module by specifying its completion, one must know which f.g. Λ̂-modules X are completions of f.g. Λ-modules. As the first step in answering this question the main result of this section shows that X is the J(R)-adic completion of some Λ-module if and only if X_{Q} is the J(R)-induced completion of some Λ_{Q}-module. The special case in which Λ is an R-order in a semisimple artinian ring, Λ is contained in a maximal order, and R is a discrete valuation ring, is often attributed to Heller [H ’61].

When Λ is an R-order in a semisimple artinian ring, as defined in (1.1.1), one can give a more complete answer to the question of when X is the completion of some Λ-module than the main result of the present section gives. We do this in Section 6 using the main result of the present section.

A difficulty in the present section is that the J(R)-induced completion Λ̂_{Q} = R̂ ⊗_{R} Λ_{Q} of Λ_{Q} is not an “adic” completion of a f.g. R_{Q}-module, in the sense of commutative algebra. For example, if Λ = R and is a discrete valuation ring, then R_{Q} is a field; so all elements of the filtration {J(R)^{n}R_{Q} | n = 1, 2, … } equal R_{Q} itself, and hence do not generate a worthwhile topology. Of course, what we are generalizing is the completion of a field with respect to a discrete valuation. It is customary to call this a p-adic completion when it arises in integral representation theory, and a generic formal fiber ring when it arises in current commutative algebra. We call it the J(R)-induced completion of Λ_{Q} and take some space to develop the basic facts we need but could not find elsewhere.

• Section 4: Normalizations. In the noncommutative part of integral representation theory it is customary to work with maximal orders rather than integral closures for two reasons. First is that integral closures in the total quotient ring usually do not exist in a noncommutative setting. The second difficulty is that, even when an integral closure exists (e.g. in the commutative case) it is not necessarily module-finite over the original ring.

Let Λ be an R-order in the semisimple artinian ring Λ_{Q}. In order to deal with the difficulties in the preceding paragraph, a normalization was defined in [GL ’89] to be a ring Γ that is maximal with respect to the properties that Λ ⊆ Γ ⊆ Λ_{Q} and Γ is integral over R. The present section briefly presents the basic properties of normalizations that will be needed subsequently. In particular, normalizations provide a crucial link between our theory and integral representation theory because Γ turns out to be a direct sum of classical maximal orders over Dedekind domains in simple artinian rings. Therefore Γ has all of the familiar ring-theoretical properties of maximal orders; for example, every left ideal is a projective Γ-module.

• Section 5: Splitting Orders. In this section R is semilocal and Λ is an Rorder in the semisimple artinian ring Λ_{Q}. One of the fundamental tools in integral representation theory is the use of a conductor ideal to relate the module theory over Λ to that of a maximal order containing Λ. Let Γ be a normalization of Λ. If Γ is not module-finite over R, there is no common “conductor” ideal C such that both rings Λ/C and Γ/C are artinian. The main result of this section is the existence of what we call a splitting order Ω between Λ and Γ. This Ω is large enough so that: (i) no splitting, ramification, or residue-growth (of maximal ideals) occurs between Ω and Γ, (ii) Ω decomposes like Γ (i.e. primitive idempotents of Ω remain primitive in Γ and Ωd ≅ Ωe ⟺ Γd ≅ Γe for all idempotents d, e ∈ Ω), and (iii) Γ is a centralizing extension of Ω, in fact Γ is generated as a ring by Ω and central elements of Γ. Since Ω is a f.g. R-module, there exist conductor ideals for Λ and Ω. In the proof that splitting orders exist (Theorem 5.2) we make use of the lying-over theorem for prime ideals of centralizing extensions of noncommutative noetherian rings.

We prove that splitting orders have the Krull-Schmidt property for f.g. projective modules. In addition, given any finite set F of torsionfree Λ-modules, Ω can be chosen large enough so that ΩM is Ω-projective for every M ∈ F.

• Section 6: Completions of Orders, Artinian Pairs. In this section R is semilocal and Λ is an R-order in the semisimple artinian ring Λ_{Q}. Let X be a f.g. module over the J(R)-adic completion Λ̂ of Λ. We return to the question of when there exists a f.g. Λ-module M such that X ≅ M̂, the J(R)-adic completion of M. Our answers sharpen the package deal theorem of Section 3 in this less general setting (i.e. orders in semisimple artinian rings) and our proofs use the results of Section 3.

Let Γ be a normalization of Λ. One difference between our situation and the familiar classical one is that, when Γ is not module-finite over R, R̂ has nilpotent elements and therefore Λ̂ = R̂ ⊗_{R} Λ has nilpotent ideals.

A basic necessary condition for the existence of M is that X_{Q} be a projective Λ_{Q}module. This holds because the ring Λ_{Q} is semisimple artinian, hence the Λ_{Q}- module M_{Q} must be projective; and this implies that M̂_{Q} = R̂⊗_{R}M_{Q} is a projective module over Λ̂_{Q} = R̂⊗_{R}Λ_{Q}. We give two answers to the question of when M exists.

The first answer focuses on the ring Γ̈ = R̂ ⊗_{R} Γ. The notation is chosen to emphasize that Γ̈ is not an “adic” completion of Γ when Γ is not module-finite over R. Nor is it noetherian. (We are grateful to W. Heinzer for showing us this latter fact in the comutative case.) However we show that Γ̈ is semiperfect: idempotents can be lifted modulo J(Γ̈), and Γ̈/J(Γ̈) is a semisimple artinian ring. Moreover, Γ̈_{Q} = Λ̂_{Q} since Γ_{Q} = Λ_{Q}; and Λ̂_{Q} is therefore the natural artinian quotient ring of both Γ̈ and Λ̂ when Λ has no artinian ring direct summands.

Let the indecomposable ring direct summands of Γ be Γ_{k} (where k runs through some finite index set). Thus Γ = ⊕_{k} Γ_{k}. As already mentioned in the discussion of normalizations, each Γ_{k} is a classical maximal order over a Dedekind domain in a simple artinian ring. Since R is semilocal, Γ_{k} is a full matrix ring over a (possibly noncommutative) principal ideal domain. Therefore Γ_{k} has, up to isomorphism, only one indecomposable f.g. projective left module. This module is isomorphic to Γ_{k}e_{k} where e_{k} is a primitive idempotent of Γ_{k}. The image ē_{k} of e_{k} in Γ̄_{k} = Γ_{k}/J(Γ_{k}) is again idempotent, but not necessarily primitive. Since the ring Γ̄_{k} is semisimple artinian, we have a decomposition Γ̄_{k}ē_{k} ≅ ⊕_{i} (S_{ki})^{s_{ki}} where each S_{ki} is a simple Γ̄_{k}-module generated by a primitive idempotent of Γ̄_{k}. We call these multiplicities s_{ki} the splitting numbers of Γ.

Let Γ̈_{k} = R̂ ⊗_{R} Γ_{k}. It turns out that Γ̈_{k}/J(Γ̈_{k}) = Γ̄_{k}. Since the ring Γ̈_{k} is semiperfect, and hence idempotents lift from Γ̄_{k} to Γ̈_{k}, each S_{ki} is the image modulo J(Γ̈_{k}) of a unique indecomposable f.g. projective Γ̈_{k}-module Ü_{ki} and every indecomposable f.g. projective Γ̈-module is isomorphic to some Ü_{ki}. Moreover, the Q(R)-localizations (Ü_{ki})_{Q} form a set of representatives of the isomorphism classes of indecomposable f.g. projective Λ̂_{Q} = Γ̈_{Q}-modules. (The proof of this last fact takes more work than one might expect.) In particular, assuming our necessary condition that X_{Q} is Λ_{Q}-projective, there is a decomposition X_{Q} ≅ ⊕_{ki} [(Ü_{ki})_{Q}]^{ρ_{ki}}. We call ((ρ_{ki})) the array of ranks of X.

Our package deal theorem for completions of orders then states that M exists if and only if each row (ρ_{k1}, ρ_{k2}, …) of the array of ranks of X is an integer multiple of the corresponding row (s_{k1}, s_{k2}, …) of the array of splitting numbers of Γ. Conceptually, this states that X_{Q} ≅ R̂ ⊗_{R} P_{Q} for some f.g. projective Γ-module P, exactly as in the case of classical orders!

Our other answer to the question about the existence of M avoids mention of the non-noetherian ring Γ̈, replacing it by Ω̂ = R̂ ⊗ Ω, the J(R)-adic completion of any convenient splitting R-order Ω ⊆ Γ. Since completions of orders are semiperfect, each simple Γ̄_{k}-module S_{ki} lifts to a unique indecomposable f.g. projective Ω̂-module U_{ki}, and the localizations (U_{ki})_{Q} again paramaterize the isomorphism classes of indecomposable f.g. projective Λ̂_{Q}-modules. Thus any f.g. Λ̂-module X such that X_{Q} is Λ_{Q}-projective yields the same array of ranks as before, via the decomposition X_{Q} ≅ ⊕_{ki} [(U_{ki})_{Q}]^{ρ_{ki}}, and the package deal theorem in the previous paragraph again applies. As before, this is equivalent to saying that M exists if and only if X_{Q} ≅ P̂_{Q} (the Q-localization of the completion of P) for some f.g. projective Ω-module.

The approach to the package deal theorem, via splitting orders, is needed in the proof of the version that involves Γ̈, and the splitting-order version is applied extensively in our analysis of the Krull-Schmidt problem in [LO].

An interesting fact about the mysterious ring Γ̈ deserves to be mentioned here: Modulo its nilradical it becomes noetherian and also a maximal order in its semisimple artinian quotient ring.

Artinian pairs. When looking only at f.g. torsionfree Λ-modules M there is a much easier tool than completions that one can use to study direct-sum relations of Λ-modules. One can associate, with M, a “module” Φ(M) over the pair of artinian rings Λ/C and Ω/C, where Ω is a splitting order such that ΩM is Ω-projective, and C is a conductor ideal for Λ and Ω. [Φ(M) consists of a pair of actual modules, one over each of the rings Λ/C and Ω/C.] As with completions, the two critical properties hold: (i) The Krull-Schmidt theorem holds for f.g. modules over artinian pairs; and (ii) Φ(M) ≅ Φ(N) if and only if M ≅ N. Ultimately, one looks at the same array of ranks, whether using completions or artinian pairs as we show at the end of Section 6.

We use artinian pairs whenever possible in [LO]. But when dealing with nontorsionfree modules, there does not seem to be any alternative to the more powerful (but more difficult to use) technique of completions.

Remark 1.2. (Artinian ring direct summands) Much of this paper does not work smoothly if the ring Λ [equivalently, its center Z(Λ)] has artinian ring direct summands.

For example, let Λ = Z ⊕ Z/5Z with Z the ring of integers. If Λ is considered as an algebra over R = Z then Λ_{Q} = Q, the rational numbers. However, if Λ is considered as an algebra over R = Λ, then Λ_{Q} = Q ⊕ Z/5Z.

Moreover, this paper has nothing new to say about artinian rings. The way we deal with this is to note that since R [usually = Z(Λ)] is noetherian it has a decomposition

R = R′ ⊕ A

in which A is artinian and R′ has no artinian direct summands ≠ 0. If M is any R-module, this decomposition of R induces a decomposition M = M′ ⊕ B where M′ and B are modules over R′ and A respectively. Whatever we are proving is usually trivially true or trivially false for B. We therefore assume that R = R′ for the rest of the proof, adding the hypothesis “if R has no artinian ring direct summands” when appropriate.

For semiprime rings (rings without nilpotent ideals) this dichotomy works in the noncommutative as well as commutative case. The following is proved in [GL ’89, (1.4.3)]:

(1.2.1) Every semiprime module-finite R-algebra Λ′, is a direct sum of two Ralgebras Λ′ = Λ ⊕ A where A is an artinian ring, Λ is an R·1_{Λ}-order in a semisimple artinian ring, and Λ has no nonzero R-submodules of finite length and no nonzero Λ-submodules of finite length.

The (known) facts about commutative rings that make the situation R = R′ work more smoothly are listed in the following lemma, in the form in which we shall need them. J(R) denotes the Jacobson radical of R.

Lemma 1.3. Suppose that R has no artinian ring direct summands ≠ 0. Then: (i) No minimal prime ideal of R is maximal; and (ii) m ≠ m^{2} for every maximal ideal m of R. (iii) Suppose that R is semilocal. Then J(R)_{Q} = R_{Q}; that is, J(R) − ⋃ Q is

nonempty. [Recall that if R has no nilpotent elements R − ⋃ Q is the set of regular elements (non-zerodivisors) of R.]

Proof. Since idempotents can be lifted modulo nil ideals, it suffices to prove (i) and (ii) for R/N, where N is the nilradical of R. So we can suppose that R has no nilpotent elements.

(i). Since R has no nonzero nilpotent elements, the intersection of the minimal primes is zero. If some minimal prime p were also maximal, we would therfore have R = I ⊕ p where I is the intersection of the minimal primes ≠ p. But then I would be a minimal ideal of R, a contradiction since R has no artinian ring direct summands.

(ii). Let m be a maximal ideal of R such that m = m^{2}. Since R is noetherian, Krull’s intersection theorem yields m = Rd for some idempotent element d. Thus m is a direct summand of R. The complementary summand is a minimal ideal, a contradiction since R has no artinian direct summands.

(iii). If (iii) were false then J(R) would be contained in the union of the minimal primes of R, hence in some minimal prime p. Therefore some maximal ideal m would be contained in p; hence m would be a minimal prime, contrary to (i).

The following lemma describes the type of ideal we use for a conductor ideal in both this paper and [LO]. Recall that “R-order” was defined in (1.1.1).

Lemma 1.4. Let R be semilocal and let Λ and Ω be R-orders in the semisimple artinian ring Λ_{Q} such that Λ ⊆ Ω ⊆ Λ_{Q}. Assume that Z(Λ) has no artinian ring direct summands. Then Λ and Ω have a common ideal C such that

(i) Λ/C and Ω/C are R-modules of finite length (hence artinian rings); (ii) C ⊆ J(Λ) [equivalently, C ⊆ J(Ω)], where J(…) denotes the Jacobson

radical; and (iii) C_{Q} = Λ_{Q} (= Ω_{Q}) (so C contains at least one central unit of Λ_{Q}).

We call any common ideal C of Λ and Ω satisfying (i) a conductor ideal for Λ and Ω. We usually choose C to take advantage of properties (ii) and (iii). Incidentally, (ii) and (iii) become false if Z(Λ) is allowed to have artinian ring direct summands, as is easily seen.

Proof. We may assume that R = Z(Λ). In particular, R has no nilpotent elements since any such element would generate a nilpotent ideal of Λ. By Lemma 1.3 there is an element c ∈ J(R) − ⋃ Q. Since R has no nilpotent elements, c is a regular element in R.

Since Λ_{Q} = Ω_{Q} and Ω is a finitely generated R-module, there is an element d ∈ R − ⋃ Q such that Ωd ⊆ Λ. Let C = Ωcd. Then C is a common ideal of Λ and Ω, and (iii) holds since C contains the unit cd of R_{Q}, hence of Λ_{Q}. Since R has Krull dimension 1 and cd is outside of every minimal prime of R, the ring R/Rcd is artinian. So (i) holds, by module-finiteness of Λ and Ω over R.

Statement (ii) holds because J(R)Λ ⊆ J(Λ). The parenthetical statement in (ii) holds because of the quasi-regularity property of the radical.

Lemma 1.5. If Λ [equivalently Z(Λ)] has no artinian ring direct summands then Λ_{Q(R)} = Λ_{Q(Z(Λ))} [i.e. both localizations invert same set of elements of Z(Λ)].

Proof. Let S = Z(Λ). It suffices to prove that S_{Q(R)} = S_{Q(S)}. In order to state this more precisely, let ν: R → S denote the natural map ν(r) = r·1_{S}. We want to show that inverting the elements of the subset ν[R − Q(R)] of S yields the same localization as inverting the elements of S − Q(S); that is, both localizations ultimately invert the same elements of S. Since the proof is more subtle than one might expect, and the truth of the lemma depends upon on the fact that R has Krull dimension 1, we begin by reviewing some basic facts.

Let H be a multiplicatively closed subset of S, and P a prime ideal of S. We say “P survives in S_{H}” if P_{H} ≠ S_{H}. This happens if and only if P ⋂ H = ∅; and when P survives, P_{H} is a prime ideal of S_{H}. The precise set of elements of S that become units in S_{H} is the complement of the union of the set of prime ideals of S that survive in S_{H}. Therefore if K is another multiplicatively closed subset of S, we have S_{H} = S_{K} if and only if the same prime ideals of S survive in both localizations.

Since S has no artinian direct summands, no maximal ideal of S is a minimal prime ideal [Lemma 1.3].

We now prove that a prime ideal of S survives in S_{Q(R)} if and only if it is a minimal prime ideal. Consider first a minimal prime ideal p of S. Since p is not a maximal ideal we have a proper inclusion p ⊂ m for some maximal ideal m of S. Since S is module-finite over R, it is integral over ν(R). Therefore, by the incomparability theorem for prime ideals we have p ⋂ ν(R) ⊂ m ⋂ ν(R). Let p′ = ν^{−1}[p ⋂ ν(R)] and m′ = ν^{−1}[m ⋂ ν(R)]. Then p′ ⊂ m′. Since R has Krull dimension 1 the prime ideal p′ is therefore a minimal prime. Now let x be any element of R − Q(R). If ν(x) ∈ p then x ∈ p′, which is impossible since p′ ∈ Q(R). Therefore p survives in S_{Q(R)}.

Now let m be a maximal ideal of S. We show that m does not survive in S_{Q(R)}. By chosing a minimal prime ideal p ⊂ m we show, as before, that m′ = ν^{−1}[m ⋂ ν(R)] is a maximal ideal of R and m′ is not a minimal prime ideal. The maximal ideal m′ cannot be contained the union of the minimal prime ideals of R, because m′ would then be contained in some minimal prime, and hence be minimal. Therefore there exists an element x ∈ m′ −⋃ Q(R). But then ν(x) ∈ m shows that m does not survive in S_{Q(R)}, as desired.

Finally, the case R = S yields the well-known fact that the prime ideals of S that survive in S_{Q(S)} are precisely the minimal primes, completing the proof of our lemma.

Lemma 1.6. Let Λ be any module-finite R-algebra, as in Notation 1.1. Then (i) The correspondence S ↪ M = ann_{Λ} S (the annihilator of S in Λ) is a bijection between the set of isomorphism classes of simple (left) Λ-modules S and the set of maximal (2-sided) ideals M of Λ. In particular, J(Λ) equals the intersection of the set of maximal ideals of Λ.

(ii) For every maximal ideal M of Λ, the simple ring Λ/M is artinian.

(iii) Every Λ-module of finite length has finite length as an R-module.

(iv) Suppose that Λ has no artinian ring direct summands. Then a f.g. Λ-module U has finite length if and only if U_{Q} = 0.

Proof. (i) and (ii). It suffices to show that an ideal M of Λ is primitive (i.e. the annihilator of a simple left module) if and only it is maximal. For the nontrivial implication suppose that M is primitive. Then Λ/M is a primitive ring that is module finite over a commutative ring, and it therefore satisfies a polynomial identity [MR ’87, (13.1.13)]. By Kaplansky’s theorem [MR ’87, (13.3.8)] Λ/M is therefore a simple artinian ring.

(iii). It suffices to show that every simple Λ-module S has finite length as an R-module. Let M = ann_{Λ} S. Then Λ/M is a simple artinian ring by statements (i) and (ii) and its simple module S is therefore isomorphic to a submodule of Λ/M. Since Λ/M is module-finite over R, so therefore is S.

(iv). We may suppose that R = Z(Λ), by Lemma 1.5. Then R has no artinian ring direct summands. We claim that S_{Q} = 0 for every simple R-module S.

We have S = R/m for some maximal ideal m; and m is not a minimal prime ideal by Lemma 1.3. Hence m is not contained in the union of the minimal prime ideals of R; and it follows that m_{Q} = R_{Q} and therefore (R/m)_{Q} = 0 as claimed.

Now suppose that the Λ-module U has finite length. Then it has finite length as an R-module, by statement (iii). To show that U_{Q} = 0 it suffices to show that S_{Q} = 0 for every composition factor S of the R-module U. This was done in the previous claim.

Conversely, suppose U_{Q} = 0. Then dU = 0 for some d ∈ R − ⋃ Q. Therefore U is a f.g. module over the ring R/Rd, which is artinian by our dimension 1 hypothesis. Consequently, U has finite length as an R-module, and hence as a Λ-module. (This half of the proof did not require the hypothesis about no artinian direct summands.)

