FINITELY GENERATED MODULES OVER PULLBACK RINGS

David M. Arnold and Reinhard C. Laubenbacher

Abstract. The purpose of this paper is to outline a new approach to the classification of finitely generated indecomposable modules over certain kinds of pullback rings. If R is the pullback of two hereditary noetherian homogeneously serial rings, finitely generated over their centers, over a common semi-simple artinian ring, then this classification can be divided into the classification of indecomposable artinian modules and those modules over the coordinate rings with no non-trivial artinian submodules. The classification of the artinian modules can be reduced to the case of a finite dimensional algebra over a semi-simple ring. This approach is then used as the key step in the case where the coordinate rings are hereditary noetherian serial rings over a common quotient which is a matrix ring over a field, resulting in a complete module classification.

1. Introduction

A focal point of recent attempts to classify finitely generated indecomposable modules over non-artinian rings has been pullback rings. Notably, there is the monumental work of L. Levy [Le3], resulting in the classification of all finitely generated indecomposable modules over Dedekind-like rings. L. Klingler extended this classification to lattices over certain non-commutative Dedekind-like rings [Kl], and Klingler and J. Haefner [HK1, HK2] classified lattices over certain non-commutative pullback rings, which they called special quasi-triads. Common to all these classifications is the reduction to a “matrix problem” over a division ring which is then solved using techniques similar to those in [KL]. However, as the rings become more complicated, the matrix problems become rather unwieldy, thereby limiting the usefulness of this matrix approach.

At the same time, the theory of finitely generated modules over finite dimensional k-algebras, k a field, has witnessed tremendous progress with the help of new and powerful functorial techniques, like the method of functorial filtrations developed by W. Crawley-Boevey [CB1-3], and almost split sequences, as well as the theory of bocses, a formalization of matrix methods [CB4]. (See [Ri] for a recent survey.) The matrix problems solved in [KL] are special cases of self-reproducing systems

1991 Mathematics Subject Classification. 16D70. Key words and phrases. pullback ring, finitely generated module, serial ring, Dedekind-like

ring.

The first author was supported, in part, by NSF grant #DMS-9101000, and Colorado College. The second author was supported, in part, by Baylor University, Colorado College, an NMSU travel grant, and the Sonderforschungsbereich at the University of Bielefeld. He would also like to thank the Cornell University Mathematics Department for its hospitality while part of this work was completed. Finally, the authors would like to thank L. Levy, M. Prest, and the referee for a careful reading of the manuscript.

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[NR1](see [NR2] for a correction), which, in turn, can be interpreted as finitely generated modules over a finite dimensional clannish k-algebra. One of the most sweeping results is Crawley-Boevey’s module classification for clannish algebras [CB2], which subsumes many earlier results, such as the classifications in [BS, SW, WW]. His functorial filtration techniques are derived from combinatorial arguments of [GP].

There are some intriguing “coincidences” between the clannish algebra situation and the non-artinian rings with local coordinate rings considered by Levy. In both cases indecomposable modules break up into two classes, called string and band modules in the artin algebra setting, and deleted cycle and block cycle modules in the non-artinian pullback setting. In fact, the deleted cycle and block cycle matrix solutions of [KL] can be described as strings and bands, respectively (see [AL]).

The purpose of this paper is to exploit these coincidences and outline a new approach to the classification of finitely generated indecomposable modules over certain types of pullback rings, including Dedekind-like rings. It remains to be explored how general the class of pullback rings is, to which our method applies.

Specifically, let R_{1}, R_{2} be noetherian rings with a semi-simple artinian ring F as common quotient. Let v_{i} : R_{i} ⟶ F, i = 1, 2, be the canonical projections, and let

R = {(r_{1}, r_{2}) ∈ R_{1} × R_{2}|v_{1}(r_{1}) = v_{2}(r_{2})}

be the pullback ring defined by these data. If R_{1} and R_{2} are Dedekind domains, and F is a common residue field, then R is an example of a Dedekind-like ring. In turn, any artinian module over a Dedekind-like ring is a module over a pullback ring with R_{1} and R_{2} discrete valuation rings [Le3, Prop. 15.1]. Also, if R_{1} and R_{2} are finite products of hereditary orders over Dedekind domains, then R is an example of a Dedekind-like order [Kl].