2. Packages of Localizations

Consider a family {X(m) | m ∈ maxspec(R)}, with each X(m) a f.g. Λ_{m}- module. In this section we consider the question of whether there is a f.g. Λ-module M such that M_{m} ≅ X(m) for all m, tht is, whether {X(m)} is the “package” of all localizations of some f.g. Λ-module at the maximal ideals m of R. Our proof forces us to answer a answer a related question for submodules of a given module M: Given a Λ_{m}-submodule X(m) ⊆ M_{m} for each m, when is there a submodule X ⊆ M such that every X(m) equals M_{m}? In both situations the spirit of the answer is that, subject to some obvious restrictions, one can specify any finite set of localizations. Moreover, if this finite set is chosen large enough, then the structure of the remaining m-localizations is determined by their Q-localization.

As an application we show that indecomposability of any f.g. Λ-module is semilocally determined.

Notation 2.1. The letter m always denotes a maximal ideal of R, and (a∀m) means “for almost all maximal ideals m of R”, that is, for all but finitely many m. We say that a set M of maximal ideals of R covers minspec(R) if for every minimal prime ideal p of R there is a maximal ideal m ∈ M such that m ⊇ p.

The following lemma will be used frequently, sometimes without explicit acknowledgement.

Lemma 2.2. (i) R_{Q} ≅ ⊕_{p∈minspec(R)}R_{p}. Therefore M_{Q} ≅ ⊕_{p∈minspec(R)}M_{p} for every Λ-module M. (ii) R_{m} ⊗_{R} R_{p} = 0 whenever m ∈ maxspec(R), p ∈ minspec(R), and m ⊉ p.

Proof. (i) holds since every commutative artinian ring is the direct sum of its localizations at its finite number of prime (necessarily maximal) ideals.

(ii) Take any x ∈ p − m. Then x becomes a unit in R_{m} and becomes nilpotent in the artinian ring R_{p}. But x^{n}/1 = 0 implies that 1⊗1 = (1/x^{n})⊗(x^{n}/1) = 0.

Lemma 2.3. (i) R − ⋃ Q localizes. In other words, (R_{Q})_{m} = (R_{m})_{Q(R_{m})}. (ii) Let X be an R_{m}-module, for some m. Then X is Q(R)-torsionfree if and only X is Q(R_{m})-torsionfree. (Hence X can unambiguously be called Q-torsionfree.)

Proof. (i) We have R_{Q} ≅ ⊕_{p∈minspec(R)}R_{p} by Lemma 2.2. Tensoring with R_{m} and using Lemma 2.2(ii) yields (R_{Q})_{m} = ⊕_{p⊆m}R_{p}⊗_{R}R_{m}. Then note that R_{p}⊗_{R}R_{m} = R_{p} when p ⊆ m.

(ii) This is an immediate consequence of (i).

The fact that R_{Q} is an artinian ring imposes the following basic finiteness restrictions on localizations.

Lemma 2.4. Let L, M, N be f.g. Λ-modules.

(i) If N ⊆ M and M_{Q} = N_{Q}, then M_{m} = N_{m} (a∀m). (ii) If L_{Q} ≅ M_{Q}, then L_{m} ≅ M_{m} (a∀m). (iii) If L_{Q} | M_{Q} (i.e. L_{Q} is isomorphic to a direct summand of M_{Q}) then L_{m} | M_{m} (a∀m). (iv) M_{m} is Q-torsionfree [equivalently, Q(R_{m})-torsionfree] (a∀m).

(v) There is a finite set M of maximal ideals of R such that the natural map M → M_{M} is one-to one, and remains one-to-one if M is replaced by any larger finite set of maximal ideals.

Proof. (i) Since M is a finitely generated Λ-module and M_{Q} = N_{Q} we have dM ⊆ N for some d ∈ R − ⋃ Q. Since d is outside of all minimal primes of our ring R of dimension 1, the ring R/Rd is artinian, and so only finitely many maximal ideals of R contain d. For any other maximal ideal m, d/1 is a unit in R_{m}, hence M_{m} = N_{m}.

(ii) and (iii) See [GL ’89, 2.1(i) and (iii)].

(iv) Let K = ker(M → M_{Q}). Then K_{Q} = ker(M_{Q} → M_{Q}) = 0. Therefore K_{m} = 0 (a∀m), by (i). But K_{m} = ker(M_{m} → (M_{m})_{Q}), and (M_{m})_{Q} can be identified with (M_{m})_{Q(R_{m})} by Lemma 2.3.

(v) Let M be the finite set of maximal ideals such that M_{m} is not Q-torsionfree, and enlarge M, if neccessary, so that it covers minspec(R). To see that the natural map M → M_{M} is one-to-one, take m ∈ M such that (m/1)_{m} = 0 for all m ∈ M. It suffices to show that (m/1)_{m} = 0 for all m ∉ M. Since M covers minspec(R) we have (m/1)_{Q} = 0. Therefore (m/1)_{m} = 0 whenever M_{m} is Q-torsionfree. This includes all m ∉ M.

Now consider a f.g. Λ-module M. We wish to change a single localization M_{n} to some specified R_{n} submodule of it, without changing any other M_{m}. What restrictions apply? The possibly surprising answer is that there are no restrictions other than the obvious one: do not change (M_{n})_{Q}, as we show in the next lemma.

Lemma 2.5. Let M be a f.g. Λ-module, n ∈ maxspec(R), and X a Λ_{n}-submodule of M_{n} such that X_{Q} = (M_{n})_{Q}. Then there is a Λ-submodule N ⊆ M such that N_{n} = X and N_{m} = M_{m} for all other maximal ideals m.

Proof. We may assume that X ≠ M_{n}. We find N by means of two approximations. First we find a submodule L ⊆ M such that

L_{n} = X and L_{Q} = M_{Q} (2.5.1)

Then, since L_{Q} = M_{Q} we have L_{m} = M_{m} (a∀m), by Lemma 2.4. Let F be the finite set of maximal ideals m ≠ n such that L_{m} ≠ M_{m}. We find an element e ∈ R such that

(2.5.2) eM_{n} ⊆ L_{n} (= X) and eM_{m} = M_{m} (∀m ∈ F)

Then N = L + eM is the desired submodule of M.

Since X_{Q} = (M_{n})_{Q}, finite generation of M_{n} yields b ∈ R − ⋃ Q such that bM_{n} ⊆ X. Since X ≠ M_{n}, (b/1)_{n} is a nonunit in the local ring R_{n}, that is, (b/1)_{n} ∈ n_{n}. Nilpotence of the radical of the artinian ring R_{n}/R_{n}b then yields a positive integer s such that n_{n}^{s} ⊆ R_{n}b. Thus

(2.5.3) bM_{n} ⊆ X and n_{n}^{s} ⊆ R_{n}b hence n^{s}M_{n} ⊆ X

Let I = ⋂{p ∈ minspec(R) | p ⊈ n}. Then n^{s} + I = R, so the Chinese Remainder Theorem yields d ∈ R such that

(2.5.4) d ≡ b (mod n^{s}) and d ≡ 1 (mod I)

Let L′ be the Λ-submodule of M generated by the numerators of some finite set of Λ_{n}-generators of X. We claim that L = L′ + dM is as required in (2.5.1).

To see that L_{n} = X it suffices to note that dM_{n} ⊆ X. To see that L_{Q} = M_{Q} it suffices to check that L_{p} = M_{p} (∀p ∈ minspec(R)). If p ⊆ n we have

L_{p} = (L_{n})_{p} = X_{p} = (X_{Q})_{p} = ((M_{n})_{Q})_{p} = M_{p}

Suppose therefore that p ⊈ n. Then d becomes a unit in R_{p} and so we again have L_{p} = M_{p}. This completes the proof of (2.5.1).

Now we find e such that (2.5.2) holds. Let b again be as in (2.5.3). Then the Chinese Remainder Theorem yields e ∈ R such that e ≡ b (mod n^{s}) and e ≡ 1 (mod m) (∀m ∈ F), completing the proof.

Lemma 2.4 together with repeated use of Lemma 2.5 immediately yields:

Package Deal Theorem 2.6. (Localizations of Submodules) Let M be a f.g. Λmodule, and for each m ∈ maxspec(R) let X(m) be a f.g. Λ_{m}-submodule of M_{m} such that X(m)_{Q} = (M_{m})_{Q}. Then there is a Λ-submodule N ⊆ M such that N_{m} = X(m) (∀m) if and only if (a∀m) X(m) = M_{m}.

The previous theorem can be sharpened if M is Q-torsionfree. In that situation every M_{m} ⊆ M_{Q}, so one can try to replace some of the submodules M_{m} by f.g. Λ_{m}-submodules X(m) ⊆ M_{Q} that are not contained in M_{m}. The following result does this, extending [GL ’88, 2.6] which was proved under the assumption that Λ is an order in a semisimple artinian ring.

Corollary 2.7. Let M be a f.g. Q-torsionfree Λ-module, and for each m ∈ maxspec(R) let X(m) be a f.g. Λ_{m}-submodule of M_{Q} such that X(m)_{Q} = (M_{m})_{Q}. Then there is a f.g. Λ-submodule N ⊆ M_{Q} such that N_{m} = X(m) (∀m), if and only if (a∀m) X(m) = M_{m}.

Proof. If. There exists d ∈ R − ⋃ Q such that (∀m) dX(m) ⊆ M_{m} because of our torsionfree and finite generation hypotheses, and the hypothesis that X(m) = M_{m} (a∀m). Since M is Q-torsionfree, d^{−1}M is a well-defined Λ-submodule of M_{Q}. Therefore we can apply the previous theorem to the Λ-module M′ = d^{−1}M and the family of Λ_{m}-submodules X(m) of M′_{m}. Since d/1 is a unit in almost all localizations R_{m} the hypothesis X(m) = M_{m} (a∀m) shows that X(m) = (M′)_{m} (a∀m); and then the previous theorem completes the proof of this half of the corollary.

Only if. Since (M_{m})_{Q} = (N_{m})_{Q} (∀m) we have M_{Q} = N_{Q}. As in the “if” part of the proof we find elements c, d ∈ R − ⋃ Q such that cM ⊆ N and dN ⊆ M. Since c and d become units in almost all localizations R_{m} we have M_{m} = N_{m} (a∀m).

Remark 2.8. Let M be a f.g. Λ-module and (∀m) X(m) a Λ_{m}-submodule of M_{m}. But do not assume that X(m)_{Q} = (M_{m})_{Q}, which was assumed in Theorem 2.6. Is there a Λ-submodule N ⊆ M such that (∀m) N_{m} = X(m)? The answer is almost the same as in Theorem 2.6 but more difficult to state. We discuss it briefly here.

There is an obviously necessary Q-consistency condition, namely that X(m)_{p} = X(n)_{p} whenever p ∈ minspec(R), m, n ∈ maxspec(R) and p ⊆ m ⋂ n. Assume that this condition is satisfied. For each p ∈ minspec(R) choose a maximal ideal m(p) ⊇ p and define X(Q) = ⊕_{p} X(m(p))_{p} ⊆ M_{Q}. Because of Q-consistency, the Λ_{Q}-submodule X(Q) ⊆ M_{Q} is independent of the particular choice of maximal ideals m(p).

Let L = {m ∈ M | (m/1)_{Q} ∈ X(Q)}. Then L_{Q} = X(Q), and we can now apply Theorem 2.6, with L in place of M. Thus N exists if and only if X(m) = L_{m} (a∀m).

We now shift our attention to localizations of modules themselves, rather than localizations of submodules of a given module. Let V be a f.g. Λ_{Q}-module. Then V = L_{Q} for some f.g. Λ-module L. If N is any other f.g. Λ-module such that N_{Q} ≅ L_{Q} then Lemma 2.4 shows that N_{m} ≅ L_{m} (a∀m). Our second package deal theorem shows that the hypothesis N_{Q} ≅ L_{Q} imposes no other restrictions on N.

Package Deal Theorem 2.9. (Localizations of Modules) Let L be a f.g. Λ-module, and for each m in some finite subset M ⊆ maxspec(R), let X(m) be a f.g. Λ_{m}- module such that X(m)_{Q} ≅ (L_{m})_{Q}. Then there is a f.g. Λ-module N such that

(i) N_{m} ≅ X(m) (∀m ∈ M), and (ii) N_{m} ≅ L_{m} for all other maximal ideals m.

Proof. We first note that the theorem is true when R is semilocal. See, for example, [W ’89, 1.11] or [HL ’87, 1.6] for the commutative case Λ = R. These proofs work without change when Λ is any module-finite, noncommutative R-algebra. (Ignore the torsionfreeness hypothesis in [HL ’87].)

Now consider the situation that R is not semilocal. We can enlarge M to any larger finite set of maximal ideals by defining X(m) = L_{m} for the new maximal ideals m. So we can assume that M covers minspec(R). By Lemma 2.4, we can further enlarge M so that L_{m} is Q-torsionfree for every m ∉ M. By the semilocal case of the theorem there is a f.g. Λ_{M}-module Y such that Y_{m} ≅ X(m) (∀m ∈ M).

We have (Y_{m})_{Q} ≅ X(m)_{Q} (∀m ∈ M) which, together with the hypothesis X(m)_{Q} ≅ (L_{m})_{Q} shows that (Y_{m})_{Q} ≅ (L_{m})_{Q}. Since M covers minspec(R) this yields, by Lemma 2.2:

L_{Q} ≅ Y_{Q} (2.9.1)

For a first approximation, let N be the Λ-submodule of Y generated by the numerators of some finite set of Λ_{M}-generators of Y. Then N_{M} = Y, and hence N_{m} ≅ X(m) (∀m ∈ M). Since M covers minspec(R), we have (L_{M})_{Q} ≅ L_{Q} and (N_{M})_{Q} ≅ N_{Q}. Therefore N_{Q} ≅ Y_{Q} which, together with (2.9.1) shows that N_{Q} ≅ L_{Q}. Therefore N_{m} ≅ L_{m} (a∀m), by Lemma 2.4.

Choose any one n of the finitely many maximal ideals such that this last isomorphism does not hold. It now suffices to show that N_{n} has a Λ_{n}-submodule X such that X ≅ L_{n} and X_{Q} = (N_{n})_{Q}. For then Lemma 2.5 allows us to replace N_{n} by X without changing any other localization of N.

Since N_{Q} ≅ L_{Q} we have (N_{n})_{Q} ≅ (L_{n})_{Q}. Since N_{n} and L_{n} are both Qtorsionfree, we can therefore assume that they are both Λ_{n}-submodules of W = (N_{n})_{Q}. Finite generation of L_{n} over Λ_{n} shows that there exists d ∈ R − ⋃ Q such that dL_{n} ⊆ N_{n}; and we may therefore take X = dL_{n}.

The previous theorem avoids the question of the structure of the infinitely many localizations that we cannot change. Our next objective is to show that, when a sufficiently large finite number of localizations are discarded, the remaining ones “decompose like” their Q-localization.

Definitions 2.10. Let X be a f.g. Λ-module. We say that X decomposes like X_{Q} if X has a decomposition X = ⊕_{i} Y_{i} where each Y_{i} is indecomposable, each (Y_{i})_{Q} is indecomposable, and Y_{i} ≅ Y_{j} whenever (Y_{i})_{Q} ≅ (Y_{j})_{Q}. Since many of our arguments involve counting the multiplicities of isomorphism classes of summands, it will be convenient to display these multiciplicites explicitly as follows.

(2.10.1) X ≅ ⊕_{i=1}^{m} (X_{i})^{e_{i}} where the X_{i} are indecomposable pairwise nonisomorphic Λ-modules whose Q-localizations (X_{i})_{Q} are indecomposable and pairwise nonisomorphic.

We call the (isomorphism classes of the) X_{i} the root summands of X. We will only be interested in these definitions when R is semilocal, namely localizations of our original R.

Note 1. Such an X can have indecomposable direct summands U such that U_{Q} is not indecomposable. Moreover, X can have indecomposable direct summands V such that V_{Q} is again indecomposable, yet V is not a root summand of X. See Examples 2.12. However, the root summands X_{i} and their multiplicities e_{i} are unique, as we show below.

Note 2. Our definition requires every (X_{i})_{Q} in (2.10.1) to be nonzero, since it is indecomposable. Therefore, if X decomposes like X_{Q}, then X is Q-torsionfree.

Lemma 2.11. Suppose that R is semilocal and let X be a f.g. Λ-module that decomposes like X_{Q}, with decomposition (2.10.1). Let X ≅ ⊕_{i=1}^{n} (Y_{i})^{f_{i}} be another decomposition of this same form. Then m = n and the Y_{i} can be renumbered in such a way that every X_{i} ≅ Y_{i} and every e_{i} = f_{i}.

Proof. We shall refer to the decomposition X ≅ ⊕_{i} (Y_{i})^{f_{i}} as decomposition (2.10.1)′. By taking the Q-localization of these two decompositions and applying the Krull-Schmidt theorem for modules of finite length, we see that m = n and, after a suitable renumbering of the Y_{i}, every (X_{i})_{Q} ≅ (Y_{i})_{Q} and e_{i} = f_{i}. Thus the nontrivial part of this proof is to show that every X_{i} ≅ Y_{i}.

Suppose m = n = 1. In other words, suppose that (X_{1})^{e} ≅ (Y_{1})^{f}. Then, as mentioned above, e = f. The desired isomorphism X_{1} ≅ Y_{1} is often expressed by saying that e^{th} roots of Λ-modules are unique, when they exist, when R is semilocal. This is easily proved in the present situation by taking the J(R)-adic completion and then applying the Krull-Schmidt theorem. To reduce to this situation, let E be the endomorphism ring of X (endomorphisms written on the right, for the rest of this proof). Then E is a module-finite R-algebra. Moreover, the given decompositions (2.10.1) and (2.10.1)′ of X induce decompositions of the left Emodule E, as displayed below,

E ≅ ⊕_{i} (E_{i})^{e_{i}} ≅ ⊕_{i} (F_{i})^{e_{i}} (2.11.1)

where E_{i} = Hom_{Λ}(X, X_{i}) and F_{i} = Hom_{Λ}(X, Y_{i}). Moreover, indecomposability of X_{i} as a Λ-module implies indecomposability of E_{i} as an E-module, X_{i} ≅ X̸_{j} when i ≠ j implies that E_{i} ≅ E̸_{j}, and the analogous statements hold for the F_{i}. If we can prove that E_{i} ≅ F_{i} as E-modules it will follow that X_{i} ≅ Y_{i} as Λ-modules. (See, e.g., [GL ’88, 1.1] for the proofs of these statements.)

Since localization of finitely presented modules commutes with forming Hom groups, taking the Q-localization of decompositions (2.11.1) and repeating the reasoning in the previous paragraph yields a pair of decompositions of left E_{Q}-modules:

E_{Q} ≅ ⊕_{i} ((E_{i})_{Q})^{e_{i}} ≅ ⊕_{i} ((F_{i})_{Q})^{e_{i}} (2.11.2)

in which each (E_{i})_{Q} is indecomposable, (E_{i})_{Q} ≅ (E_{j})_{Q} when i ≠ j, and the analogous statements hold for the (F_{i})_{Q}.

Identify the decompositions in (2.11.1) and (2.11.2) with internal direct-sum decompositions of E and E_{Q}. Then these decompositions arise by writing the identity element of E in two different ways as a sum of primitive orthogonal idempotents. Let W denote the nilradical of E. Since our decompositions of E and E_{Q} arise from orthogonal idempotents they induce direct-sum decompositions of E/W and E_{Q}/W_{Q}. Recall that idempotents can be lifted modulo nil ideals, and the left ideals that they generate are isomorphic modules after the lifting if and only if they were isomorphic before the lifting. Therefore, after replacing E and E_{Q} by E/W and E_{Q}/W_{Q} respectively, we can assume that E is semiprime. [Caution. It will unfold that E_{Q} is semiprime, but this is not yet obvious.]

Since E is semiprime and its Q-torsion submodule T = ker(E → E_{Q}) is a 2-sided ideal that has finite length as an R-module hence as an E-module, T is generated by a central idempotent of E (see the proof of [GL ’89, (1.4.3)]). It follows that every indecomposable left ideal of E is either Q-torsion (i.e. its Q-localization equals zero) or Q-torsionfree. Since each (E_{i})_{Q} is nonzero, E_{i} must therefore be Q-torsionfree, hence E itself is Q-torsionfree. Therefore we can regard E as an R-subalgebra of E_{Q}. It now follows that E_{Q} is semiprime (otherwise the intersection with E, of any nilpotent ideal of E_{Q} would be a nilpotent ideal of E). In other words, E_{Q} is semisimple artinian ring.

Still identifying our first direct-sum decomposition of E_{Q} with an internal directsum decomposition of E_{Q}, we see that the terms ((E_{i})^{e_{i}})_{Q} become the set of minimal (2-sided) ideals of E_{Q}, and the terms (E_{i})^{e_{i}} are their intersections with E. This proves that each (E_{i})^{e_{i}} = (F_{i})^{e_{i}} after the above-mentioned identification, and therefore (E_{i})^{e_{i}} ≅ (F_{i})^{e_{i}} before the identification.

The previous discussion of e^{th} roots now shows that E_{i} ≅ F_{i}.

Examples 2.12. (Root summands). Suppose that R is semilocal and X is a f.g. Λ-module that decomposes like X_{Q}.

(i) X can have indecomposable direct summands U such that U_{Q} is not indecomposable.

To see this, let n be any positive integer. Then there is a (noncommutative) integral domain Λ that is module-finite over a discrete valuation ring and such that Λ^{n} can be written as the direct sum of two indecomposable modules [L ’89, 2.4]. On the other hand, since Λ is an integral domain, Λ^{n} is necessarily decomposes like Λ_{Q}, and its unique root sumand is Λ.

(ii) X can have an indecomposable direct summand P such that P_{Q} is also indecomposable, but P is not a root summand of X [i.e. P is not isomorphic to any X_{i} in any decomposition of the form (2.10.1)].

We present the example briefly, using the terminology and machinery in [L ’89]. Let Γ be a (noncommutative) principal ideal domain that is module-finite and torsionfree over a discrete valuation ring, and such that there is a surjective ring homomorphism f: Γ → → B = Δ_{2×2}, the ring of two-by-two matrices over a division ring Δ with H := ker(f) = rad(Γ), as in [L ’89, 2.1]. Let A be the subring of B consisting of diagonal matrices, and let Λ be the pullback Λ = {x ∈ Γ | f(x) ∈ A}. Note that A has two indecomposable projective modules, namely X = Δ × 0 and Y = 0 × Δ. Moreover, an arbitrary f.g. projective A-module X^{a} ⊕ Y^{b} has the form P/HP for some f.g. projective Λ-module P if and only if a + b is an even number [L ’89, 2.2]; and P ≅ P′ for f.g. projective Λ-modules P, P′ if and only if P/HP ≅ P′/HP′ [L ’89, 1.2(i)]. Note that Λ/H ≅ X ⊕ Y.

It now follows that the integral domain Λ has projective modules P_{1}, P_{2} such that P_{1}/HP_{1} ≅ X^{2} and P_{2}/HP_{2} ≅ Y^{2}, and Λ^{2} ≅ P_{1} ⊕ P_{2}. Thus Λ^{2} decomposes like (Λ^{2})_{Q} and its unique root summand is Λ. But since Λ_{Q} is a division Ralgebra, the P_{i} are indecomposable and each (P_{i})_{Q} is again indecomposable. In fact, (P_{1})_{Q} ≅ Λ_{Q} ≅ (P_{2})_{Q}. And neither P_{i} is isomorphic to Λ because these isomorphisms fail modulo H.

We return to the general, non-semilocal R. Let M be an R-module and m a maximal ideal of R. Then, by Lemma 2.3 we have (M_{m})_{Q} = (M_{m})_{Q(R_{m})}, where the unadorned Q denotes Q(R). We take advantage of this notational simplification in what follows.

Lemma 2.13. Let N, L be f.g. Λ-modules such that every indecomposable direct summand of N_{Q} is isomorphic to a direct summand of L_{Q}. Then (a∀m), N_{m} and L_{m} decompose like (N_{m})_{Q} and (L_{m})_{Q} respectively, and the root summands of N_{m} are among those of L_{m}.

Proof. By the Krull-Schmidt theorem for modules of finite length we have N_{Q} ≅ ⊕_{i} (V_{i})^{e_{i}} and L_{Q} ≅ ⊕_{i} (V_{i})^{d_{i}} where the V_{i} are non-isomorphic indecomposable f.g. modules over the artinian ring Λ_{Q}, each e_{i} ≥ 0 and each d_{i} > 0. For each i let X_{i} be a finitely generated Λ-module such that (X_{i})_{Q} ≅ V_{i}, and let X = ⊕_{i} (X_{i})^{e_{i}} and W = ⊕_{i} (X_{i})^{d_{i}}. Then X_{Q} ≅ N_{Q} and W_{Q} ≅ L_{Q}. Therefore, (a∀m), X_{m} ≅ N_{m} and W_{m} ≅ L_{m} by Lemma 2.4. To complete the proof we show that, except for finitely many m, the X_{i} provide the desired root summands, that is, (X_{i})_{m} is either zero (and can be ignored), or it is indecomposable and ((X_{i})_{m})_{Q} ≅ V_{i}.

Let M be the finite (possibly empty) set of maximal ideals m such that, when m ∉ M, we have X_{m} ≅ N_{m} and W_{m} ≅ L_{m} and both of X_{m} and W_{m} are Qtorsionfree. Fix any m ∉ M, and any i.

Since V_{i} is indecomposable we have V_{i} ≅ (V_{i})_{p} for some minimal prime p, by Lemma 2.2. We consider two cases.

Case 1: m ⊉ p. Then (V_{i})_{m} ≅ ((V_{i})_{p})_{m} = 0 by Lemma 2.2(ii). Since (X_{i})_{m} is Q-torsionfree we have (X_{i})_{m} ⊆ ((X_{i})_{m})_{Q} ≅ (V_{i})_{m} = 0, and we can therefore ignore (X_{i})_{m}.

Case 2: m ⊇ p. Then ((X_{i})_{m})_{Q} ≅ ((X_{i})_{Q})_{m} ≅ (V_{i})_{m} ≅ ((V_{i})_{p})_{m} ≅ (V_{i})_{p} ≅ V_{i} as desired. To see that (X_{i})_{m} is indecomposable, suppose that (X_{i})_{m} = A ⊕ B with A ≠ 0. Since V_{i} is indecomposable and (X_{i})_{m} is Q-torsionfree, localization at Q shows that B_{Q} = 0. Since (X_{i})_{m} is Q-torsionfree, so is B. Therefore B = 0.

Remark 2.14. We do not know whether the finite, exceptional set of maximal ideals m in the previous lemma can be chosen large enough so that the Krull-Schmidt theorem holds for decompositions of all remaining N_{m}. As shown in Examples 2.12, uniqueness of the root summands and their multiplicities can be a weaker condition than Krull-Schmidt uniqueness.

However, we now have enough information to show that almost all localizations of f.g. projective Λ-modules behave as well as free modules over commutative integral domains in two important ways. See Definitions 2.10 for terminology.

Theorem 2.15. Let P, P′ be a f.g. projective Λ-modules. Then

(i) (a∀m) Λ_{m} and P_{m} decompose like (Λ_{m})_{Q} and (P_{m})_{Q} respectively, and the root summands of P_{m} are among those of Λ_{m} (up to isomorphism). (ii) Suppose that R is semilocal and P and P′ are direct sums of root summands

of Λ. If P_{Q} ≅ P′_{Q} then P ≅ P′.

Proof. Statement (i) is a special case of Lemma 2.13. To prove statement (ii), write P and P′ as direct sums of root summands of Λ, say

P ≅ ⊕_{i} (X_{i})^{d(i)} and P′ ≅ ⊕_{i} (X_{i})^{e(i)}

where, as usual, the X_{i} are pairwise non-isomorphic. It suffices to prove that every d(i) = e(i). Take the Q-localization of these decompositions, remembering that root summands remain indecomposable after Q-localization, and non-isomorphic root summands of Λ remain non-isomorphic after this localization. Sinc P_{Q} ≅ P′_{Q} we can apply the Krull-Schmidt theorem for modules of finite length to conclude that every d(i) = e(i) as desired.

Definition 2.16. Return to our basic package deal question: For each m let X(m) be a f.g. Λ_{m}-module. Is there a f.g. Λ-module N such that (∀m) N_{m} ≅ X(m)? Theorem 2.9 answers the question affirmatively if we already know a Λ-module L whose localizations L_{m} we can conveniently compare with the X(m). The statement of the next theorem avoids the necessity of such a global comparison when Λ has no nilpotent ideals. Our first step is to isolate the following Q-consistency condition that is obviously necessary for the existence of N.

Consider a family {X(m) | m ∈ maxspec(R)} where each X(m) is a f.g. Λ_{m}- module. We call {X(m)} a Q-consistent family of f.g. {Λ_{m}}-modules if (compare with Remark 2.8):

p ⊆ m ⋂ n ⇒ X(m)_{p} ≅ X(n)_{p} (2.16.1)

for p ∈ minspec(R) and m, n ∈ maxspec(R). For each p ∈ minspec(R) choose a maximal ideal m(p) ⊇ p and then set

(2.16.2) X(Q) = ⊕_{p∈minspec(R)} X(m(p))_{p}

Because of Q-consistency, the isomorphism class of the Λ_{Q}-module X(Q) is independent of the particular choice of maximal ideals m(p). Moreover, if the desired Λ-module N exists, we necessarily have N_{Q} ≅ X(Q).

We now answer the package deal question for localizations of modules,as posed in Definition 2.16, if certain Λ_{Q}-modules are projective. This hypothesis is always satisfied if the ring Λ is semiprime, for then the ring Λ_{Q} is semisimple artinian and therefore all of its modules are projective. See Definitions 2.10 for additional terminology.

Theorem 2.17. Let {X(m): m ∈ maxspec(R)} be a Q-consistent family of f.g. {Λ_{m}}-modules, and suppose that every X(m)_{Q} is a projective Λ_{Q}-module. Then the following statements are equivalent.

(i) There is a f.g. Λ-module N such that N_{m} ≅ X(m) (∀m). (ii) (a∀m) such that Λ_{m} decomposes like Λ_{Q}, X(m) is a direct sum of root summands of Λ_{m}.

Proof. Both implications use the fact that, since the Krull-Schmidt theorem holds for f.g. modules over the artinian ring Λ_{Q}, every indecomposable f.g. projective Λ_{Q}-module is isomorphic to a direct summand of Λ_{Q}.

(i) ⇒ (ii). Let L = Λ, considered as a left Λ-module. In view of the preceeding paragraph every indecomposable direct summand of N_{Q} is isomorphic to a direct summand of L_{Q}. Therefore the implication (i) ⇒ (ii) is a special case of Lemma 2.13.

(ii) ⇒ (i). We first find a module M that has the required localizations almost everywhere, and then change the finite number of remaining localizations. Because of the Q-consistency hypothesis we can form the Λ_{Q}-module X(Q) defined in Definition 2.16. Let M be the Λ-submodule of X(Q) generated by some finite set of Λ_{Q}-generators of X(Q). Then M_{Q} ≅ X(Q) and M_{Q} is a projective Λ_{Q}-module.

By the first paragraph of this proof, every indecomposable direct summand of the projective Λ_{Q}-module M_{Q} is isomorphic to a direct summand of Λ_{Q}. Therefore we can apply Lemma 2.13 to the pair of modules M, Λ. We conclude that (a∀m) M_{m} and Λ_{m} decompose like (M_{m})_{Q} and (Λ_{m})_{Q} respectively, and the root summands of M_{m} are among those of Λ_{m}.

We claim that M_{m} ≅ X(m) (a∀m). Consider any m that satisfies the conditions in the final sentence of the previous paragraph. After discarding finitely many more m we have that X(m) is a direct sum of root summands of Λ_{m}. By Theorem 2.15(ii) it now suffices to check that (M_{m})_{Q} ≅ X(m)_{Q}. Since M_{Q} ≅ X(Q) it suffices to prove X(Q)_{m} ≅ X(m)_{Q} which, with the help of Lemma 2.2, follows easily from the definition of X(Q).

Now that the claim is proved, we obtain the desired module N by applying the package deal theorem for localizations of modules 2.9 to change the remaining finite number of localizations M_{m} to X(m). This is possible since, as already shown, (M_{m})_{Q} ≅ X(m)_{Q}.

The following special case of Theorem 2.17 is particularly easy to understand, and extends a result mention in the introduction to this paper.

Corollary 2.18. Assume that R = Λ and Λ has no nilpotent elements. For every m ∈ maxspec(R) let X(m) be a f.g. Λ_{m}-module. Then there exists a f.g. Λ-module N such that N_{m} ≅ X(m) (∀m) if and only if

(i) X(m) is a free Λ_{m}-module (a∀m); and (ii) Q-consistency condition (2.16.1) holds.

Proof. If. By hypothesis (ii) the X(m) form a Q-consistent family in the sense of Definition 2.16. By hypothesis (i) X(m) is free for almost all m such that Λ_{m} decomposes like (Λ_{m})_{Q}. For such m, X(m) is clearly a direct sum of root summands of Λ_{m} (namely Λ_{m} itself). Therefore N exists by Theorem 2.17.

Only if. Since N exists, the family {X(m)} is Q-consistent, that is, condition (ii) holds. By Theorem 2.15 Λ_{m} decomposes like (Λ_{m})_{Q} (a∀m). By Theorem 2.17 X(m) is a direct sum of root summands of Λ_{m} for almost all of the m in the previous sentence, hence for almost m. In particular, X(m) is projective over the local ring Λ_{m} = R_{m} and is therefore free, as desired.

Example 2.19. The preceeding corollary has some content even in the following very simple situation. Assume that R = Λ is an integral domain with infinitely many maximal ideals m and no Λ_{m} is a discrete valuation ring. Thus every localization Λ_{m} has nonzero ideals not isomorphic to Λ_{m} as Λ_{m}-modules. However, the preceding corollary (with Λ = R) shows that a family {X(m): m ∈ maxspec(R)} of nonzero ideals X(m) of R_{m} is isomorphic to the set of localizations of a nonzero ideal of R if and only if X(m) ≅ R_{m} (a∀m). (Note that M. Hochster has given an example of such a domain [H ’73].)

Our final main result in this section is that indecomposability of Λ-modules is semilocally determined.

Theorem 2.20. Let M be an indecomposable f.g. Λ-module. Then there is a finite set M of maximal ideals of R such that M_{M} is an indecomposable Λ_{M}-module, and remains indecomposable if M is replaced by any larger finite set of maximal ideals.

Proof. (a∀m) M_{m} decomposes like (M_{m})_{Q}, by Lemma 2.13. Let M be the finite set of m for which this is not true, and enlarge M if necessary (keeping it finite) so that so that M covers minspec(R), and perhaps contains additional maximal ideals. We claim that, for any such M, M_{M} is indecomposable.

Let X be a Λ_{M}-module such that X | M_{M} (i.e. X is isomorphic to a direct summand of M_{M}). We want to show that X = 0 or X = M_{M} (i.e. the image of X in M_{M} is M_{M} itself). We have X ≅ N_{M} for some f.g. Λ-module N. Then N_{m} | M_{m} (∀m ∈ M). We claim that N_{m} | M_{m} also holds for almost all m ∉ M.

Since M covers minspec(R), we have N_{p} | M_{p} for every minimal prime of R, and hence, N_{Q} | M_{Q} by Lemma 2.2. Therefore N_{m} | M_{m} (a∀m) by Lemma 2.4(iii), as claimed.

We now wish to change the finite number of remaining localizations N_{m} to achieve N_{m} | M_{m} there, too. Recall that M_{m} decomposes like (M_{m})_{Q} at all these localizations. The package deal theorem for localizations of modules 2.9 therefore allows us to make the needed changes: Change N_{m} a direct sum of root summands of M_{m}, choosing whatever number of root summands are needed so as not to change the isomorphism class of (N_{m})_{Q}.

[GL ’89, 4.1] states: If N and M are f.g. Λ-modules such that N_{m} | M_{m} for every maximal ideal m, then N′ | M for some N′ such that N′_{m} ≅ N_{m} (∀m). Since M is indecomposable, we have either N′ = 0 (in which case 0 = N′_{M} ≅ N_{M} = X) or N′ = M (in which case X = M_{M} because this is true locally).

3. Packages of Completions

In this section we determine when a f.g. Λ̂-module is the completion of some f.g. Λ-module. We assume that R is semilocal (and, as always, noetherian of dimension 1) and Λ denotes any module-finite R-algebra.

Definitions, Notation 3.1. (Completions). Let J = J(R), the Jacobson radical of R. We denote the J-adic completion of R by R̂ and the J-adic completion of any f.g. R-module M by M̂. Since M is f.g. and R is noetherian and semilocal, the Krull intersection theorem shows that M ⊆ M̂ [B ’72, Chap. III §3.3 Prop. 6]. In addition, R̂ is a faithfully flat R-module, M̂ = R̂M, and the canonical map R̂ ⊗_{R} M → M̂ is a bijection [B ’72, Chap. III §3.4 Thm. 3 and Cor. 1, and §3.5 Prop. 9]. We regard this bijection as equality R̂ ⊗_{R} M = R̂M = M̂ whenever convenient. In particular, M̂ is a f.g. R̂-module.

Since Λ is a module-finite R-algebra the foregoing discussion shows that Λ̂ = R̂ ⊗_{R} Λ; and if M is any f.g. Λ-module we have M̂ = R̂ ⊗_{R} M, a f.g. Λ̂-module.

Caution. R̂ has nilpotent elements whenever the integral closure of R in its total quotient ring is not a f.g. R-module, even if R itself has no nilpotent elements [Mtl ’73, 7.1]. Nevertheless, the crucial properties that make completions useful to us still hold: (i) the Krull-Schmidt theorem holds for f.g. Λ̂-modules [CR ’81, (6.12)]; and (ii) for f.g. Λ-modules M and N we have M ≅ N as Λ-modules if and only if M̂ ≅ N̂ as Λ̂-modules. (Statement (ii) is a special case of Grothendieck’s theorem on isomorphism by faithfully flat descent over semilocal rings [G ’65, 2.5.8]).

Let R̂_{Q} = R̂ ⊗_{R} R_{Q}; that is, let R̂_{Q} be the Q(R)-localization of R̂. We also call R̂_{Q} the J(R)-induced completion of R_{Q} (but see the Caution statement below). We claim:

(3.1.1)  R̂_{Q} is an artinian ring

R̂ is noetherian of Krull dimension 1 [N ’62, (17.2)], and its maximal ideals are the ideals R̂m, where m runs through the maximal ideals of R [Mts ’80, p. 173]. Since m_{Q} = R_{Q} for every maximal ideal m of R that is not a minimal prime, the preceding sentence shows that all prime ideals of R̂_{Q} = (R̂)_{Q} are minimal primes, proving the claim.

Caution. As mentioned in the introduction to this paper, the J(R)-induced completion R̂_{Q} is not an “adic” completion of R_{Q} with respect to some ideal of R_{Q}. But it is the completion of R_{Q} with respect to a suitable topology determined by J(R), as explainded in Remark 3.6. However, what really interests us is the flatness properties of completions, and these can be obtained more quickly by ignoring the topology and proceeding as explained below.

Since R̂ is a faithfully flat R-module, it is easy to see that R̂_{Q} is a faithfully flat R_{Q}-module.

For a f.g. Λ_{Q}-module V, set V̂ = R̂⊗_{R} V and call it the J(R)-induced completion of V. Note that, by slight abuse of notation, the notation X̂ now has two meanings: the J(R)-adic completion X̂ = R̂⊗_{R}X if X is a f.g. Λ-module, and the J(R)-induced completion X̂ = R̂⊗_{R} X if X is a f.g. Λ_{Q}-module. We shall never have any occasion to refer to R̂ ⊗_{R} X for an arbitrary non-f.g. Λ-module X.

Note that Λ̂_{Q} = R_{Q} ⊗_{R} R̂ ⊗_{R} Λ is a module-finite R̂_{Q}-algebra; and V̂ is a f.g. Λ̂_{Q}-module when V is a f.g. Λ_{Q}-module. Since Λ̂_{Q} is a module-finite R̂_{Q}-algebra, statement (3.1.1) shows:

(3.1.2)  Λ̂_{Q} is an artinian ring

Since R̂ is a flat R-module, we have V ⊆ V̂ via v → 1 ⊗ v [B ’72, Chap. I §3.5 Prop. 9]. We always make this identification. One consequence is that R̂V = V̂. Another consequence of this identification is that, if M is a Λ-submodule of V, we can also regard it as a Λ-submodule of V̂. Therefore R̂M ⋂ V is a well-defined Λ-submodule of V and clearly contains M. Since R̂ is a faithfully flat R̂-module, we have the following “up-down” result [B ’72, Chap. I §3.5 Prop. 10(ii)].

Lemma 3.2. Let V be a f.g. Λ_{Q}-module, and M a Λ-submodule of V. Then (R̂M) ⋂ V = M (intersection in R̂V = V̂).

The corresponding “down-up” result is false; and this is the reason that we need a package deal theorem for completions. We begin with the following key situation.