For M a finitely generated R-module, define A(M) to be the unique maximal artinian submodule of M. After deriving some properties of the functor A in Section 2, and some results on pullbacks of serial rings in Section 3, we will show in Section 4 (Theorem 4.1) that, if R_{1} and R_{2} are hereditary noetherian homogeneously serial rings, finitely generated over their centers, and M is indecomposable, then A(M) is also indecomposable, and M is determined, up to isomorphism, by A(M) and M/A(M). For instance, finite products of maximal orders over complete discrete valuation rings are noetherian serial [Wa, Theorem 6.8]. In this case, M/A(M) is an R-lattice, hence easily classified up to isomorphism. Thus, classification of indecomposables is reduced to the classification of artinian indecomposables.

In Section 5 we establish a bijection between isomorphism classes of indecomposable R-modules, for a pullback ring R, whose coordinate rings are finitely generated over their respective centers, and isomorphism classes of indecomposable modules over the associated graded ring G(R) of R. The ring G(R) is also a pullback ring.

In particular, if R_{1} and R_{2} are hereditary noetherian serial, finitely generated over their centers, and F is a matrix ring over a field k, then G(R) is Morita equivalent to the Gelfand algebra k[x, y]/⟨xy⟩, localized at the maximal ideal ⟨x, y⟩, where k[x, y] is the polynomial ring in commuting variables.

Finally, in Section 6, we classify the indecomposable modules over a pullback ring of hereditary noetherian serial rings, finitely generated over their centers, with F a matrix ring over a field. This classification uses the complete list of indecomposables in [BS] for the truncations

Λ_{pq} = k[x, y]_{⟨x,y⟩}/⟨xy, x^{p+1} − y^{q+1}⟩.

These algebras are classical examples of Crawley-Boevey’s clannish algebras, being realized as the path algebra of a quiver with one vertex and two loops x and y, modulo the relations xy = 0 = x^{p+1} − y^{q+1} [CB2]. Thus, our approach, for instance, provides an alternative derivation of the local case of Levy’s classification [Le2] (Example at the end of Sect. 6).

Several problems remain, which we hope to address in a later paper:

(1) What conditions on R_{1}, R_{2}, and F guarantee that the artinian modules over the associated graded ring G(R) of the pullback ring R can be classified? For instance, if G(R) has suitable quotients which are clannish algebras, then the artinian modules are known from the work of Crawley-Boevey. Theorem 5.1 then leads to a classification of the artinian modules over R.

(2) Describe more general pullback rings for which an indecomposable module M can be reconstructed, up to isomorphism, from A(M) and M/A(M).

Answers to these two questions would provide conditions on the rings defining the pullback ring R, for which the above program can be carried out to classify finitely generated indecomposable modules by splitting the problem into an artinian and a non-artinian part. The artinian problem is then a problem to be settled through the representation theory of k-algebras, and the non-artinian problem reduces to classifying non-artinian modules over the coordinate rings. For instance, it seems likely that much of this program can be carried out for Dedekind-like orders [Kl], or, more generally, for certain orders defined by conductor squares.

2. Separated representations and the functor A

In this section we study the functor which assigns to a finitely generated module over a pullback ring its maximal artinian submodule, as it relates to separated representations.