Lemma 3.3. Let V be a f.g. Λ_{Q}-module, and W a f.g. Λ̂-submodule of V̂ such that W_{Q} = V̂. Then W ⋂ V is a f.g. Λ-module such that R̂(W ⋂ V) = W and (W ⋂ V)_{Q} = V. Moreover, W is a J-adic completion of W ⋂ V.

[figure: V̂ (Λ̂_{Q}-module); W (Λ̂-module); V (Λ_{Q}-module); W ⋂ V (Λ-module)]

(3.3.1)

Proof. Let L be the Λ-module generated by some finite set of Λ_{Q} generators of V. Then L_{Q} = V, and hence L̂_{Q} = V̂. Since L̂ and W are Λ̂-f.g. and L̂_{Q} = V̂ = W_{Q}, there exist c, d ∈ R − ⋃ Q such that dW ⊆ L̂ and cL̂ ⊆ dW. After replacing L by d^{−1}L we have cL̂ ⊆ W ⊆ L̂. Using Lemma 3.2 and recalling that L̂ = R̂L yields

cL ⊆ W ⋂ V ⊆ L̂ ⋂ V = L [⊆ V] (3.3.2)

Since L is a noetherian Λ-module we see that W ⋂ V is a f.g. Λ-module; and localizing (3.3.2) at Q shows that (W ⋂ V)_{Q} = V.

Next we show that W = R̂(W ⋂ V). But since the R-algebra R̂ is a faithfully flat R-module we have R̂(W ⋂ V) = R̂W ⋂ R̂V = W ⋂ V̂ = W [B ’72, Chap. I §3.5 Prop. 10].

In preparation for the proof of the final assertion of the lemma we note that the following natural map is one-to-one since R̂ is a flat R-module.

R̂ ⊗_{R} (W ⋂ V) ⟶ R̂ ⊗_{R} V = V̂ (3.3.3)

Now we prove that W is a J-adic completion of W ⋂ V. It suffices to prove that the natural surjection R̂ ⊗_{R} (W ⋂ V) → R̂(W ⋂ V) = W is an isomorphism. But any element of the kernel of this map is in the kernel of the monomorphism of the map in (3.3.3), and therefore equals zero as desired.

Package Deal Theorem 3.4. (Completions) The following conditions are equivalent for a f.g. Λ̂-module W.

(i) W ≅ M̂ for some f.g. Λ-module M. (ii) W_{Q} ≅ V̂ for some f.g. Λ_{Q}-module V.

Proof. To prove the nontrivial implication, assume (ii) holds. First we consider the Q-torsionfree case [the canonical map W → W_{Q} is one-to-one]. Hence we can suppose W ⊆ W_{Q} = V̂. Then, by Lemma 3.2 we have that W is a J-adic completion of the f.g. Λ-module W ⋂ V.

Next we consider the Q-torsion case, W_{Q} = 0. In this case dW = 0 for some d ∈ R − ⋃ Q. So W is a finitely generated module over the artinian ring R/dR ≅ (R/dR)ˆ ≅ R̂/dR̂. Therefore W has finite length as an R-module and hence is the J-adic completion of itself.

Now we put the torsion and torsionfree cases together. Consider the short exact sequence

E: T ↪ W → → X

where X is the image of W under the canonical map W → W_{Q} and T = ker(W → X) = ker(W → W_{Q}), the Q-torsion submodule of W.

Since T_{Q} = 0, localizing E at Q shows that X_{Q} ≅ W_{Q} ≅ V̂, so the torsionfree case shows that X ≅ N̂ for some f.g. Λ-module N. The torsion case shows that T = T̂.

Localizing Ext_{Λ}^{1}(N, T) at Q, and using the fact that T_{Q} = 0 shows that Ext_{Λ}^{1}(N, T)_{Q} = 0. So Ext_{Λ}^{1}(N, T), like T, is an R-module of finite length and therefore equals its own J-adic completion. Thus, after making the identification X = N̂, we have the canonical isomorphisms

(3.4.1) Ext_{Λ}^{1}(N, T) = [Ext_{Λ}^{1}(N, T)]ˆ = Ext1 Λ̂(N̂, T̂) = Ext1 Λ̂(X, T)

Let D be the element of Ext_{Λ}^{1}(N, T) that is carried to E by the isomorphisms in (3.4.1). Then the middle term of D is a f.g. Λ-module M such that M̂ ≅ W.

Remark 3.5. (i) Let W be a f.g. Λ̂-module and let t(W) = ker(W → W_{Q}), the Qtorsion submodule of W. An immediate corollary of the preceding theorem is that W is the completion of some Λ-module if and only if the Q-torsionfree Λ̂-module W/t(W) is the completion of some Λ-module [since t(W)_{Q} = 0].

(ii) In order to use our package deal theorem we need a convenient way of verifying condition (ii). Thus it would be helpful to have an easily-described family of Λ-modules P such that, for every Λ̂_{Q}-module V, V is the completion of some Λ_{Q}-module if and only if V ≅ P̂_{Q} for some P in the set. In integral representation theory, the modules P that play this role are the projective modules over some maximal order containing Λ. In Section 6 we extend this to orders Λ that are not contained in a maximal order, by showing that the f.g. projective modules P over any convenient splitting order, or any normalization of Λ, can be used for this purpose.

We limit our investigation of this type of question to the situation that Λ is an R-order in a semisimple artinian ring, although the preceding package deal theorem does not make such a restriction. See Theorem 6.15.

Remark 3.6. (Topological version of V̂). Let V be a f.g. Λ_{Q}-module. It is possible to view the J(R)-induced completion V̂ as a topological completion as follows. Choose a f.g. Λ-submodule of L ⊆ V such that L_{Q} = V. Then the family of Λ-submodules {J(R)^{n}L} of V has intersection zero; so forming Cauchy sequences with respect to this filtration yields a Hausdorff topology on V. It can be verified that this topology, hence the resulting completion of V, is independent of the choice of L; and this completion is canonically isomorphic to V̂ = R̂ ⊗_{R} V, as defined in Notation 3.1. See [B ’72, III §2.1, 2.6] for the necessary machinery, but not the actual result. This procedure generalizes the familiar completion of an F -vector space with respect to a discrete valuation of the field F, when F = R_{Q} for some discrete valuation ring R. See [R ’75].

We close this section by extending two simple results (needed later, and familiar in the settings of commutative algebra and integral representation theory) to our situation.

Lemma 3.7. Every Λ̂-module of finite length has finite length as an R-module (not just as an R̂-module), and its R-submodules coincide with its R̂-submodules.

Proof. Every Λ̂-module of finite length has finite length as an R̂-module (Lemma 1.6, applied to the R̂-algebra Λ̂). Also, the canonical isomorphism R̂/J(R̂) = R/J shows that every R̂-module of finite length has the same length as an R-module. This proves the first assertion of the Lemma.

For the second assertion let M be an R-submodule of some R̂-module W of finite length. Then we have M ⊆ R̂M ⊆ W. By the previous paragraph W has finite R-length and we therefore have λ(M) ≤ λ(R̂M), where λ(…) denotes R-length. It now suffices to prove the opposite inequality. Note that every R-module of finite length is its own J-adic completion. Since M̂ = R̂ ⊗ M maps onto the submodule R̂M of X we therefore have λ(M) = λ(M̂) ≥ λ(R̂M) as desired.

Proposition 3.8. Let M be a f.g. Λ-module. Then the set of Λ-submodules N ⊆ M such that M/N has finite length is in one-to-one correspondence with the set of Λ̂- submodules X ⊆ M̂ such that M̂/X has finite Λ̂-length via

N → X = R̂N and X → N = X ⋂ M

Moreover, M̂/X ≅ M/N as Λ-modules.

Proof. We have N = R̂N ⋂ M for all submodules of M by Lemma 3.2. Since M/N has finite Λ-length, it has finite R-length by Lemma 1.6, and is therefore its own J-adic completion. Hence M/N ≅ R̂ ⊗ M/N ≅ M̂/R̂N, as claimed.

Conversely, suppose that M̂/X has finite Λ̂-length. We need to prove that R̂(X ⋂ M) = X. By Lemma 3.7 the Λ̂-module M̂/X of finite length has finite R-length, and is therefore annihilated by some power J^{s} of J = J(R). Hence X ⊇ J^{s}M̂. As in the previous paragraph the completion functor M → M̂ induces an isomorphism M/J^{s}M ≅ M̂/J^{s}M̂ which we regard as equality. Then X/J^{s}M̂ = N/J^{s}M for some Λ-submodule N (J^{s}M ⊆ N ⊆ M) such that R̂N = X. Therefore N = R̂N ⋂ M = X ⋂ M by Lemma 3.2. Therefore R̂(X ⋂ M) = R̂N = X, as desired.

4. Normalizations

In this section we assume that R and Z(Λ) (not necessarily semilocal) have no artinian ring-direct summands and no nonzero nilpotent ideals; and Λ denotes an R-order in the semisimple artinian ring Λ_{Q}, as defined in (1.1.1). In this situation R_{Q} becomes the total quotient ring of R, and R itself is a Q-torsionfree R-module. As mentioned in Notation 1.1 we therefore use the more traditional terminology torsionfree module instead of “Q-torsionfree module”.

This section surveys the basic properties of noncommutative normalizations and our critical link with integral representation theory, the relation between normalizations of R-orders and maximal orders over the normalization of R.

Definitions, Notation 4.1. Throughout the rest of this paper Γ denotes a normalization of Λ; that is, any maximal element Γ of the family of rings such that Λ ⊆ Γ ⊆ Λ_{Q} and Γ is integral over R. Such a Γ obviously exists, by Zorn’s Lemma. Note that the property that Γ is a normalization of Λ is a ring-theoretic property of the inclusion Λ ⊆ Γ. That is, it does not depend upon the particular ring R over which Λ is a module-finite algebra, because an element of Γ is integral over R if and only if it is integral over the center Z(Λ).

(4.1.1) Every f.g. R-subalgebra of Γ is a f.g. R-module. Since every element of Γ is integral over R, this is a special case of Sirsov’s theorem [MR ’87, 13.8.8].

We now summarize the link with integral representation theory that is provided by normalizations.

Let R̃ denote the integral closure of R·1_{Λ} in Z(Λ_{Q}), which equals the integral closure of Z(Λ) in Z(Λ_{Q}). We claim that R̃ is a direct sum of finitely many Dedekind domains R̃_{k} and none of these Dedekind domains is a field.

Proof. Since Z(Λ) has no nilpotent elements, Z(Λ_{Q}) is a finite direct sum of fields F_{k} and the ideals p_{k} = ker(Z(Λ) → F_{k}) are the minimal prime ideals of Z(Λ). Since Z(Λ) has no artinian direct summands, Lemma 1.3 shows that no minimal prime ideal of Z(Λ) is a maximal ideal. This shows that, for each k, the zero ideal of S_{k} is not a maximal ideal. Hence S_{k} has dimension exactly one; in particular it is not a field. Since S_{k} is noetherian of dimension 1, the Krull-Akizuki theorem [N ’62, (33.2)] shows that its normalization R̃_{k} is again noetherian of dimension one, hence a Dedekind domain. Since R̃ = ⊕_{k} R̃_{k} the claim is proved.

(4.1.2) R̃ = Z(Γ), and Γ is a maximal R̃-order in Λ_{Q}. (4.1.3) There is a (finite) decomposition Γ = ⊕_{k} Γ_{k} where each ring Γ_{k} is a central

maximal order over a Dedekind domain (one of the direct summands of R̃) in one of the simple artinian ring-direct summand of Γ_{Q}.

(4.1.4) Every maximal R̃-order in Λ_{Q} containing Λ is a normalization of Λ. Statements (4.1.2)–(4.1.4) are proved in [GL ’89, 2.5]. However, the proof given there omitted mention of Sirsov’s theorem (4.1.1).

Note that (4.1.3) guarantees that normalizations have the familiar ring-theoretic properties of maximal orders. For example, all left ideals are projective modules [R ’75, (21.4)], and a left ideal L is indecomposable if and only if L_{Q} is a simple Λ_{Q}-module [R ’75, (21.5)]. We make frequent use of the following part of statement (4.1.2).

(4.1.5) Γ is a f.g. R̃-module.

(4.1.6) The sets of maximal (2-sided) ideals of the rings Γ and R̃ are in one-to-one

correspondence via contraction; that is, via A → (A ⋂ R̃) [R ’75, (22.4)]. (4.1.7) Let Γ′ be any other normalization of Λ. Then Γ and Γ′ are Morita equivalent.

Indeed, there is an invertible (Γ, Γ′)-subbimodule M of Λ_{Q} such that Γ′ = H^{−1}ΓH. In particular, if R is semilocal, any two normalizations of Λ are conjugate in Λ_{Q}.

To prove (4.1.7) let Λ_{k} be the projection of Λ in the simple ring-direct summand (Γ_{k})_{Q} of Λ_{Q}. In view of (4.1.2) every normalization of Λ is also a normalization of ⊕_{k} Λ_{k}. Therefore, to prove (4.1.7), we can work one coordinate ring Λ_{k} at a time. In this situation the desired result is proved in [R ’75, (22.21)]. When R is semilocal, so is R̃, by Lemma 4.2 below. Therefore the invertible bimodule M becomes principal (on each side), yielding the claimed conjugacy. (4.1.8) If R is semilocal, then each Γ_{k} is a full matrix ring (of size usually denoted

by n_{k}×n_{k}) over a semilocal principal ideal domain (possibly noncommutative). Hence the Krull-Schmidt theorem holds for f.g. Γ-modules.

As already mentioned, R̃ is semilocal when R is. Now see the proofs of [R ’75, (18.7) and (21.6)], noting that N in (21.6) is free in the semilocal case.

(4.1.9)

(i) Every Γ-module of finite length has finite length as an R-module.

(ii) For every regular element c of Z(Λ), Γ/cΓ has finite length as an R-module. (iii) J(Z(Λ)) ⊆ J(Γ).

We prove (4.1.9) by reducing to the situation that Γ is module-finite over R, using the part of our link with integral representation theory that states that Γ is a module-finite R̃-algebra [see (4.1.2)].

(i). Since Γ is a module-finite R̃-algebra, the given Γ-module of finite length has finite length as an R̃-module [Lemma 1.6]. It therefore suffices to show that every simple R̃-module R̃/A has finite length as an R-module. If R and hence R̃ are integral domains, this is part of the conclusion of the Krull-Akizuki theorem [N ’62, (33.2)]. Since R has no nilpotent elements, the general case is easily reduced to this.

(ii). By statement (i) it suffices to show that Γ/Γc is an artinian ring. Since R is now out of the picture, we can redefine R to be Z(Γ), so that Γ becomes module-finite over R by (4.1.2). Since c is a regular element of R, and R is has dimension 1, the ring R/cR is artinian. Therefore the module-finite R/cR-algebra Γ/Γc is an artinian ring, as desired.

(iii). We may take R = Z(Λ). Since R̃ = Z(Γ) is an integral extension of R, the lying-over and going up theorems of commutative algebra [K ’70, Theorem 44] show that every maximal ideal of R has a maximal ideal of R̃ lying over it and every maximal ideal R̃ contracts to a maximal ideal of R. Therefore J(R) ⊆ J(R̃). Since Γ is module-finite over R̃, we have J(R̃) ⊆ J(Γ) [every simple Γ-module S is module-finite over R̃, hence J(R̃)·S = 0 by Nakayama’s Lemma]. Therefore statement (iii) holds.

A special case of the next result is that our assumption that Λ has no artinian ring-direct summands carries over to Γ.

(4.1.10) Γ has no Γ-submodules of finite length and no R-submodules of finite length.

Proof. Let U ⊆ Γ be an R-submodule of finite length. Since Γ_{Q} = Λ_{Q} we have cU ⊆ Λ for c ∈ R−⋃ Q. Since Λ is an R-order in Λ_{Q} we have U ≅ cU as Λ-modules. Since Λ has no nonzero R-submodules of finite length [see (1.2.1)], we have U = 0 as desired.

The assertion about Γ-submodules now follows from (4.1.9)(i).

Lemma 4.2. Let S be a ring, integral over R, such that R ⊆ S ⊆ R_{Q} (and recall that we are assuming that R has no nilpotent elements ≠ 0 and no artinian ringdirect summands). Let M be a finite set of prime ideals of R, and N the (necessarily finite) set of prime ideals P of S such that P lies over some prime ideal in M. Then (R − ⋃ M)^{−1}S = S_{N} (i.e. both localizations invert the same set of elements of S).

Proof. Since R has no nilpotent elements, R_{Q} is the total quotient ring of R, hence of S. By (a minor extension of) the Krull-Akizuki theorem [N ’62, (33.2)] S is Therefore noetherian of Krull dimension 1.

Next we prove that N is finite. Since the noetherian ring S of Krull dimension 1 has only finitely many minimal primes it suffices to show that N contains only finitely many maximal ideals that are not minimal primes. Each such maximal ideal n ∈ N contracts to a maximal ideal of R, because n properly contains a minimal prime p, and n and p must contract to distinct primes of R by the incomparability theorem of commutative algebra [K ’70, Theorem 44]. Therefore it suffices to fix a maximal ideal m ∈ M and prove that m is contained in only finitely many maximal ideals of S.

Since R has no nilpotent elements, ⋃ Q(R) is the set of zero-divisors of R. Since m is not a minimal prime, it therefore contains a regular element d of R. This element d is a unit in R_{Q} and therefore is regular in S. Therefore d is outside of all minimal primes of S, and therefore the noetherian ring S/Sd has dimension 0 and is therefore artinian. It follows that d, and therefore m, is contained in only finitely many maximal ideals of S, as claimed.

Now we prove equality of the two localizations of S in the statement of the lemma. As in the proof of Lemma 1.5 it suffices to show that the same prime ideals of S both localizations of S.

Suppose that a prime ideal p of S survives in S_{N}. Then p ⊆ n for some n ∈ N. Since n ⋂ R ∈ M we see that p survives in (R − ⋃ M)^{−1}S.

Conversely, suppose that p is a prime ideal of S that survives in (R − ⋃ M)^{−1}S. We consider two cases. Case 1: p ⋂ R ∈ M. In this case P ∈ N; and therefore p survives in S_{N}. Case 2. p ⋂ R ∉ M. Since p survives in (R − ⋃ M)^{−1}S, p (and hence p ⋂ R) must have empty intersection with the multiplicatively closed set R − ⋃ M with respect to which we are localizing. Therefore there is a prime ideal m ∈ M such that p ⋂ R ⊂ m. Since S is integral over R, the going up theorem of commutative algebra [K ’70, Theorem 44] yields a prime ideal n of S that contains p and lies over m. But then p ⊂ S − ⋃ N and hence p survives in S_{N}.

Lemma 4.3. (Normalizations Localize) For every finite set M of maximal ideals of R the localization Γ_{M} is a normalization of the R_{M}-algebra Λ_{M} [in (Λ_{M})_{Q}].

Note. The notation (Λ_{M})_{Q} at the end of the lemma should really be (Λ_{M})_{Q(R_{M})}. But by a slight modification of Lemma 2.3(i) these two localizations of Λ_{M} are the same, and we therefore use the shorter notation.

Proof. By the note above and Lemma 1.5 we may assume that R ⊆ Λ. Observe that this lemma generalizes the familiar fact that classical maximal orders localize, and the slight extension of this fact in [GL ’88, 2.7], but its proof uses these facts. We give the short proof of these known facts, since it takes less space than explaining the difference between two sets of notations and subtleties. Thus, suppose that Γ is a maximal R-order in Λ_{Q} and Γ_{M} ⊆ Υ ⊆ (Γ_{M})_{Q} for some R_{M}-order Υ. For each maximal ideal m of R we define a Λ_{m}-submodule X(m) ⊆ (Γ_{m})_{Q} = (Γ_{Q})_{m} as follows. If m ∈ M let X(m) = Υ_{m}, but if m ∉ M let X(m) = Γ_{m}. Since Γ_{M} ⊆ Υ ⊆ (Γ_{M})_{Q} we have X(m)_{Q} = (Γ_{m})_{Q} for all m. And since M is a finite set we have X(m) = Γ_{m} for all but finitely many m. Therefore Corollary 2.7 to our package deal theorem for localizations of submodules yields a f.g. Λ-submodule X ⊆ Γ_{Q} such that X_{m} = X(m) for every m. It is easily checked, locally, that Γ ⊆ X, X^{2} = X and 1 ∈ X. Therefore X is an R-suborder of Γ_{Q} = Λ_{Q} containing Γ. Maximality of Γ then shows that Γ = X; and localizing at M then shows that Υ = Γ_{M}, proving the lemma when Γ is an R-order; that is, maximal orders localize to maximal orders.

To extend the lemma to the non-module-finite case, we use our link with integral representation theory. Let N be the set of maximal ideals n of R̃ such that n lies over some maximal ideal of R. By Lemma 4.2, N is a finite set. Since maximal orders localize to maximal orders, we therefore have that Γ_{N} is a maximal R̃_{N} -order in (Γ_{N})_{Q}. By Lemma 4.2, localizing R̃ and Γ at N is the same as localizing them at the multiplicatively closed subset R − ⋃ M. Indicating this latter localization by the subscript M we have that Γ_{M} is a maximal R̃_{M}-order in (Γ_{M})_{Q}. Since, by standard commutative algebra, R̃_{M} is the normalization of R_{M} [in its total quotient ring (R_{M})_{Q}], the proof is completed by (4.1.4).

Proposition 4.4. Suppose that R is semilocal and complete in its J(R)-adic topology. Then any normalization Γ of Λ is a f.g. R-module, hence is a maximal R-order in Λ_{Q}.

Proof. Since Λ is a f.g. R-module and Γ_{Q} = Λ_{Q}, Γ_{Q} is a f.g. R_{Q}-module, and hence Z(Γ_{Q}) is a finitely generated R_{Q}-module. Since R is complete, the integral closure R̃ of R in Z(Γ_{Q}) is a finitely generated R-module, by Nagata’s theorem [Mts ’80, Cor. 2 to Theorem 31C]. Since Γ is a f.g. R̃-module, by (4.1.2), the proof is complete.

5. Splitting Orders

In this section R is semilocal and has no nilpotent elements ≠ 0, R and Z(Λ) have no artinian ring direct summands, and Λ is an R-order in a semisimple artinian ring as defined in (1.1.1). Γ denotes a fixed normalization of Λ, and R̃ denotes the integral closure of R·1_{Λ} in Z(Λ_{Q}). Recall that R̃ = Z(Γ), by (4.1.5). We make frequent use of the decomposition Γ = ⊕_{k} Γ_{k} in (4.1.3), in which each Γ_{k} is a full matrix ring over a semilocal principal ideal domain [see (4.1.8)].

Since R is semilocal, so is R̃ [Lemma 4.2], and hence Γ has only finitely many maximal ideals [by (4.1.6)].

Definition 5.1. A splitting order is an R-order Ω such that Ω ⊆ Γ and Ω_{Q} = Γ_{Q} (= Λ_{Q}), and such that Ω has the properties (5.1.1)–(5.1.4) below.

(5.1.1)  R̃ Ω = Γ

(5.1.2) The ring Ω decomposes like Γ, that is, Ω = ⊕_{k} Ω_{k} (2-sided decomposition) where Ω_{k} ⊆ Γ_{k} for each k [decomposition of Γ as in (4.1.3)] and Ω_{k} and Γ_{k} are full matrix rings of the same size over integral domains.

(5.1.3) R̃(A ⋂ Ω) = A (∀ maximal ideal A of Γ) and R̃(m′ ⋂ Z(Ω)) = m′ (∀ maximal ideal m′ of R̃); and each maximal ideal of Ω is the contraction of some maximal ideal of Γ.

(5.1.4) For every maximal ideal A of Γ, Ω/(A ⋂ Ω) = Γ/A; that is, the natural monomorphism from the left-hand side to the right-hand side is a surjection. (In particular, A ⋂ Ω is a maximal ideal of Ω.)

One can obtain an equivalent restatement of (5.1.4) by replacing Γ, Ω by Γ_{k}, Ω_{k} respectively, and adding “for every k”.

Theorem 5.2. (i) Let F be a finite subset of Γ and S a finite set of torsionfree (left) Λ-modules. Then there exists a splitting order Ω (Λ ⊆ Ω ⊆ Γ) such that Ω ⊇ F and such that ΩL is Ω-projective for every L ∈ S.

(ii) If Ω is a splitting order and Ω′ is an R-order such that Ω ⊆ Ω′ ⊆ Γ, then

Ω′ is again a splitting order. (iii) If Ω and Ω′ are splitting orders contained in Γ, then some splitting order

Ω′′ ⊆ Γ contains both Ω and Ω′.

Proof. We may assume that R = Z(Λ). Recall that by Sirsov’s theorem (4.1.1) every f.g. R-subalgebra of Γ is a f.g. R-module. We construct Ω by a sequence of successive enlargements of Λ, keeping Ω ⊆ Γ at each step. At each stage of the enlargement process, Ω will be a ring that is module-finite over R and therefore an R-order in Λ_{Q} = Γ_{Q}.

Property (5.1.1). By (4.1.5) there is a finite generating set F′ for the R̃-module Γ. Let F′′ be a finite generating set of the R-module Λ, and then let Ω be the Ralgebra generated by F′ ⋃ F′′. By Sirsov’s theorem, Ω is a module-finite R-algebra, and clearly Λ ⊆ Ω ⊆ Γ. We now have property (5.1.1), and any further enlargement of Ω obviously preserves this property.

Recall that for a ring S and a positive integer n, we have S ≅ M_{n}(T) for some ring T if and only if S contains a set of n × n matrix units; that is, a set of n^{2} elements e_{ij} (1 ≤ i, j ≤ n) such that e_{ij}e_{jk} = e_{ik} for all i, j, k, e_{ij}e_{kl} = 0 whenever j ≠ k, and ∑_{i} e_{ii} = 1. T can then be taken to be the centralizer of the e_{ij}.

Property (5.1.2). Enlarge Ω so that it contains all of the (finitely many) central idempotents of Γ, and a set of matrix units (of appropriate size) of each Γ_{k}. Again, further enlargements of Ω preserve these properties.

Property (5.1.4). Let d be a regular element of J(R), which exists by Lemma 1.3(iii) since R = Z(Λ) has no artinian ring direct summands. Since every maximal ideal of the integral extension R̃ of R = Z(Λ) lies over a maximal ideal of R [K ’70, Theorem 44], we have d ∈ J(R̃). Since Γ is a module-finite R̃-algebra [by (4.1.5)], we have J(R̃)·Γ ⊆ J(Γ), and therefore d ∈ J(Γ). By the Krull-Akizuki theorem [N ’62, (33.2)] the artinian ring R̃/R̃d is f.g. R-module. Therefore the module-finite R̃/R̃d-algebra Γ/Γd is a f.g. R-module. Therefore a finite enlargement of Ω makes the natural map Ω → Γ/Γd a surjection. This enlarged Ω satisfies (5.1.4), and further enlargements preserve this property.

Property (5.1.3). Each of the finitely many maximal ideals m′ of R̃ has a finite, (in fact, 1-element) generating set. Including these generators in Ω assures us that R̃(m′ ⋂ Z(Ω)) = m′.

Similarly we obtain R̃(A ⋂ Ω) = A by including, in Ω, a finite set of Γ-generators of each of the finitely many of maximal ideals A of Γ.

The more difficult part of the proof of (5.1.3) is to show that every maximal ideal B of Ω is the contraction of some maximal ideal A of Γ. It suffices to show that ΓBΓ ≠ Γ. For then some maximal ideal A of Γ contains ΓBΓ and therefore A ⋂ Ω ⊇ B, whence maximality of B shows that A ⋂ Ω = B. So suppose that ΓBΓ = Γ.

Recall that Γ is a centralizing extension of Ω [property (5.1.1)]. In [MR ’87, Theorem 10.2.9] a lying-over theorem is proved for finite centralizing extensions, when the rings involved are left noetherian. However, our Γ need not be a f.g. Ω-module. Before reducing to the finite case, we recall the relevant definitions. A ring T is a finite centralizing extension of a subring S if there exist finitely many elements x_{i} ∈ T such that T = ∑_{i} Sx_{i} and x_{i}s = sx_{i} for every s ∈ S.

First note that since R̃Ω = Γ our assumption ΓBΓ = Γ can be rewritten R̃B = Γ. This yields an expression of the form ∑_{i} x_{i}b_{i} = 1 with each x_{i} ∈ R̃ and b_{i} ∈ B. Let Ω′ be the R-subalgebra of Γ generated by Ω and the x_{i}. By Sirsov’s theorem Ω′ is a f.g. R-module. Also, Ω′BΩ′ = Ω′. Since Ω′ is a finite centralizing extension of Ω, the lying-over theorem for finite centralizing extensions yields a prime ideal B′ of Ω′ such that B′ ⋂ Ω = B, and this contradicts the relation Ω′BΩ′ = Ω′, completing our proof of (5.1.3).

This completes the proof of parts (i) and (ii), except for the statement that we can choose Ω large enough to make each ΩL Ω-projective. This is straightforward, and is done in [GL ’89, 2.8]. Part (iii) is an immediate consequence of Sirsov’s theorem.

Lemma 5.3. Let Ω be a splitting order. Then

(i) The sets of maximal (2-sided) ideals of all four rings Γ, Ω, Z(Γ), Z(Ω) are in one-to-one correspondence via contraction.

(ii) J(Γ) ⋂ Ω = J(Ω) and R̃·[J(Ω)] = J(Γ).

Proof. (i) The statement about Γ and Z(Γ) was already noted in (4.1.6). The one-to-one correspondence of maximal ideals between Γ and Ω is part of (5.1.3) and (5.1.4). Most of this correspondence for R̃ and Z(Ω) was stated in (5.1.3). We need to show that every maximal ideal of Z(Ω) is the contraction of some maximal ideal of R̃. Since R̃ is integral over Z(Ω), this follows from the lyingover and incomparability theorems of commutative algebra [K ’70, Theorem 44]. Finally, the relation between the maximal ideals of Ω and its center follows from the situations already considered.

(ii) The first equality follows from statement (i) as soon as we know that the Jacobson radical of each ring is the intersection of its maximal ideals. This follows from Lemma 1.6(i), applied to the R-algebra Ω and the R̃-algebra Γ.

We define the symmetric product P(D_{1}, …, D_{n}) of ideals D_{1}, …, D_{n} of a ring S. This consists of the sum of all n! possible terms, where each term is the product of all n of the ideals D_{k}, taken in some order without repetition. Thus P(D_{1}, D_{2}) = D_{1}D_{2} + D_{2}D_{1}. We claim that, if S = D_{i} + D_{j} whenever i ≠ j, then:

⋂_{i} D_{i} = P(D_{1}, …, D_{n}) (5.3.1)

We have S = (D_{1} + D_{2}) ⋯ (D_{1} + D_{n}) = D_{1} + (D_{2} ⋯ D_{n}). Left-multiplying this by I = ⋂_{i} D_{i} yields I = ID_{1} + I(D_{2} ⋯ D_{n}). Since I ⊆ ⋂_{i≥2} D_{i}, induction shows that ID_{1} ⊆ P(D_{1}, …, D_{n}). The term I(D_{2} ⋯ D_{n}) in our expression for I is obviously contained in P(D_{1}, …, D_{n}), and this proves the inclusion (⊆) in (5.3.1). The opposite inclusion is obvious.

To prove the second equality in statement (ii), let A_{1}, …, A_{n} be the maximal ideals of Γ, and let D_{i} = A_{i} ⋂ Ω. Then, by statement (i) the distinct maximal ideals of Ω are D_{1}, …, D_{n}; and R̃D_{i} = A_{i} by (5.1.3). We have J(Ω) = ⋂_{i} D_{i}. Therefore (5.3.1) yields

R̃·J(Ω) = R̃(⋂_{i} D_{i}) = R̃P(D_{1}, …, D_{n}) = P(A_{1}, …, A_{n}) = J(Γ)

completing the proof.

Lemma 5.4. Let Ω be a splitting order, and let S be a simple (left) Γ-module. Then:

(i) S is a simple Ω-module, and all simple Ω-modules are of this form. (ii) Nonisomorphic simple Γ-modules remain non-isomorphic as Ω-modules. (iii) Γ ⊗_{Ω} S ≅ S as Γ-modules.

Proof. The annihilator A of S in Γ is a maximal ideal of Γ, by Lemma 1.6 applied to Γ as an R̃-order. Since Γ/A is a simple artinian ring, we have Γ/A ≅ S^{m} as Γ-modules, for some m. By (5.1.4) we have Ω/(A ⋂ Ω) ≅ S^{m} as Ω-modules, and S is a simple Ω-module. Also, Γ(A ⋂ Ω) = A by (5.1.3) and (5.1.1). Therefore

(Γ ⊗_{Ω} S)^{m} ≅ Γ ⊗_{Ω} S^{m} ≅ Γ ⊗_{Ω} Ω/(A ⋂ Ω) ≅ Γ/Γ(A ⋂ Ω) = Γ/A ≅ S^{m}

Since S is a simple Γ-module, this implies statement (iii).

To complete the proof of (i), let T be a simple Ω-module. We show that T ≅ S for some S as above. By Lemma 1.6, the annihilator D of S in Ω is a maximal ideal of Ω. Therefore T is the unique simple module over the simple artinian ring Ω/D. By Lemma 5.3 we have D = A ⋂ Ω for some maximal ideal A of Γ. It follows that T is also the unique simple Γ/A-module, hence a simple Γ-module.

To prove (ii), note that the annihilators of non-isomorphic simple Γ-modules S, S′ are distinct maximal ideals of Γ, say A, A′ respectively. The annihilators of S and S′ in Ω are A ⋂ Ω and A′ ⋂ Ω, respectively, and these are distinct maximal ideals of Ω by Lemma 5.3. Therefore S and S′ are not isomorphic Ω-modules.

Now we obtain the second main result of this section: the structure of projective Ω-modules is the same as the structure of projective Γ-modules (equivalently, the same as the structure of projective Ω_{Q}-modules).

Theorem 5.5. Let Ω be a splitting order, and let P, P′ be f.g. projective Ω-modules. Then:

(i) P decomposes like P_{Q}; that is, P = ⊕_{i} P_{i} where each P_{i} is indecomposable, each (P_{i})_{Q} is an indecomposable (hence simple) Ω_{Q}-module, and P_{i} ≅ P_{j} if and only if (P_{i})_{Q} ≅ (P_{j})_{Q}.

(ii) P ≅ P′ ⟺ P_{Q} ≅ P′_{Q}. Moreover, every f.g. (necessarily projective) Ω_{Q}- module is isomorphic to P_{Q} for some f.g. projective Ω-module P.

(iii) The Krull-Schmidt theorem holds for projective Ω-modules.

Proof. By (5.1.2) we may suppose that Ω = Ω_{1} and Γ = Γ_{1}; that is, Ω and Γ are R-orders in the simple artinian ring Ω_{Q} = Γ_{Q}. Then Ω and Γ are full n × n matrix rings over integral domains. Let L = Ωe_{11} where e_{11} is the n × n matrix with 1 in its (1,1)-position and 0 elsewhere. If we can show that

(5.5.1) P ≅ L^{s} for some unique integer s ≥ 0

then all three assertions of the theorem follow immediately. Uniqueness of s holds because L is a noetherian module.

To show the existence of s, first note that Γ ⊗_{Ω} L ≅ ΓL = Γe_{11}. Since Γ ⊗_{Ω} P is Γ-projective and Γ is an n×n matrix ring over a principal ideal domain, Γe_{11} is the unique indecomposable projective Γ-module, and we therefore have Γ⊗P ≅ (Γ⊗L)^{s} for some s. To complete the proof we cancel Γ from this isomorphism. Reducing both sides modulo J(Γ) = R̃J(Ω) [Lemma 5.3] and using right-exactness of ⊗ yields

Γ ⊗_{Ω} P/(J(Ω)·P) ≅ Γ ⊗_{Ω} (L/J(Ω)·L)^{s} (5.5.2)

Since P/(J(Ω)·P) and (L/J(Ω)·L)^{s} are semisimple Ω-modules, Lemma 5.4 allows us to cancel Γ from the isomorphism in (5.5.2). Therefore P/(J(Ω)·P) ≅ (L/J(Ω)·L)^{s} as Ω-modules. Uniqueness of the projective cover (see, e.g. [LO, 8.8]) then yields P ≅ L^{s}.

Corollary 5.6. Each prime ring-direct summand Ω_{k} of Ω has, up to isomorphism, exactly one indecomposable f.g. projective module—which we call B_{k} from now on.

Proof. Since (Ω_{k})_{Q} is a simple artinian ring, the result follows immediately from the theorem.

6. Completions of Orders, Artinian Pairs

Throughout this section R is semilocal and has no nilpotent elements ≠ 0, and Λ is an R-order in the semisimple artinian ring Λ_{Q}, as in (1.1.1). We also make the nontriviality assumption that Λ [equivalently, Z(Λ)] has no artinian ring-direct summands. By (4.1.10), Γ and Z(Γ) likewise have no artinian ring-direct summands.

According to the “abstract” Package Deal Theorem for completions 3.4, a f.g. Λ̂-module X is the completion of some f.g. Λ-module if and only if X_{Q} is the completion of some Λ_{Q}-module. In order to make use of this, we need an easily verifiable criterion for the latter condition to be fulfilled. Let Γ denote a normalization of Λ. The condition is that there exists a f.g. projective module P over Γ [alternatively, over some conveniently chosen splitting order Ω] such that R̂ ⊗_{R} P_{Q} ≅ X_{Q}. The Package Deal Theorem for completions of orders, in this section, proves this and reduces its verification, for a specific X, to the comparison of two arrays of nonnegative integers called “ranks” (which describe the decomposition of X_{Q}) and “splitting numbers”, which describe how the decomposition of Γ is refined by passing to Γ̈ = R̂ ⊗_{R} Γ.

Caution. As mentioned in the discussion of §6, in the Introduction to this paper, the notation Γ̈ is chosen to emphasize that the ring Γ̈ = R̂ ⊗_{R} Γ is not an “adic” completion of Γ when Γ is not module-finite over R. Nor is it noetherian in this situation.

We prepare for the proof of our Package Deal Theorem for completions of orders by giving detailed descriptions of the projective modules over the rings Γ̈, Ω̂ (Ω any splitting R-order contained in Γ), and their localizations Γ̈_{Q} = Ω̂_{Q}. An unexpected technical difficulty is the proof that indecomposable projective Ω_{Q}- and Γ_{Q}-modules remain indecomposable after tensoring with R̂ (equivalently, primitive idempotents of these rings remain primitive in Ω̂_{Q} = Γ̈_{Q}).

When dealing only with torsionfree Λ-modules, there is an alternative to working with Λ̂. One can work with a simpler structure involving modules over an associated pair of artinian rings. In the last part of this section we obtain the package deal theorem associated with artinian pairs.

Notation 6.1. The results and proofs of this section require a great deal of technical notation, much of which we collect here. Subscript Q — as in Λ_{Q} or M_{Q} — always denotes the Q(R)-localization, that is, the localization formed by using the denominator set R − ⋃ Q(R). In particular, the notation Λ̂_{Q} denotes the localization formed by using the denominator set R − ⋃ Q(R) rather than its counterpart R̂ − ⋃ Q(R̂) for R̂. Recall that the ring Λ̂_{Q} = R̂ ⊗_{R} R_{Q} ⊗_{R} Λ is artinian, by (3.1.2).

Fix a normalization Γ of Λ for the rest of this section. The notation Ω always denotes a splitting R-order contained in Γ (but not always the same splitting order). By (5.1.2) we have 2-sided decompositions:

Ω = ⊕_{k} Ω_{k} and Γ = ⊕_{k} Γ_{k} (6.1.1)

where each Ω_{k} ⊆ Γ_{k} and (Ω_{k})_{Q} = (Γ_{k})_{Q}, a simple artinian ring.

Simple modules S_{ki} and maximal ideals M′_{ki}. The four main rings dealt with in this section — Γ, Γ̈, Ω, Ω̂ — all turn out to have the same finite family of simple (left) modules. For our formal definition, we begin with Γ. For each index k in (6.1.1) let S_{k1}, S_{k2}, … be a set of representatives of the isomorphism classes of simple (left) Γ_{k}-modules. Then the set {S_{ki}} (where both k and i are allowed to vary) is a set of representatives of the isomorphism classes of simple Γ-modules. Since R is semilocal in this section, {S_{ki}} is a finite set as we observe in the next paragraph.

Since Γ is a module-finite R̃-algebra [see (4.1.2)], its set of isomorphism classes of simple left modules S corresponds bijectively with its set of maximal ideals via S → ann_{Γ} S [Lemma 1.6]. Moreover, since R is semilocal, Γ has only finitely many maximal ideals, as explained immediately before Definition 5.1. Therefore each Γ_{k} has only finitely many simple left modules S_{ki}, as mentioned above. We index the maximal ideals of each Γ_{k} with the same set of ordered pairs (k, i) used for simple modules by setting M′_{ki} = ann_{Γ_{k}} S_{ki}.