As in the previous section, let R_{1} and R_{2} be noetherian rings and v_{i} : R_{i} → F, i = 1, 2, be ring homomorphisms onto a semi-simple artinian ring F. Let R = {(r_{1}, r_{2}) ∈ R_{1} × R_{2} | v_{1}(r_{1}) = v_{2}(r_{2})}, the pullback ring defined by v_{1} and v_{2}. Let I_{i} = ker(v_{i}), i = 1, 2, and I = I_{1} ×I_{2}. Then R is a noetherian ring and R/I ≅ F ≅ R_{i}/I_{i}, i = 1, 2. Furthermore, for i ≠ j, the sequence 0 ⟶ I_{i} ⟶ R ⟶ R_{j} ⟶ 0 is an exact sequence of R-modules. Also, note that the Jacobson radical J(R) is contained in I, and J(R_{i}) ⊂ I_{i}. Alternatively, one may think of R as a subdirect product of R_{1} and R_{2}, that is, R is a subring of R_{1} × R_{2} such that the projections R ↠ R_{i} are onto. In particular, each R_{i} is an R-module.

If R_{1} and R_{2} are finite products of hereditary orders over a Dedekind domain, such that F is a finite product of fields, then R is a special case of a Dedekind-like pullback ring. These have been studied by Levy [Le3] (for the commutative case) and Klingler [Kl]. If R_{1} and R_{2} are Dedekind domains and F is a field, then the resulting pullback ring was called a 2-pullback ring in [AL].

Now let R be a pullback ring, and denote by Mod(R) the category of finitely generated R-modules. A module S in Mod(R) is called separated if there exist S_{i} in Mod(R_{i}), i = 1, 2, such that S is a submodule of S_{1} ⊕ S_{2}. Alternatively, S is separated if it is isomorphic to the pullback of the diagram

S/I_{1}S ⟶ S/IS ⟵ S/I_{2}S

[Le1, Cor. 3.3].

Two separated R-modules S and S′ are isomorphic if and only if there exists a commutative diagram of R-module homomorphisms

S/I_{1}S ⟶ S/IS ⟵ S/I_{2}S
≅↓  ≅↓  ≅↓
S′/I_{1}S′ ⟶ S′/IS ⟵ S′/I_{2}S

in which the vertical maps are isomorphisms. This follows immediately from the universal property of pullback diagrams.

A separated representation of a module M in Mod(R) is an R-module epimorphism φ : S ↠ M such that S is separated and such that, if φ admits a factorization

φ : S ⟶^{f} S′ ↠ M

with S′ separated, then f is one-to-one [Le1, p. 51].

We recall some properties of separated representations. The module K = ker(φ) is an F -module, since F = R/I and IK = 0 [Le1, Prop. 2.4]. An exact sequence

0 ⟶ K ⟶ S ⟶ M ⟶ 0

of R-modules with S separated and K an R/I-module is a separated representation of M if and only if I_{i}S ∩ K = 0 for each i and K ⊂ IS [Le1, Prop. 2.3].

Every module in Mod(R) has a separated representation, which is unique up to isomorphism [Le1, Theorem 2.8]. More generally, R-homomorphisms lift to separated representations, preserving epis and monos [Le1, Theorem 2.6]. That is, separated representations are “almost functorial”.

Let A : Mod(R) ⟶ Mod(R) be the functor which sends an R-module M to its (unique) maximal artinian submodule A(M). The following proposition shows that A “almost” preserves separated representations.

Lemma 2.1. Let R be a pullback ring, and let

0 → K → S ⟶^{φ} M → 0

be a separated representation of M in Mod(R). Then (1) the sequence

0 ⟶ K ⟶ A(S) ⟶^{A(φ)} A(M) ⟶ 0

is exact; (2) M/A(M) ≅ S/A(S) is separated, in fact, isomorphic to an R-submodule of

S_{1} ⊕ S_{2}, such that S_{i} has no non-trivial artinian submodules;

(3) the module A(S) is separated;

(4) the sequence

0 ⟶ K ∩ IA(S) ⟶ T ⟶^{A(φ)} A(M) ⟶ 0

is a separated representation of A(M), where K = (K ∩ IA(S)) ⊕ L and A(S) = T ⊕ L.

Proof. 1. Since K is a finitely generated F -module and F is artinian it follows that K ⊂ A(S). Moreover, A(S) is the preimage of A(M) in S, since extensions of artinian modules are artinian. This shows that

0 ⟶ K ⟶ A(S) ⟶ A(M) ⟶ 0

is exact.