Splitting numbers s_{ki}. Each Γ_{k} is a full matrix ring over a principal ideal domain, by (4.1.8), and therefore has a unique (up to isomorphism) indecomposable f.g. projective left module Γ_{k}e_{k}, where e_{k} is an arbitrary primitive idempotent of Γ_{k}. Since the ring Γ_{k}/M′_{ki} is simple artinian, its module Γ_{k}e_{k}/M′_{ki}e_{k} is the direct sum of some number s_{ki} of copies of the simple Γ_{k}-module annihilated by M′_{ki}. That is, (6.1.2) Γ_{k}e_{k}/M′_{ki}e_{k} ≅ (S_{ki})^{s_{ki}} where M′_{ki} = ann_{Γ_{k}} S_{ki}. We call s_{ki} the splitting number of M′_{ki}, or the (k, i)-splitting number of Γ. We often write Γ̄ = Γ/J(Γ) and Γ̄_{k} = Γ_{k}/J(Γ_{k}). Since J(Γ_{k}) is the intersection of the maximal ideals of Γ_{k}, we can rewrite (6.1.2) in the form

Γ̄_{k}ē_{k} ≅ ⊕_{i} (S_{ki})^{s_{ki}} (6.1.3)

The simple Γ-modules coincide with the simple Ω-modules (for every Ω). This follows from the relation

Ω/J(Ω) = Γ/J(Γ) = Γ̄ (6.1.4)

where the first equality means that the natural map is a bijection. Relation (6.1.4) follows imediately from (5.1.4) and Lemma 5.3.

Caution. When dealing with artinian pairs, at the end of this section, we use the overbar notation Ω̄ to denote reduction modulo a conductor ideal. Whenever we use the overbar notation we will include reminders of what is intended.

Maximal ideals M_{ki}. By Corollary 5.6, Ω_{k} has a unique indecomposable f.g. projective left module that we call B_{k}; and by Lemma 5.3 the maximal ideals of Ω_{k} are the ideals M_{ki} = M′_{ki} ⋂ Ω_{k}, with distinct subscripts (k, i) corresponding to distinct ideals M_{ki}. The following facts refine (6.1.4)

(6.1.5) B_{k}/M_{ki}B_{k} ≅ (S_{ki})^{s_{ki}} where M_{ki} = ann_{Ω_{k}} S_{ki} = M′_{ki} ⋂ Ω_{k} and therefore we also refer to s_{ki} as the splitting number of M_{ki}.

To prove (6.1.5), note first that the natural monomorphism Ω_{ki}/M_{ki} → Γ_{k}/M′_{ki} is a bijection by (5.1.4). Therefore the simple Γ_{k}-modules coincide with the simple Ω_{k}-modules, as claimed. By (5.1.2), Ω_{k} and Γ_{k} are full matrix rings of the same size n_{k} × n_{k} over integral domains. It follows that (B_{k}/M_{ki}B_{k})^{n_{k}} ≅ Ω_{k}/M_{ki} and (Γ_{k}e_{k}/M′_{ki})^{n_{k}} ≅ Γ_{k}/M′_{ki}. The natural isomorphism Ω_{ki}/M_{ki} ≅ Γ_{k}/M′_{ki}, together with (6.1.2) now yields (6.1.5).

Matrix size. Let Δ be an integral domain (perhaps a division ring). We define the matrix size of any ring isomorphic to the full n × n matrix ring M_{n}(Δ) to be the integer n. In particular, the matrix sizes of the rings Γ_{k} and Ω_{k} will often be denoted by n_{k}, as in the preceding paragraph.

The foregoing definitions of splitting numbers were stated in terms of left modules. We claim that using right modules would yield the same splitting numbers. This is an immediate consequence of the following easily derived formula.

(6.1.6) s_{ki} = \frac{matrix size of Γ_{k}/M′_{ki}}{matrix size of Γ_{k}}

Finally, we note that the set of splitting numbers s_{ki} is independent of which normalization Γ we choose because, since R is semilocal, Γ is unique up to conjugacy, by (4.1.7).

Rings Γ̂, Γ̈. The notation Γ̂ denotes the completion of Γ with respect J(R̃), where, as in (4.1.2), R̃ = Z(Γ). Since, in our semilocal situation, R̃ is a direct sum of principal ideal domains and Γ is a direct sum of classical maximal orders over the ring-direct summands of R̃ in simple algebras [see (4.1.3)], Γ̂ is a direct sum of classical maximal orders over discrete valuation rings in simple algebras, and we can make the identification Γ̂ = (R̃)ˆ ⊗R̃ Γ as (R̃)ˆ -algebras.

Γ̂ is also the J(R)-adic completion of Γ, since J(R)·1_{Γ} ⊆ J(R̃) and J(R̃) is nilpotent modulo J(R)·R̃ (see the proof of Lemma 6.2). But this fact does not seem particularly useful in the situations that interest us, because when Γ is not module-finite over R, the rings Γ̂ and Γ̈ = R̂ ⊗_{R} Γ are never isomorphic. This follows, for example, from the fact that the ring Γ̈ is not noetherian in this situation [Lemma 6.7]. Furthermore, it has nilpotent ideals since R̂ does.

The ring Γ̈ = R̂ ⊗_{R} Γ, despite being nonnoetherian, has a number of finiteness conditions that we need [see Lemma 6.7 below], and plays a more central role in this section than Γ̂ does because it is more directly related to the flatness properties of R̂ as an R-module.

We shall use the following simple fact often.

(6.1.7) For every R-module U of finite length we have U = R̂ ⊗_{R} U (that is, the natural map from the left-hand side to the right-hand side is a bijection). In particular, U is an R̂-module. Moreover, every R-submodule of U is an R̂-submodule.

Proof. Since U is f.g. as an R-module, R̂ ⊗_{R} U is a J(R)-adic completion of U; and since U has finite length, this can be identified with U itself, proving the first assertion. To prove the last assertion, let V be an R-submodule of U. Then, by the first assertion, R̂ ⊗_{R} V = V; and therefore V is an R̂-module. Moreover, since R̂ is a flat R-module, the inclusion V ⊆ U induces an inclusion R̂ ⊗ V ↪ R̂ ⊗_{R} U, showing that V is an R̂-submodule of U.

Lemma 6.2. The J(R)-adic completion of Λ equals the J(Z(Λ))-adic completion of Λ.

Proof. Let S = Z(Λ). It suffices to show that the ideals J(R)·S and J(S) generate the same adic topology on S. To prove this it suffices to show that each of these ideals contains a power of the other. Since S is a module-finite R-algebra, J(R) annihilates every simple S-module (as in the proof of Lemma 1.6), and hence J(R)·S ⊆ J(S). On the other hand, S/(J(R)·S) is a module-finite algebra over the artinian ring R/J(R), and its radical is therefore nilpotent. Therefore some power of J(S) is contained in J(R)·S.

Lemma 6.3. Z(Λ̂) = Z(Λ)ˆ .

Proof. We sketch a proof of this known result because we do not know a reference to it. Let a_{1}, a_{2}, … be a finite set of generators of Λ as an R-module. Then they also generate Λ̂ as an R̂-module. Let A_{i} = {xa_{i} − a_{i}x | x ∈ Λ}. Then let

f: Λ → ⊕_{i} A_{i} (6.3.1)

be the R-homomorphism whose i^{th} coordinate map is x → xa_{i} − a_{i}x. The kernel of this map is Z(Λ); and since the a_{i} generate Λ̂ the kernel of the induced map f̂: Λ̂ → ⊕_{i} Â_{i} is Z(Λ̂). Since J(R)-adic completion is accomplished by tensoring with the flat R-module R̂, the kernel of f̂ = 1 ⊗ f equals (ker f)ˆ , as claimed.

Corollary 6.4. Λ̂ is an indecomposable ring if and only if Z(Λ) is a local ring.

Proof. By Lemma 6.2 we may assume that R = Z(Λ). Ring-direct-sum decompositions of Λ̂ arise from writing its identity element as a sum of orthogonal, central idempotents. Therefore Λ̂ is an indecomposable ring if and only if Z(Λ̂) is. Since Z(Λ̂) = Z(Λ)ˆ [Lemma 6.3], our corollary is reduced to a problem in commutative rings. The proof is completed by the fact that if R = Z(Λ) is semilocal with, say, m maximal ideals, then R̂ is the direct sum of m local rings. [B ’72, Chap III, §2.13, final Cor.].

As in [LO], we call a ring T matrix-local if T/J(T) is a simple artinian ring.

Lemma 6.5. Let m_{ki} = M_{ki} ⋂ Z(Ω_{k}). Then we have the ring-direct sum

Ω̂ = ⊕_{ki} (Ω_{k}) ^{ˆ (ki)} (6.5.1)

where (Ω_{k}) ^{ˆ (ki)} denotes the m_{ki}-adic completion of (the m_{ki}-localization of) Ω_{k}, and is a matrix-local, hence indecomposable ring with J((Ω_{k}) ^{ˆ (ki)}) = M_{ki} ^{ˆ (ki)}.

Proof. By Lemma 6.2 we may assume that R = Z(Ω). Then R = ⊕_{k} R_{k} where R_{k} = Z(Ω_{k}). Since the sets of maximal ideals of Ω and its center are in one-to-one correspondence via contraction [Lemma 5.3], {m_{ki}} is the set of maximal ideals of R_{k}. As with all semilocal commutative rings we therefore have R̂_{k} = ⊕_{i} R_{k} ^{ˆ (ki)} [B ’72, Chap III, §2.13, final Cor.], and this yields decomposition (6.5.1). So it only remains to prove that each term (Ω_{k}) ^{ˆ (ki)} is matrix-local with the indicated Jacobson radical. After localizaing at m_{ki} and changing notation, R = Z(Ω) becomes a local ring and Ω has only one maximal ideal, namely J(Ω), and Ω is matrix-local. What we want to prove is that J(Ω̂) = J(Ω)ˆ and the ring Ω̂/J(Ω̂) is simple artinian. This follows from Proposition 3.8.

Theorem 6.6. (Projective Ω̂-modules)

(i) For each (k, i) there is a unique indecomposable f.g. projective (left) Ω̂_{k}- module U_{ki} that maps Ω̂-linearly onto S_{ki}. Every indecomposable f.g. projective Ω̂-module is isomorphic to some U_{ki}.

(ii) We have B̂_{k} ≅ ⊕_{i} (U_{ki})^{s_{ki}}, where B_{k} denotes the unique indecomposable f.g. projective Ω_{k}-module; and Ω̂_{k} ≅ ⊕_{i} (U_{ki})^{n_{k}s_{ki}} where n_{k} is the matrix size of Γ_{k} (equivalently, of Ω_{k}).

Proof. By Lemma 6.2 we may assume that R = Z(Ω). We have B_{k}/M_{ki}B_{k} ≅ (S_{ki})^{s_{ki}} by (6.1.5). Since Ω-modules of finite length have finite R-length [Lemma 3.7], they equal their completions, and therefore tensoring with R ^{ˆ (ki)} yields

(B_{k}) ^{ˆ (ki)}/(M_{ki}) ^{ˆ (ki)}B_{k} ^{ˆ (ki)} ≅ (S_{ki})^{s_{ki}} (6.6.1)

Since the complete ring Ω ^{ˆ (ki)} is semiperfect, its simple module S_{ki} has a projective cover U_{ki} → S_{ki}, and this yields a projective cover (U_{ki})^{s_{ki}} → (S_{ki})^{s_{ki}}. Since (M_{ki}) ^{ˆ (ki)} is the unique maximal ideal of Ω ^{ˆ (ki)} [Lemma 6.5], the natural homomorphism (B_{k}) ^{ˆ (ki)} → (S_{ki})^{s_{ki}} given by (6.6.1) is also a projective cover. Therefore uniqueness of the projective cover (see, e.g. [LO, 8.8]) shows that (B_{k}) ^{ˆ (ki)} ≅ (U_{ki})^{s_{ki}}. The rest of the proof is straightforward.

We use the notation N(T) for the sum of all nilpotent ideals of a ring T. For all the rings we consider, this turns out to be the nilradical of T, in the sense that N(T) is the largest nilpotent ideal of T and T/N(T) has no nilpotent ideals.

Lemma 6.7. (i) The ring Γ̈ is neither left nor right noetherian if Γ is not a

f.g. R-module. (ii) N(Γ̈) is a nilpotent ideal, and J(Γ̈) = Γ̈δ where δ is any element of Γ such

that J(Γ) = Γδ. (iii) Γ̈/N(Γ̈) is a noetherian ring. In fact, it is a maximal R′ = R̂/N(R̂)-order

in the semisimple artinian ring (Γ̈/N(Γ̈))_{Q(R′)} = (Γ̈/N(Γ̈))_{Q(R)}.

Proof. We may suppose that R ⊆ Λ. We first establish some facts that will be needed again in the proof of Theorem 6.12.

J(R) ⊆ J(Γ̈) (6.7.1)

It suffices to show that, for all c ∈ J(R) and all x ∈ Γ̈, the element 1 − xc is a unit of Γ̈. After writing x = ∑ r̂_{i}γ_{i} (r̂_{i} ∈ R̂, γ_{i} ∈ Γ) we see that there is a splitting R-order Ω ⊆ Γ such that x ∈ Ω̂ [Theorem 5.2]. We have J(R) ⊆ J(Ω) since Ω is module-finite over R; and J(Ω) ⊆ J(Ω̂) by Proposition 3.8. Therefore 1 − xc is invertible in the subring Ω̂ = R̂ ⊗_{R} Ω of Γ̈ and hence in Γ̈ itself, proving (6.7.1).

Next we show:

J(Γ̈) = R̂J(Γ) (6.7.2)

Throughout this section we are assuming that Z(Λ) has no artinian ring-direct summands. Therefore J(R) contains a regular element c of R [Lemma 1.4(iii)]. Since c becomes a unit in R_{Q}, hence in Γ_{Q}, it remains regular in Γ. Moreover, by (4.1.9), c ∈ J(Γ).

Since c is a regular element of R, Γ/Γc has finite length as an R-module, by (4.1.9). Therefore Γ/Γc = Γ̈/Γ̈c and every R-submodule of the right-hand side is an R̂-submodule, by (6.1.7). Since c ∈ J(Γ) the radical of the left-hand side, considered as a Γ-module, is J(Γ)/Γc. Therefore the radical of the right-hand side, considered as a Γ-module is X = [J(Γ) + Γ̈c]/Γ̈c. Since every R-submodule of the right-hand side is an R̂-module, X equals the radical of Γ̈/Γ̈c as an R̂Γ = Γ̈- module, and X = [R̂J(Γ) + Γ̈c]/Γ̈c. It follows that J(Γ̈) = R̂J(Γ) + Γ̈c. To complete the proof of the claim it therefore suffices to show that Γ̈c ⊆ R̂J(Γ). Since Γ̈c = R̂Γc it suffices to show that Γc ⊆ J(Γ). Since Λ is module-finite over R we have c ∈ J(R) ⊆ J(Z(Λ)), and J(Z(Λ)) ⊆ J(Γ) by (4.1.9). This completes the proof of (6.7.2).

Let Γ̄ = Γ/J(Γ). We claim that Γ̄ has finite length as an R-module and

Γ̈/J(Γ̈) = Γ̄ (6.7.3)

Choose c as in the proof of (6.7.2). Then Γ/Γc has finite length as an R-module, as already observed. Moreover, Γc ⊆ J(Γ), as shown near the end of the proof of (6.7.2). Therefore the R-algebra Γ/Γc can be mapped onto Γ̄, and therefore Γ̄ has finite length as claimed. Because of this finite length, (6.1.7) shows that Γ̄ = R̂ ⊗_{R} Γ̄ = Γ̈/R̂J(Γ), and (6.7.2) completes the proof of (6.7.3).

Statement (ii), the assertion about J(Γ̈). This follows immediately from (6.7.2) together with the facts that Γ̈ = R̂Γ and J(Γ) = Γδ [see (4.1.8)].

(i). Since Γ is not a f.g. R-module but is a f.g. R̃-module, the ring R̃ is not a f.g. R-module. We first prove the commutative case of statement (i), that is, R̈ = R̂ ⊗_{R} R̃ is not a noetherian ring. Let J(R̃) = R̃π.

Claim. π is a regular element of R̈; that is, the natural image 1 ⊗ π of π in R̈ is regular. Our assumption that Z(Λ) has no artinian ring-direct summands, implies that Z(Γ) has no artinian ring-direct summands, by (4.1.10). Therefore J(R̃) contains regular elements of R̃ [Lemma 1.3], and therefore its generator π is regular. Therefore the multiplication map by π on R̃ is one-to-one. By flatness of R̂, multiplication by 1 ⊗ π is one-to-one on R̈, as claimed.

By (6.7.2), applied to R̃ in place of Γ, we have J(R̈) = R̈π, and the preceding claim showed that π is a regular element of R̈. Therefore, if R̈ were noetherian it would follow easily that all of its localizations at maximal ideals are integral domains (in fact, discrete valuation rings), and therefore R̈ has no nilpotent elements. On the other hand, R̈ contains the subring R̂ ⊗ R = R̂, which has nilpotent elements because we are assuming that R̃ is not module-finite over R [Mtl ’73, 7.1]. This completes the proof of the commutative case.

Now suppose, by way of contradiction, that Γ̈ is noetherian. Since Γ is modulefinite over R̃, it follows that Γ̈ is module-finite over its central subring R̈, and therefore R̈ is noetherian [MR ’87, 10.1.11], contrary to what was proved above.

(iii) and the rest of (ii). Tensoring the inclusions R ⊆ Ω ⊆ Γ ⊆ Ω_{Q} with the flat R-module R̂ yields

R̂ ⊆ Ω̂ ⊆ Γ̈ ⊆ Ω̂_{Q} (6.7.4)

Recall that the ring Ω̂_{Q} is artinian [see (3.1.2)]. It follows from (6.7.4) that N(Γ̈)_{Q} is a sum of nilpotent ideals of the artinian ring Ω̂_{Q}, and is therefore nilpotent. Therefore (ii) holds.

Let R′, Ω′, Γ′ be the natural images of R̂, Ω̂, Γ̈ respectively, in the semisimple artinian ring A′ = Ω̂_{Q}/N(Ω̂_{Q}). Then the inclusions in (6.7.4) yield:

R′ ⊆ Ω′ ⊆ Γ′ ⊆ A′  and  (Ω′)_{Q} = (Γ′)_{Q} = A′ (6.7.5)

We claim that none of these rings has nilpotent ideals ≠ 0, that is, R′ = R̂/N(R̂), Ω′ = Ω̂/N(Ω̂), and Γ′ = Γ̈/N(Γ̈). For any nilpotent ideal X of either Ω′ or Γ′, the localization X_{Q} is a nilpotent ideal of A′ and therefore equals 0. Hence X = 0. On the other hand, if x is any nilpotent element of R′, then centrality of x in Ω′ shows that xA′ is a nilpotent ideal of A′ and hence x = 0, proving the claim in this situation, too.

Next we claim that the natural map R′ → (R′)_{Q} (where Q = Q(R)) is oneto-one, that is, every regular element d of R has a regular image in R′. Since regular elements of R remain regular in R̂ it suffices to show that regular elements of R̂ remain regular in R′. But this follows since, by the previous paragraph, R′ = R̂/N(R̂), and regular elements of a commutative ring remain regular modulo the nil radical.

Although the unadorned Q in (6.7.5) denotes Q(R), we claim that (R′)_{Q} = (R′)_{Q(R′)}; that is (since R′ has no nilpotent elements), (R′)_{Q} is the total quotient ring of R′. Since R ⊆ (R′)_{Q} by the preceding paragraph, it suffices to show that (R′)_{Q} is artinian. But R̂_{Q} is artinian by (3.1.1), and (R′)_{Q} = R̂_{Q}/N(R̂)_{Q}, completing the proof of the claim.

By the claim in the previous paragraph we have (Ω′)_{Q(R′)} = (Ω′)_{Q} = A′, and therefore Ω′ is an R′-order in the semisimple artinan ring A′.

Moreover, the semilocal ring R′ is complete in its J(R′)-adic topology. [Proof. This topology coincides with the J(R̂)-adic topology of R′, and the completion of R′ in this topology is R̂ ⊗R̂ R′ = R′.]

Next we claim that Γ′ is integral over R′. It suffices to prove that Γ′ is a set-theoretic union of module-finite R′-algebras. And for this it suffices to prove that Γ̈ is a set-theoretic union of module-finite R̂-algebras. Since tensor products commute with direct limits, it suffices to show that Γ is a directed set-theoretic union of module-finite R-algebras; and this is proved in (4.1.1).

Next we claim that Γ′ is a module-finite R′-algebra. Since Γ′ is integral over R′, a simple application of Zorn’s Lemma shows that some normalization Γ′′ of Ω′ contains Γ′. (In fact, when the proof of the lemma is complete, we will know that Γ′′ = Γ′.) Since R′ is complete in its J(R′)-adic topology, it follows from Proposition 4.4 that Γ′′ is a f.g. R′-module, hence so is Γ′, as claimed.

Since R′ ≅ R̂/N(R̂) and Γ′ ≅ Γ̈/N(Γ̈), statement (iii) of our lemma is now proved, except for the assertion that the R′-order Γ′ is a maximal order.

Let δ be as in statement (ii), and δ′ its image in Γ′. We claim that δ′ is a regular element of Γ′.

Since N(Γ̈) ⊆ J(Γ̈) it follows from statement (ii) that J(Γ′) = Γ′δ′. As in the proof of (6.7.2), let c be a regular element in J(R). Then the natural image c′ of c in R′ is a non-zero-divisor in Γ′, as proved two paragraphs below (6.7.5). Also, it follows from (6.7.1) that c′ ∈ J(Γ′). Since J(Γ′) = Γ′δ′ contains the non-zero-divisior c′, it follows that δ′ is not a left zero divisor in Γ′, and therefore in (Γ′)_{Q}. Since (Γ′)_{Q} is an artinian ring, δ′ is therefore a unit in (Γ′)_{Q} and therefore a regular element of Γ′, as claimed.

Next we claim that Γ′ is a hereditary R′-order (i.e., all left and right ideals are projective Γ′-modules). Since δ′ is a regular element of Γ′, the left ideal J(Γ′) = Γ′δ′ is a free, hence projective Γ′-module. In the situation that R′ is a discrete valuation ring, a well-known theorem of Auslander and Goldman states that our claim is true. In fact, the proof of this result given in [R ’75, (39.1)] works in our more general situation, with the following very minor change. Instead of choosing the element π, in that proof, to be a generator of J(R′), which might not be principal in our situation, choose it to be the regular element c′ ∈ J(R′). The proof then works without additional changes.

Since Γ′ is a hereditary, noetherian semiprime ring, it is easy to see that Γ′ is a direct sum of hereditary noetherian prime rings [MR ’87, 5.4.6]. Since Γ′ is modulefinite over a commutative ring, it satisfies a polynomial identity. Therefore, by a theorem of Robson and Small [MR ’87, 13.9.16; RS ’74], the centers of the prime summands of Γ′ are Dedekind domains (complete discrete valuation rings, in our situation), and therefore Γ′ is a direct sum of hereditary classical orders. It now suffices to show that all of these prime summands are maximal orders.

The structure theorem for hereditary orders over complete discrete valuation rings in simple algebras [R ’75, (39.14)] states that every such order is isomorphic to a tiled block lower pseudo-triangular matrix ring of the form displayed below, which we now explain.

(6.7.6) T = [ \begin{matrix}(Δ) & (p) & (p) & … & (p) \\ (Δ) & (Δ) & (p) & … & (p) \\ (Δ) & (Δ) & (Δ) & … & (p) \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ (Δ) & (Δ) & (Δ) & … & (Δ) \end{matrix} ]^{\{n_{1},... ,n_{r}\}}

The superscript {n_{1}, …, n_{r}} denotes a sequence of r ≥ 1 positive integers n_{i}. For each (i, j), the symbol (Δ) or (p) in the (i, j)-position of the matrix in (6.7.6) indicates the set of all n_{i} × n_{j} matrices with enries in (Δ) or (p) respectively. Moreover, Δ denotes a maximal R′-order in a division algebra, and p = J(Δ).

The structure theorem quoted above also states that T is a maximal order if and only if r = 1, that is, T = M_{n_{1}}(Δ).

It is easy to see that one can form J(T) by changing every n_{i} × n_{i} (“main diagonal”) block from (Δ) to (p). Note that the only central idempotents of T are {0, 1}. Therefore, if r ≠ 1, then T/J(T) has central idempotents that cannot be lifted to central idempotents of T. We conclude: If every primitive central idempotent of every T/J(T) can be lifted to a central idempotent of T, then Γ′ is a maximal order.

We now verify that every primitive central idempotent of Γ′/J(Γ′) can be lifted to a central idempotent of Γ̈.

We have Γ′/J(Γ′) ≅ Γ̈/J(Γ̈) ≅ Γ/J(Γ), the last isomorphism by (6.7.3). Therefore, to complete the proof of the lemma it suffices to verify that every primitive central idempotent of Γ̄ = Γ/J(Γ) can be lifted to a central idempotent of Γ̈.

Let Ω be any splitting R-order contained in Γ. We have Γ̄ = Ω/J(Ω) = Ω̂/J(Ω̂) by (6.1.4) and Proposition 3.8. It is obvious, from Lemma 6.5, that every indecomposable ring-direct summand of Ω̂/J(Ω̂) is the image of an indecomposable ring-direct summand of Ω̂. This is equivalent to saying that every primitive central idempotent of the former ring can be lifted to a primitive central idempotent of the latter ring. In view of inclusions (6.7.4), our proof is complete.

The next results show that primitive idempotents of Ω̂ remain primitive in Ω̂_{Q}. The commutative case of this was proved by D. Katz [Kt ’86], but from a somewhat different point-of-view. Katz’s results, in dimension 1, are that if R is a local domain, then the kernel of the natural map R̂ → (R̃)ˆ is nilpotent and both rings have the same number of minimal prime ideals, namely, the number of maximal ideals of R̃. Our proof is an adaptation of Katz’s proof, but must be modified to avoid his use of integral closures of ideals (which do not exist in our noncommutative setting), and to accomodate the presence of non-central idempotents (which are not related to prime ideals). We are grateful to W. Heinzer for informing us about Katz’s work, and to L. Klingler for a key idea that enabled us to use this alternative approach. See also Remark 6.10.

Lemma 6.8. The kernel of the natural map ν: Ω̂ → Γ̂ is nilpotent, where Γ̂ denotes the J(R̃)-adic completion of Γ.

Note. To see that the kernel can be nonzero, consider the simplest commutative situation, in which R = Ω and is a local domain whose normalization R̃ = Γ is again local (hence a discrete valuation ring) but not module-finite over R. Then the J(R)-adic completion R̂ of R contains nilpotent elements [Mtl ’73, 7.1], while the J(R̃)-adic completion Γ̂ of R̃ is a local domain, in fact a discrete valuation ring. Then the natural map in the statement of the lemma is clearly a nonzero nilpotent ideal of R̂ = Ω̂.

Proof of the lemma. We may assume that Λ = Ω, since Λ itself does not appear in the statement of the lemma. We also may assume that R = Z(Ω), by Lemma 6.2. By Lemma 1.3(iii), J(R) contains regular elements of R. Fix such an element a for the rest of this proof. Since R/Ra is an artinian ring, each of the ideals Ra and J(R) contains some power of the other; and therefore the J(R)-adic completion R̂ can be identified with the completion of R with respect to the filter of ideals {Ra^{n} : n = 1, 2, … }. Consequently Ω̂ can be identified with the completion of Ω with respect to the filter of ideals {Ωa^{n}}.

Since R̃ is integral over R, the commutative lying-over and going-up theorems show that a ∈ J(R̃); and since a is regular in R it is a unit in R_{Q} and therefore regular in R̃. We conclude, as in the previous paragraph, that the J(R̃)-adic completion Γ̂ can be identified with the completion of Γ with respect to the filter of ideals {Γa^{n}}.

Let Γ′ = Γ̈/N(Γ̈), as in (6.7.5) and the paragraph below it, and let a′ be the natural image of a in Γ′. We claim that

⋂_{n=1}^{∞} Γ′(a′)^{n} = 0 (6.8.1)

Call this intersection X. Since a is a unit in Ω_{Q}, hence in Ω̂_{Q}, a′ is a unit in A′ = Ω̂_{Q}/N(Ω̂_{Q}) and hence is a non-zerodivisor on Γ′. Therefore X = a′X. Since Γ′ is a f.g. module over the noetherian ring R̂/N(R̂), by Lemma 6.7(iii), and a′ ∈ J(R′), it follows from Nakayama’s Lemma that X = 0 as claimed.

We now apply our adaptation of Katz’s main step [Kt ’86, last paragraph of proof of Theorem 4].

Choose any element x^{∗} ∈ ker(ν). We have x^{∗} = lim_{n} x_{n} for some Cauchy sequence {x_{n}} in Ω. After passing to a suitable subsequence we have x^{∗} −x_{n} ∈ Ω̂a^{n} for every n. On the other hand, ν(x^{∗}) = 0 shows that {x_{n}} is a null sequence in Γ. Therefore, after passing to a suitable subsequence, we have x_{n} ∈ Γa^{n} for every n.

Note that Γ and Ω̂ are subrings of Γ̈. Therefore we may write x^{∗} = x_{n} +(x^{∗} −x_{n}) obtaining an expression of the form

x^{∗} = x_{n} + y_{n} with \begin{cases}x_{n} ∈ Γa^{n} \\ y_{n} ∈ Ω̂a^{n} \end{cases} (6.8.2)

for every n. Therefore x^{∗} ∈ Γ̈a^{n} for every n. It follows from (6.8.1) that the image of x^{∗} in Γ′ equals zero. Therefore

x^{∗} ∈ ker(Γ̈ → Γ′) = N(Γ̈) (6.8.3)

Since N(Γ̈) is a nilpotent ideal [by Lemma 6.7] we conclude that ker(ν) is nilpotent.

Theorem 6.9. Primitive idempotents of Ω̂ remain primitive in Ω̂_{Q} (hence in its subring Γ̈) and Γ̂_{Q} (hence in its subring Γ̂).

Proof. First we recall that Γ̈ is indeed a subring of Ω̂_{Q}, as shown in inclusions (6.7.4).

Next we note that primitive idempotents of Γ̂ remain primitive in Γ̂_{Q}. This is a restatement of a well-known fact about classical orders. For Γ is a maximal R̃-order and is therefore a direct sum of maximal orders, over principal ideal domains, in simple algebras. [See (4.1.2) and (4.1.3).] It follows that Γ̂ is a direct sum of full matrix rings over valuation rings in division rings. See [R ’75, bottom of p. 170]. The claim now follows easily.

Let d be a primitive idempotent in Ω̂. We claim that d remains primitive in Γ̂. If we knew that the natural map Ω̂ → Γ̂ is a surjection, this would follow from nilpotence of its kernel. Instead, we proceed less directly. By completeness, an idempotent d of Ω̂ is primitive in Ω̂ if and only its natural image in Ω̂/J(Ω̂) is primitive there (because idempotents can be lifted modulo the radical, in the complete case [CR ’81, (6.7)]). Since Ω is a splitting order, we have Ω/J(Ω) = Γ/J(Γ), and hence Ω̂/J(Ω̂) = Γ̂/J(Γ̂). Thus, if d is primitive in Ω̂, it remains primitive in Ω̂/J(Ω̂) = Γ̂/J(Γ̂), and hence remains primitive in Γ̂. Therefore, by the previous paragaph, d also remains primitive in Γ̂_{Q}.

Now suppose that d = e + f with e, f orthogonal, nonzero idempotents in Ω̂_{Q} and d primitive in Ω̂. Let ν denote both the natural homomorphism Ω̂ → Γ̂ and its extension ν_{Q}: Ω̂_{Q} → Γ̂_{Q}. Then we have ν(d) = ν(e) + ν(f), where the terms on the right-hand side are orthogonal idempotents of Γ̂_{Q}. Since ker(Ω̂ → Γ̂) is a nilpotent ideal [Lemma 6.8], so is ker(Ω̂_{Q} → Γ̂_{Q}). Therefore both terms ν(e), ν(f) are nonzero. But, as shown above, ν(d) is primitive in Γ̂, and hence in Γ̂_{Q}, a contradiction. Therefore d remains primitive in Ω̂_{Q}.

Remark 6.10. We now know as much as we need to know about Ω̂ versus Γ̂ versus Γ̈, in order to obtain our package deal theorem for completions. But our results are not completely satisfying. By passing to the direct limit, the natural maps Ω̂ → Γ̂ induce a map Γ̈ → Γ̂ whose kernel is nilpotent by Lemma 6.8. Is this map a surjection? If so, then the maximal order Γ̈/N(Γ̈) in Lemma 6.7 can presumably be identified with Γ̂. If the map is not a surjection, is the induced map Γ̈_{Q} → Γ̂_{Q} of artinian rings a surjection?

Theorem 6.11. (Projective Ω̂_{Q}-modules)

(i) Every f.g. projective Ω̂_{Q}-module is isomorphic to U_{Q} for some f.g. projective Ω̂-module U that is unique up to isomorphism.

(ii) (In more detail:) Let U_{ki} be the unique indecomposable f.g. projective Ω̂_{k} module that maps onto the simple module S_{ki}, as in Theorem 6.6. Then the indexed set of modules (U_{ki})_{Q} forms a set of representatives for the isomorphism classes of indecomposable f.g. projective Ω̂_{Q}-modules.

Proof. Write the identity element of Ω̂ as a sum 1 = ∑_{i} e_{i} of orthogonal primitive idempotents e_{i}. Since the Krull-Schmidt theorem holds for f.g. modules over the complete ring Ω̂, every indecomposable f.g. projective Ω̂-module is isomorphic to some Ω̂e_{i}. These idempotents e_{i} remain primitive in Ω̂_{Q} [Theorem 6.9]; and since the Krull-Schmidt theorem holds for f.g. modules over the artinian ring Ω̂_{Q}, we see that every indecomposable f.g. projective Ω̂_{Q}-module is isomorphic to some Ω̂_{Q}e_{i} = (Ω̂e_{i})_{Q}.

Thus it suffices to show that if indecomposable f.g. projective Ω̂-modules satisfy U ≅ V̸, then we have U_{Q} ≅ V̸_{Q}. This follows from Theorem 6.6 and Lemma 6.5, which show that U ≅ U_{ki} and V ≅ U_{k′i′} where U_{ki} and U_{k′i′} are isomorphic to left ideals of distinct ring-direct summands of Ω̂, hence of Ω̂_{Q}.

Proposition 6.12. Γ̈ is a semiperfect ring.

Proof. We need to show that the ring Γ̈ becomes artinian modulo J(Γ̈) and idempotents of Γ̈/J(Γ̈) can be lifted to idempotents of Γ̈. We may assume that R = Z(Λ).

Let Γ̄ = Γ/J(Γ), and recall that Γ/J(Γ) = Γ̈/J(Γ̈) by (6.7.3). It follows that the ring Γ̈ becomes artinian modulo J(Γ̈). Thus it suffices to show that every idempotent ē ∈ Γ̄ can be lifted to an idempotent e ∈ Γ̈.

Choose any splitting R-order Ω ⊆ Γ. Then Γ̄ = Ω/J(Ω) = Ω̂/J(Ω̂), the first equality by (6.1.4) and the second by (6.1.7). By completeness of Ω̂, ē can be lifted to an idempotent e ∈ Ω̂. But, by flatness of R̂ as an R-module, Ω̂ = R̂ ⊗_{R} Ω is a subring of Γ̈ = R̂ ⊗_{R} Γ, and e is therefore the desired idempotent of Γ̈ that lifts ē.

By (6.7.3) we have Γ̈/J(Γ̈) = Γ̄. Therefore the simple Γ-modules S_{ki} form a set of representatives of the isomorphism classes of simple (left) Γ̈-modules.

Theorem 6.13. (Projective Γ̈- and Γ̈_{Q}-modules)

(i) For each (k, i) there is a unique indecomposable f.g. projective Γ̈-module Ü_{ki} that can be mapped onto S_{ki}. The indexed family of modules Ü_{ki} forms a set of representatives of the isomorphism classes of indecomposable, f.g. projective Γ̈-modules.

(ii) The indexed family of modules (Ü_{ki})_{Q} forms a set of representatives of the isomorphism classes of indecomposable f.g. projective Γ̈_{Q}-modules.

(iii) Let Ω be a splitting R-order (Ω ⊆ Γ) and let U_{ki} the indecomposable f.g. projective Ω̂-module that maps onto S_{ki}, as in Theorem 6.6. Then (U_{ki})_{Q} ≅ (Ü_{ki})_{Q}.

Proof. (i). Since Γ̄ is a semisimple artinian ring, each S_{ki} ≅ Γ̄ē_{ki} for some primitive idempotent ē_{ki} ∈ Γ̄. Since Γ̄ = Γ̈/J(Γ̈) and Γ̈ is a semiperfect ring [Proposition 6.12], each ē_{ki} lifts to a primitive idempotent e_{ki} of Γ̈, and the indexed family of modules Ü_{ki} = Γ̈e_{ki} forms a set of representatives of the isomorphism classes of indecomposable f.g. projective Γ̈-modules.

(ii). Write Γ̈ = ⊕_{ν} Γ̈e_{ν} where the e_{ν} are primitive orthogonal idempotents with ∑_{ν} e_{ν} = 1. Then we have Γ̈_{Q} = ⊕_{ν} Γ̈_{Q}e_{ν}. Since primitive idempotents of Γ̈ remain primitive in the artinian ring Γ̈_{Q} [Theorem 6.9], every isomorphism class of indecomposable projective Γ̈_{Q}-module is isomorphic to some term Γ̈_{Q}e_{ν}. Thus it suffices to show that non-isomorphic terms Γ̈e_{ν} remain non-isomorphic after Qlocalization; and to show this it suffices to show that non-isomorphic terms Γ̈e_{ν} are contained in distinct ring-direct summands of Γ̈_{Q}. To do this, note that every finite set of elements of Γ̈ = R̂ ⊗_{R} Γ is contained in a ring of the form Ω̂ = R̂ ⊗_{R} Ω where Ω is a splitting R-order contained in Γ [Theorem 5.2]. Therefore we can choose a splitting order Ω such that Ω̂ contains every e_{ν}. By Lemma 6.5 non-isomorphic Ω̂- modules Ω̂e_{ν} are left ideals of distinct ring-direct summands of Ω̂. Therefore their Q-localizations (Ω̂e_{ν})_{Q} = (Γ̈e_{ν})_{Q} are contained in distinct ring-direct summands of Ω̂_{Q} = Γ̈_{Q}, as desired.

(iii) Let S_{ki} ≅ Γ̄ē_{ki}, as in the proof of part (i). Since Ω̂ is complete, ē_{ki} lifts to a primitive idempotent e_{ki} ∈ Ω̂. Since Ω̂e_{ki} is indecomposable and maps onto S_{ki}, the uniqueness part of Theorem 6.6(i) shows that U_{ki} ≅ Ω̂e_{ki}. Since Ω̂ = R̂ ⊗_{R} Ω is a subring of Γ̈, e_{ki} is also an idempotent in Γ̈. Moreover, since Γ̈ is semiperfect and (6.7.3) holds, e_{ki} is a primitive idempotent of Γ̈. Therefore our notation e_{ki} is consistent with its use in the proof of part (i). In particular, Ü_{ki} ≅ Γ̈e_{ki}. But Ω̂_{Q} = Γ̈_{Q}, as one sees by localizing the inclusions Ω ⊆ Γ ⊆ Γ_{Q} = Ω_{Q} at Q and then tensoring with R̂. Therefore (Ω̂e_{ki})_{Q} = (Γ̈e_{ki})_{Q}, completing the proof of part (iii).

Notation 6.14. (Ranks) Let X be a f.g. Λ̂-module such that the Λ̂_{Q}-module X_{Q} is projective. We have Γ̈_{Q} = Λ̂_{Q}, as one can see by localizing the inclusions Λ ⊆ Γ ⊆ Λ_{Q} at Q and then tensoring with R̂. Since the ring Γ̈_{Q} = Λ̂_{Q} is artinian [see (3.1.2)] we can express X_{Q} uniquely as a direct sum of indecomposable projective Γ̈_{Q}- modules. Thus, in the notation of Theorem 6.13(ii), there exist unique nonnegative integers ρ_{ki}(X) such that

X_{Q} ≅ ⊕_{ki} [(Ü_{ki})_{Q}]^{ρ_{ki}(X)} (6.14.1)

We call the array ρ(X) = ((ρ_{ki})) = ((ρ_{ki}(X) )) the array of ranks of X.

Alternatively, this definition can be phrased in terms of splitting orders. Let Ω be a splitting R-order (Ω ⊆ Γ). There is a unique indecomposable projective Ω̂-module U_{ki} such that (U_{ki})_{Q} ≅ (Ü_{ki})_{Q} [Theorem 6.13(iii)]. If one wishes to avoid mentioning Γ̈-modules, U_{ki} can be described as the unique indecomposable f.g. projective Ω̂-module that maps onto the simple module S_{ki} (over any of the rings Ω, Ω̂, Γ, Γ̄). Then (6.14.1) can be rewritten:

X_{Q} ≅ ⊕_{ki} [(U_{ki})_{Q}]^{ρ_{ki}(X)} (6.14.2)

Note that the array of ranks ρ(X) is not a matrix, because its rows need not all have the same length. The subscript Q denotes Q(R), as usual.