2. Using the Five-Lemma it follows from Part 1 that M/A(M) ≅ S/A(S), which is separated since, if S ⊂ S_{1} ⊕ S_{2}, then S/A(S) ⊂ S_{1}/A(S_{1}) ⊕ S_{2}/A(S_{2}).

3. Clearly, A(S) ⊂ A(S_{1}) ⊕ A(S_{2}), hence is separated.

4. Since F is semi-simple and K is an F -module, it follows that

K = (K ∩ IA(S)) ⊕ L

for some F -module L. Since L ∩ IA(S) = 0, it follows from [Le1, Lemma 2.2] that L is a summand of A(S). Let A(S) = T ⊕ L. The exact sequence in Part 1 yields the exact sequence

0 ⟶ K ∩ IA(S) ⟶ T ⟶ A(M) ⟶ 0.

The module T is separated, being a direct summand of a separated module, and K ∩ IA(S) ⊂ IA(S) = IT. Furthermore, (K ∩ IA(S)) ∩ I_{i}T = 0 for each i = 1, 2. This follows from the fact that K ∩ I_{i}S = 0, and

K ∩ IA(S) ∩ I_{i}T ⊂ K ∩ IA(S) ∩ I_{i}S = 0.

Assertion 4 now follows from [Le1, Prop. 2.3].

Lemma 2.2. Let R be a pullback ring, and let S in Mod(R) be separated. Assume further that each coordinate module S/I_{i}S is either artinian or has no non-trivial artinian submodules.

(1) If A(S/I_{i}S) = S/I_{i}S for each i, then A(S) = S, that is, if the coordinate modules of S are artinian, then so is S.

(2) If A(S/I_{i}S) = 0 for each i, then A(S) = 0.

(3) If 0 ≠ A(S) ≠ S, then A(S) = I_{j}S ≅ I_{j}(S/I_{i}S) for some i ≠ j.

Proof. 1. This follows immediately, since S ⊂ S/I_{1}S ⊕ S/I_{2}S, which is artinian by hypothesis.

2. Observe that A(S) is isomorphic to the pullback of the diagram

(A(S) + I_{1}S) /I_{1}S ⟶ (A(S) + IS) /IS ⟵ (A(S) + I_{2}S) /I_{2}S,

because this pullback is an artinian submodule of S and contains A(S). Furthermore, each (A(S) + I_{i}S) /I_{i}S is contained in A(S/I_{i}S), which is zero by hypothesis.

3. In light of Parts 1 and 2, we may assume without loss of generality that A(S/I_{1}S) = 0 and A(S/I_{2}S) = S/I_{2}S. In particular, (A(S) + I_{1}S) /I_{1}S = 0, hence A(S) ⊂ I_{1}S. Conversely, I_{1}S is artinian, since it is contained in the artinian module S/I_{2}S. Observe furthermore that I_{1}S ∩ I_{2}S = 0. Thus A(S) = I_{1}S = I_{1}(S/I_{2}S). This finishes the proof.

3. Pullbacks of Serial Rings

First we prove a general result about pullback rings. Throughout this section we shall assume that the Jacobson radical I of R is equal to I_{1} × I_{2}, where I_{i} = ker(R_{i} ↠ F) is the Jacobson radical of R_{i}, and R/I = F is semi-simple artinian. Recall [Wa, Lemma-Definition 1.3] that, if R is a ring with Jacobson radical I, then R is semiperfect if R/I is artinian and idempotents of R/I lift to idempotents of R.

Theorem 3.1. If R is a pullback ring with semiperfect coordinate rings, then R is semiperfect.

Proof. It is sufficient to show that an idempotent f ∈ F lifts to an idempotent in R. But each R_{i} is semiperfect, hence there are idempotents e_{i} ∈ R_{i} which lift f. Then e = (e_{1}, e_{2}) ∈ R is an idempotent which lifts f.

We now recall the definition of basic rings and basic idempotents [AF, pp. 308– 309]. Let R be a semiperfect ring with R/I = F semi-simple artinian. We can write 1 = f_{1} + ⋯ + f_{n} ∈ F as a sum of primitive orthogonal idempotents. Then 1 = e_{1} + ⋯ + e_{n} ∈ R is a sum of primitive orthogonal idempotents, with the property that e_{i} lifts f_{i} and Re_{i} ≅ Re_{j} if and only if F f_{i} ≅ F f_{j}. After suitably relabeling the e_{i}, choose a representative Re_{1}, …, Re_{m} for each isomorphism class of the modules Re_{1}, …, Re_{n}, and let

e = e_{1} + ⋯ + e_{m}.

Then e is called a basic idempotent of R, and eRe is a called a basic ring of R. If f ∈ F is an idempotent, which lifts to e ∈ R, then f is a basic idempotent of F and fF f is a product of division rings [AF, Prop. 27.15].

Corollary 3.2. Let R be a pullback ring, with semiperfect coordinate rings R_{1}, R_{2}. Let e ∈ R be an idempotent such that eRe is a basic ring of R. Then

(1) R is Morita equivalent to eRe;

(2) if e_{i} is the image of e in R_{i}, and f ∈ F is the image of e_{i}, then e_{i}Re_{i} is a basic ring of R_{i} and fF f is a product of division rings; eRe is the pullback of e_{i}Re_{i} ⟶ fF f for i = 1, 2.

Proof. 1. This is [AF, Prop. 27.14].

2. Since each R_{i} is semiperfect, with R_{i}/I_{i} = F, a basic idempotent of F lifts to basic idempotents in R_{i} which, in turn, lift to a basic idempotent of R. This implies (2).

We next impose some conditions on the coordinate rings of R to guarantee that indecomposable separated modules in Mod(R) satisfy the hypotheses of Lemma 2.2.

Let A be a ring. Recall that an A-module is serial if its submodules are linearly ordered under inclusion. The ring A is serial if it is both a finite direct sum of left serial modules and a finite direct sum of right serial modules. An A-module is local if it has a unique maximal submodule.

Lemma 3.3. Let R be a pullback ring with noetherian serial coordinate rings R_{i}. Let S ∈ Mod(R) be separated and indecomposable.

(1) For each i ≠ j, S/I_{i}S is a serial R_{j}-module, and S/IS is simple.

(2) If 0 ≠ A(S) ≠ S, then A(S) = I_{j}S is indecomposable for some i ≠ j, S/A(S) = S/I_{j}S is an indecomposable projective R_{i}-module, and S/I_{i}S is an artinian indecomposable R_{j}-module.

Proof. 1. First recall that S/I_{i}S is an R_{j}-module, for i ≠ j. Since R_{j} is noetherian serial, it follows from [Wa, Cor. 3.4 and Theorem 2.6] that S/I_{i}S is a finite direct sum of serial, hence indecomposable, modules:

S/I_{i}S = A_{1} ⊕ ⋯ ⊕ A_{n}.

Call x ∈ S/I_{i}S a local element if it generates a local submodule. Thus, the summands in the above decomposition are generated by local elements, and these form a minimal generating set of local elements for S/I_{i}S. By [Wa, Lemma 1.11] these local elements map bijectively to a minimal set of local generators for S/IS. Hence, S/IS = F_{1} ⊕ ⋯ ⊕ F_{n} decomposes as a direct sum of n simple F -modules, with A_{k} ↠ F_{k}. It follows from [Wa, Cor. 3.4] that

S/I_{j}S = B_{1} ⊕ ⋯ ⊕ B_{n}

where each B_{k} is an indecomposable local cyclic serial R_{i}-module, which maps onto F_{k}. Taking pullbacks of the diagrams

A_{k} ⟶ F_{k} ⟵ B_{k},

we conclude that S is isomorphic to the direct sum of these pullbacks. Since S was assumed to be indecomposable, it follows that n = 1, which completes the proof of Part 1.