Package Deal Theorem 6.15. (Completions of orders) Let X be a f.g. left Λ̂- module and Ω a splitting R-order (Ω ⊆ Γ). Then the following conditions are equivalent.

(i) X ≅ Â for some f.g. Λ-module A.

(ii) X_{Q} is Λ̂_{Q}-projective and, for every k, the array of ranks ρ(X) = ((ρ_{ki})) satisfies

(ρ_{k1}, ρ_{k2}, …) ∈ N·(s_{k1}, s_{k2}, …) (6.15.1)

where N denotes the nonnegative integers, and s_{k1}, s_{k2}, … are the splitting numbers of Γ, as defined in (6.1.2) and (6.1.3).

(iii) X_{Q} ≅ P̂_{Q} for some f.g. projective Ω-module P. [Alternatively: X_{Q} ≅ R̂ ⊗ N_{Q} for some f.g. projective Γ-module N.]

Proof. (i) ⇒ (iii). Condition (i) implies that X_{Q} ≅ Â_{Q} = R̂⊗_{R} A⊗_{R} R_{Q}. Since Λ_{Q} is a semisimple artinian ring, the Λ_{Q}-module A ⊗_{R} R_{Q} is projective. It now suffices to prove that every f.g. (necessarily projective) Λ_{Q}-module is isomorphic to P_{Q} for some f.g. projective Ω-module P. This holds since Ω is a splitting order [Theorem 5.5].

To obtain the alternative form of (iii) first recall that Ω ⊆ Γ ⊆ Ω_{Q} and hence Ω_{Q} = Γ_{Q}. Therefore it suffices to show that if we are given a f.g. projective Ωmodule P there exists a f.g. projective Γ-module N such that P_{Q} ≅ N_{Q}; and conversely, if N is given then P exists such that P_{Q} ≅ N_{Q}.

Given P, let N = Γ ⊗_{Ω} P. Conversely, suppose N is given. Then N_{Q} is a f.g. projective Γ_{Q} = Ω_{Q}-module, and therefore Theorem 5.5 yields P such that P_{Q} ≅ N_{Q}.

(iii) ⇒ (ii). Write P as a direct sum of copies of the indecomposable projective Ω-modules B_{k}, yielding an expression P = ⊕_{k} (B_{k})^{m_{k}}. Take the completion, and substitute B̂_{k} = ⊕_{i} (U_{ki})^{s_{ki}} [Theorem 6.6]. Then localize at Q. This yields

X_{Q} ≅ P̂_{Q} ≅ ⊕_{ki} ((U_{ki})_{Q})^{m_{k}s_{ki}}

Therefore X has the array of ranks whose k^{th} row is m_{k}(s_{k1}, s_{k2}, …) as required.

(ii) ⇒ (i). By (6.15.1) we can let ρ_{ki} = m_{k}s_{ki} for all k, i. Set P = ⊕_{k} (B_{k})^{m_{k}}, a projective Ω-module. Then, by Theorem 6.6, P̂ ≅ ⊕_{ki} (U_{ki})^{m_{k}s_{ki}} and therefore P̂_{Q} ≅ X_{Q}. Therefore, by the abstract form of the package deal theorem for completions, Theorem 3.4, X is the J(R)-adic completion of some f.g. Λ-module.

Corollary 6.16. Suppose that

(i) Every row of the array of splitting numbers of Γ consists of the single

number s_{k1}, and s_{k1} = 1. Then:

(ii) Every f.g. Λ̂-module X such that X_{Q} is Λ̂_{Q}-projective is the completion of

some f.g. Λ-module. This, in turn, implies that:

(iii) The Krull-Schmidt theorem holds for f.g. Λ-modules.

Proof. (i) ⇒ (ii). This is the trivial case of implication (ii) ⇒ (i) of Theorem 6.15.

(ii) ⇒ (iii). (This observation is credited to I. Reiner in [H ’61].) The Krull-Schmidt theorem holds for f.g. modules over the complete ring Λ̂ [CR ’81, (6.12)]. Since f.g. Λ-modules are isomorphic if and only if their completions are isomorphic Λ̂-modules, the Krull-Schmidt theorem therefore descends to f.g. Λ-modules when condition (ii) holds.

Remarks 6.17. (i) A less technical restatement of condition (i) of Corollary 6.16 is:

(6.17.1) Every Γ_{k} is matrix-local [i.e. Γ_{k}/J(Γ_{k}) is a simple artinian ring] and

idempotents of Γ_{k}/J(Γ_{k}) can be lifted to Γ_{k}. Yet another equivalent condition is:

(6.17.2) (Heller’s condition) The J(R)-induced completion of every simple (left)

Γ_{Q}-module is a simple Γ̂_{Q}-module. To prove this latter equivalence, first note that the J(R)-adic completion of any Γmodule is the same as its J(R̃)-adic completion [see the paragraphs before (6.1.7)]; and similarly, the J(R)-induced completion of any Γ_{Q}-module is the same as its J(R̃)-induced completion. [For the definition of “J(R)-induced completion”, see either the comments about it in the discussion of §3 in the Introduction to this paper, or else the cautionary statement below statement (3.1.1)] Moreover, Γ is a direct sum of classical maximal orders over semilocal principal ideals domains (namely, the indecomposable ring-direct summands of R̃), by (4.1.3). Therefore Γ̂_{Q} is a semisimple artinian ring, and therefore the simple Γ̂_{Q}-modules, in Heller’s condition, coincide with the indecomposable projective Γ̂_{Q}-modules.

In the situation R = R̃ in which we may now work, Γ is a maximal, hence splitting R-order. Hence we may take Ω = Γ in Theorem 6.6. Then Theorem 6.6 shows that we have B̂_{k} ≅ ⊕_{i} (U_{ki})^{s_{ki}} where B_{k} denotes (in this situation) the indecomposable projective Γ_{k}-module and the U_{ki} denote the indecomposable projective Γ̂_{k}-modules. Since each (U_{ki})_{Q} is an indecomposable Γ_{Q}-module [by Theorem 6.11) our claim is now obvious, by comparing Heller’s condition with Condition (i) of Corollary 6.16. [Of course, one does not need the complicated machinery of the present paper to prove the present claim about classical orders.]

(ii) (Heller’s theorem) The implication (6.17.2) ⇒ Corollary 6.16(iii) generalizes a well-known theorem of Heller. His original result assumes that Λ is contained in a maximal R-order in Λ_{Q} (i.e. Γ is module-finite over R) and R is a discrete valuation ring of characteristic zero. The substance (but not statement) of this is contained in [H ’61, paragraph before 2.5 and proof of 2.6]. The “characteristic zero” restriction was removed in [CR ’81, (30.18)], but only for torsionfree modules.

In [LO] we refine Heller’s theorem in such a way that it provides a necessary and sufficient condition for the Krull-Schmidt Theorem to hold for f.g. Λ-modules.

(iii) If Γ is a f.g. R-module, the statements of both Theorem 6.15 and Corollary 6.16 can be simplified by not mentioning that X_{Q} is Λ̂_{Q}-projective [because the ring Λ̂_{Q} is semisimple artinian, and hence all of its modules are projective] and by setting Ω = Γ. In this form the theorem is well-known in integral representation theory, although we do not know a precise reference to it.

Artinian Pairs

We now compare the preceding package deal theorem with its counterpart involving artinian pairs. This counterpart has the advantage of being simpler to apply, because it avoids the cumbersome machinery of completions. But it applies only to torsionfree Λ-modules.

Notation 6.18. Let Λ ⊆ Ω ⊆ Λ_{Q}, where Λ is an R-order in the semisimple ring Λ_{Q} and Ω is a splitting R-order, and R is semilocal. Let C ⊆ J(Λ) ⋂ J(Ω) be a conductor ideal for Λ and Ω; that is, an ideal of both rings such that Λ̄ = Λ/C and Ω̄ = Ω/C are R-modules of finite length, and therefore artinian rings. Since we are assuming that Z(Λ) has no artinian ring-direct summands, C exists by Lemma 1.4.

We call the ordered pair of rings (Λ̄, Ω̄) an artinian pair. This is a slight generalization of what is called an artinian pair in [W ’89, GL ’89, O ’89], where Ω̄ is required to be a principal ideal ring. (This holds, in [W ’89] and [GL ’89] because Ω is a maximal order.) The decomposition (6.1.1) of Ω yields decompositions C = ⊕_{k} C_{k} and Ω̄ = ⊕_{k} Ω̄_{k} where Ω̄_{k} = Ω_{k}/C_{k}. Similarly, we let B̄_{k} = B_{k}/C_{k}B_{k}, where B_{k} is the unique indecomposable projective f.g. Ω_{k}-module. (See Corollary 5.6.) The notation for maximal ideals M_{ki} ⊆ Ω_{k} and splitting numbers s_{ki} retains its meaning, as defined in Notation 6.1, especially (6.1.5).

Caution. Throughout our discussion of artinian pairs, we use an overbar to denote some version of reduction modulo C [e.g. B̄_{k} = B_{k}/C_{k}B_{k}]. This differs from our earlier use of an overbar to denote reduction modulo the Jacobson radical, in the discussion of completions.

(6.18.1) Each Ω̄_{k} = ⊕_{i} Ω̄_{ki} where each Ω̄_{ki} is a matrix-local ring [i.e. Ω̄_{ki}/J(Ω̄_{ki}) is a simple artinian ring] with maximal ideal M̄_{ki}, where M_{ki} is the maximal ideal of Ω_{k} defined in (6.1.5).

Proof. We have Ω/C ≅ Ω̂/Ĉ, since Ω/C has finite length as an R-module [Proposition 3.8]; so the claim follows quickly from Lemma 6.5. But, since the spirit of artinian pairs is to avoid completions, we sketch an alternative proof. We can suppose that Ω = Ω_{1} and R = Z(Ω). The commutative ring R̄ = R/(C ⋂ R) is artinian, since it is an R-subalgebra of the R-module Ω/C of finite length. Therefore it is a direct sum of m local rings, where m is the number of maximal ideals of R̄. This induces a decomposition of the R̄-algebra Ω̄ into a direct sum of m rings Ω̄_{i}.

Thus Ω̄ has at least m maximal ideals. But since the sets of maximal ideals of the rings Ω and Z(Ω) are in one-to-one correspondence via contraction [Lemma 5.3], the ring Ω̄ has the same number of maximal ideals as the ring R̄. Therefore each Ω̄_{i} must be matrix-local.

Since C_{k} ⊆ J(Ω_{k}), the simple (left) Ω̄_{k}-modules coincide with the simple Ω_{k}- modules S_{ki} [see (6.1.5)]. Therefore the following description of the projective Ω̄-modules is essentially the same as the description of the projective Ω̂-modules given in Theorem 6.6.

Theorem 6.19. (Projective Ω̄-modules)

(i) For each (k, i) there is a unique indecomposable f.g. projective (left) Ω̄_{k}- module B̄_{ki} that maps Ω̄-linearly onto S_{ki}. Every indecomposable f.g. projective Ω̄-module is isomorphic to some B̄_{ki}

(ii) B̄_{k} ≅ ⊕_{i} (B̄_{ki})^{s_{ki}}, and Ω̄_{k} ≅ ⊕_{i} (B̄_{ki})^{n_{k}s_{ki}} where n_{k} is the matrix size of Ω_{k} [equivalently, of Γ_{k}]. [See (5.1.2) and (4.1.8).]

Proof. This and some other proofs that follow make use of projective covers of modules. See e.g. [LO, 8.8] for a review of their basic properties. Since the ring Ω̄_{k} is artinian, each S_{ki} has a projective cover B̄_{ki} → S_{ki} as an Ω̄_{k}-module, and {B̄_{ki}} is a full set of nonisomorphic indecomposable projective left Ω̄_{k}-modules. Thus statement (i) holds.

To obtain (ii) note that by (6.1.5) there is a surjective Ω̄-module homomorphism: B̄_{k} → → ⊕_{i} (S_{ki})^{s_{ki}} with kernel J(Ω̄_{k})B̄_{k}. This is a projective cover, as is ⊕_{i} (B̄_{ki})^{s_{ki}} → ⊕_{i} (S_{ki})^{s_{ki}}, the direct sum of the projective covers constructed earlier. Uniqueness of the projective cover now yields the first statement of (ii). The second statement of (ii) is obvious.

Notation 6.20. We define a (Λ̄, Ω̄)-module [X ⊆ Y] to be a f.g. projective left Ω̄-module Y, together with a Λ̄-submodule X such that Ω̄X = Y. We say that [X ⊆ Y] ≅ [X′ ⊆ Y′] if there is an Ω̄-isomorphism Y ≅ Y′ that carries X onto X′.

Every f.g. torsionfree Λ-module A such that ΩA is Ω-projective yields a (Λ̄, Ω̄)- module

Φ(A) = [A/CA ⊆ ΩA/CA]

In order to state which (Λ̄, Ω̄)-modules arise from such an A, we define the array of ranks ρ[X ⊆ Y] = ((ρ_{ki})) to be the array of nonnegative integers ρ_{ki} such that Y ≅ ⊕_{ki} (B̄_{ki})^{ρ_{ki}}. See Remark 6.23, below, for the relationship between this and the “array of ranks” used in connection with the package deal theorem for completions. As with that definition, this array is not a matrix because it need not have the same number of entries in every row. Note that ρ[X ⊆ Y] has one row for each prime ring-direct summand Ω_{k} of Ω. In addition, ρ[X ⊆ Y] is completely determined by Y.

Package Deal Theorem 6.21. (Artinian pairs) The following statements about a (Λ̄, Ω̄)-module [X ⊆ Y] and a splitting order Ω ⊇ Λ are equivalent.

(i) [X ⊆ Y] ≅ Φ(A) for some f.g. torsionfree Λ-module A such that ΩA is Ω-projective.

(ii) For each k, the array of ranks ((ρ_{ki})) = ρ[X ⊆ Y] satisfies

(ρ_{k1}, ρ_{k2}, …) ∈ N·(s_{k1}, s_{k2}, …)

where N denotes the nonnegative integers. (iii) Y ≅ P/J(Ω)·P for some f.g. projective Ω-module P.

Moreover, A is unique up to isomorphism when it exists.

Proof. (i) ⇒ (ii). When A exists, Y ≅ ΩA/CA. Since ΩA is Ω-projective, we have ΩA ≅ ⊕_{k} (B_{k})^{m_{k}} for suitable nonnegative integers m_{k}. Theorem 6.19(ii) then shows that row k of the resulting array of ranks is m_{k}(s_{k1}, s_{k2}, …), as desired.

(ii) ⇒ (i). For each k define m_{k} by (ρ_{k1}, ρ_{k2}, …) = m_{k}(s_{k1}, s_{k2}, …), and let P = ⊕_{k} (B_{k})^{m_{k}}. Then P/CP ≅ Y, and we can take A to be the inverse image of X in P.

(ii) ⟺ (iii). Statement (iii) is clearly just a way of restating (ii). It is included to emphasize the parallel between the present theorem and its counterpart for completions, Theorem 6.15.

Supplementary statement. Let Φ(A) = [X ⊆ Y] and Φ(A′) = [X′ ⊆ Y′], and suppose that Φ(A) ≅ Φ(A′), that is, some Ω̄-isomorphism θ: Y ≅ Y′ carries X onto X′. As already observed, the natural surjection ΩA → → Y is a projective cover. A similar statement applies to A′. By uniqueness of the projective cover, θ can be lifted to an isomorphism Θ: ΩA ≅ ΩA′. Since θ(X) = X′ we have Θ(A) = A′, as desired.

Proposition 6.22. The Krull-Schmidt theorem holds for modules over the artinian pair (Λ̄, Ω̄).

Proof. We can identify a (Λ̄, Ω̄)-module [X ⊆ Y] with the external direct sum X ⊕ Y, its elements written in the form [ \begin{matrix}x \\ y \end{matrix}]. Then the collection of elements of the form [ \begin{matrix}x \\ y \end{matrix}]. can be regarded as a f.g. left module over the artinian ring ( \begin{matrix}Λ̄ & 0 \\ Λ̄ & Ω̄ \end{matrix} ). Therefore the desired result follows from the Krull-Schmidt theorem for f.g. modules over artinian rings.

Remark 6.23. We have given two definitions of “array of ranks”, one in connection with artinian pairs, and one in connection with completions [Notation 6.14]. The following fact shows that the two definitions define the same collection of arrays when they arise from a torsionfree Λ-module.

(6.23.1) Let A be a torsionfree Λ-module such that ΩA is Ω-projective, and let [A/CA ⊆ ΩA/CA] = [X ⊆ Y] be its associated (Λ̄, Ω̄)-module. Then ρ[X ⊆ Y] = ρ(Â).

To prove this, first observe that ΩA/CA ≅ Ω̂Â/ĈÂ, since the left-hand side is an R-module of finite length [Lemma 1.6]. Then note that, since C ⊆ J(Ω), the natural maps ΩA → → ΩA/CA and Ω̂A → → Ω̂A/ĈÂ are projective covers, with respect to Ω and Ω̂, respectively. Uniqueness of projective covers, together with the structure of projective Ω̄ and Ω̂-modules given in theorems 6.19 and 6.6, then finishes the proof.

7. References

[B ’72] N. Bourbaki, Commutative Algebra, Hermann, Paris, 1972. [CR ’81] C. W. Curtis and I Reiner, Methods of Representation Theory, with Applications to

Finite Groups and Orders, vol. 1, Wiley-Interscience, New York, 1981 [G ’65] A. Grothendieck, “Élements de Géométry Algébrique, IV”, in Publ. Math. IHES 24 (1965) [GL ’88] R. M. Guralnick and L. S. Levy, “Presentations of modules when ideals need not be

principal”, Ill. J. Math 32 (1988), 593–653. [GL ’89] Guralnick, R. M., and Levy, L. S., “Cancellation and direct summands in dimension 1”,

J. Alg. 142 (1991), 310–347. [H ’61] A. Heller, “On group representations over a valuation ring”, Proc. Nat. Acad. of Sci. 47

(1961), 1194–1197. [HL ’87] J. Haefner and L. S. Levy, “Commutative orders whose lattices are direct sums of ideals”,

J. Pure and Appl. Algebra, 50 (1988), 1–20. [H ’73] Hochster, M., “Non-openness of loci in noetherian rings”, Duke Math J. 40 (1973), 215–219. [J ’56] N. Jacobson, Structure of Rings, Amer. Math. Soc. Colloq. Publ. 37, New York, 1956. [K ’70] I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston, 1970. [Kt ’86] D. Katz, “On the number of minimal prime ideals in the completion of a local domain”,

Rocky Mountain J. Math. 16 (1986), 575–578. [L ’89] “Projectives of large uniform-rank, in Krull dimension 1”, Bulletin London Math. Soc. 21

(1989), 57–64.

[LO] L. S. Levy and C. J. Odenthal, “Krull-Schmidt Theorems in Dimension 1”, Trans. Amer. Math. Soc. (********).

[MR ’87] J. C. McConnell and J.C. Robson, Noncommutative Noetherian Rings, Wiley-Interscience,

Chichester, 1987. [Mtl ’73] E. Matlis, 1-Dimensional Cohen-Macaulay Rings, Lecture Notes in Math 327, Springer,

New York, 1973. [Mts ’80] H. Matsumura, Commutative Algebra, second edition, Bemjamin-Cummings, Reading,

Mass., 1980. [N ’62] M. Nagata, “Local Rings”, Interscience, New York, 1962. [O ’89] C. J. Odenthal, “Presentations over HNP rings with enough invertible ideals and torsionfree

cancellation over neoclassical orders”, J. Algebra 127 (1989), 290–319. [R ’75] I. Reiner, Maximal Orders, Academic Press, New York, 1975. [RS ’74] J. C. Robson and L. Small, “Hereditary prime P.I. rings are classical orders”, J. London

Math. Soc. (2) 8 (1974), 499–503. [W ’89] R. Wiegand, “Noetherian Rings of bounded representation type”, in Proceedings of Mi-

croprogram on Commutative Algebra, MSRI, Berkeley, 1987 (Springer, New York, 1989), 497–516.

Acknowledgements. We wish to thank Robert Guralnick, William Heinzer, Lee Klingler, and Nikolaus Vonessen for their valuable suggestions which helped us avoid the standard hypothesis that orders are contained in maximal orders. Without these suggestions our Krull-Schmidt results in [LO] would not have been as complete as they now are.

L. S. Levy, Mathematics Department, University of Wisconsin, Madison, WI 53706-1388, USA

E-mail address: levy@math.wisc.edu

C. J. Odenthal, Mathematics Department, University of Toledo, Toledo, Ohio 43606-3390, USA

E-mail address: codentha@math.utoledo.edu