2. We verify the hypotheses of Lemma 2.2. Since S/I_{i}S is a serial module over the serial ring R_{j}, we have that S/I_{i}S = P/L for some serial (hence indecomposable) projective R_{j}-module P [Wa, Cor. 3.4 and Theorem 2.6]. If S/I_{i}S is not artinian, then, for all k ≥ 1, I_{j}^{k}P is not contained in L. Since P is serial, L ⊂ ⋂_{k} I_{j}^{k}P. But ⋂_{k} I_{j}^{k}P = 0 [Wa, proof of Theorem 5.11], hence L = 0. Thus, S/I_{i}S = P is projective and indecomposable. A noetherian serial ring is the product of an artinian serial ring and a hereditary serial ring with no simple left or right ideals [Wa, Theorem 5.11]. Since S/I_{i}S is indecomposable projective and non-artinian, it follows therefore that A(S/I_{i}S) = 0.

The rest of the proof follows from Lemma 2.2 and the fact that A(S) = I_{j}S ≅ I_{j}(S/I_{i}S) is indecomposable, since it is a submodule of the serial module S/I_{i}S.

As mentioned above, a noetherian serial ring A is the product of an artinian serial ring and a hereditary serial ring with no simple left or right ideals [Wa, Theorem 5.11].

Lemma 3.4. Let A be a noetherian hereditary serial ring with Jacobson radical J. Assume furthermore that A is a finitely generated module over its center. Let M and M′ be finitely generated artinian A-modules. Let ϕ : JM ⟶ JM′ be an isomorphism, and M/JM ≅ M′/JM′. Then ϕ extends to an isomorphism μ : M ⟶ M′.

Proof. We first reduce to the case where M = M′ is projective. Let

0 ⟶ L ⟶ P ⟶ M ⟶ 0

be a projective cover of M, such that L ⊂ JP and the induced map P/JP ⟶ M/JM is an isomorphism. Such a projective cover exists [Wa, Lemma 1.6], since serial rings are semiperfect. Since A is hereditary, JP is projective, as a submodule of a projective module. We obtain a short exact sequence

0 ⟶ L ⟶ JP ⟶ JM ⟶ 0.

Let P′ and L′ be defined similarly. Then the isomorphism ϕ : JM ⟶ JM′ extends to an isomorphism ϕ′ : JP ⟶ JP′. To see this observe that A is semi-perfect, so that one can explicitly characterize projective covers [AF, 27.13]. It follows from this characterization that the isomorphism P/JP ≅ P′/JP′ can be extended to an isomorphism P ≅ P′, which restricts to an isomorphism JP ⟶^{Ψ} JP′. Now let Q, resp. Q′, be a projective cover of JM, resp. JM′. Then the isomorphism ϕ : JM ⟶ JM′ extends to an isomorphism Φ : Q ⟶ Q′. Furthermore, JP ≅ Q ⊕ Z and JP′ ≅ Q′ ⊕ Z′, and Z, resp. Z′, is contained in the kernel of the projection Q ⟶ JM, resp. Q′ ⟶ JM′ [AF, Lemma 17.17]. We can now construct a commutative diagram

\begin{aligned}0 & ⟶ & Q & ⟶ & Q ⊕ Z & ⟶ & Z & ⟶ & 0 \\ & & ^{Φ}↓ & & ^{Ψ}↓ & & ≅↓ \\ 0 & ⟶ & Q′ & ⟶ & Q′ ⊕ Z′ & ⟶ & Z′ & ⟶ & 0 \end{aligned}.

Using the isomorphism Z ⟶^{Γ} Z′, we can now lift ϕ : JM ⟶ JM′ to an isomorphism

JP ⟶^{Φ⊕Γ} JP′.

By [Wa, Theorem 5.15] A is a product of rings which are Morita equivalent to either an upper triangular matrix ring over a division ring (which then is artinian hereditary serial) or a (D : J)-upper triangular matrix ring over a discrete valuation ring D with Jacobson radical J (in which case A(A) = 0). We may assume that A is a ring of one of the two types.

CASE 1: If A is Morita equivalent to an upper triangular matrix ring over a discrete valuation ring, then A(A) = 0. Since A is assumed to be finitely generated over its center, it follows from [Wa, Theorems 5.15 and 6.7] that A is actually equal to a hereditary order over a commutative discrete valuation domain D with maximal ideal ℘. By [Re, Theorem 18.2.3] we may assume that D is complete. Thus, we are in a special case of [Re, Theorem 39.14], and A has the form

( \begin{matrix}D & ℘ & ℘ & … & ℘ \\ D & D & ℘ & … & ℘ \\ ⋮ & ⋮ & ... & & ℘ \\ D & D & … & & D \end{matrix} ) .

Since A is hereditary, A-lattices and A-projectives agree. By [Re, Theorem 39.23], the indecomposable A-lattices are, up to isomorphism, all of the form J^{i}C, i = 0, …, n − 1, where C is the space of column vectors of length n, with entries in D. Thus,

J^{i}C = [\underbrace{℘, ℘, …, ℘}_{i}, D, …, D]^{t}.

Hence, we may assume that our projective P is of the form ⨁_{i}(J^{i}C)^{n_{i}}, with n_{i} nonnegative integers, and JP = ⨁_{i}(J^{i+1}C)^{n_{i}}. Our goal is to extend an automorphism ϕ of JP to P. Thus, ϕ can be thought of as a matrix whose entries are A-homomorphisms f : J^{i+1}C ⟶ J^{j+1}C. In turn, f can be represented as a matrix whose entries are elements of D. Since f is A-linear, this matrix must commute with all elements of A. Therefore f is multiplication by an element of D, of the form uπ^{k}, where π is a generator of ℘. It is now clear that f can be extended to a homomorphism J^{i}C ⟶ J^{j}C, which is multiplication by u if k = 0 and multiplication by uπ^{k−1} if k > 0.

By extending all appropriate maps in the matrix representing ϕ in this way, we obtain an extension of ϕ to P. Applying the same procedure to the inverse of ϕ, it follows that this extension is also an automorphism.

CASE 2: If A is an upper triangular matrix ring over a division ring then we are in the situation of [Re, Theorem 39.7], which implies that the indecomposable projectives are of the same form as in Case 1. The proof now continues as before. This completes the proof of the lemma.

Lemma 3.5. Let R be a pullback ring with serial coordinate rings R_{1}, R_{2}. Assume, furthermore, that R_{i}, i = 1, 2, is finitely generated as a module over its center, and that R/I is artinian. Let S be a separated R-module. Then S is indecomposable and cyclic if and only if S/I_{1}S, resp. S/I_{2}S, is an indecomposable R_{2}-, resp. R_{1}- module.

Proof. First assume that S is indecomposable. Since S is separated, it can be represented as the pullback of S_{1} = S/I_{2}S and S_{2} = S/I_{1}S over S/IS. First decompose S_{1} into a direct sum of serial (hence cyclic) summands:

S_{1} = T_{1} ⊕ ⋯ ⊕ T_{n},

and pick generators x_{i} for T_{i}. Then \{\overline{x_{1}}, …, \overline{x_{n}}\} is a minimal set of local generators for S/IS.

Now observe that S/IS = S_{2}/I_{2}S_{2}, and that R_{2} is serial, hence semiperfect, by hypothesis. By [Wa, Lemma 1.11], the set \{\overline{x_{1}}, …, \overline{x_{n}}\} lifts to a minimal set of local generators {y_{1}, …, y_{n}} of S_{2}. Consequently, we get a decomposition

S_{2} = R_{2}y_{1} ⊕ ⋯ ⊕ R_{2}y_{n}.

Hence S composes as a direct sum of the n modules obtained by pulling back R_{1}x_{i} and R_{2}y_{i} over (R/I) · \overline{x_{i}}. Since S was assumed indecomposable it follows that n = 1, so that S_{1} and S_{2} are indecomposable. Since they are serial they are also cyclic, which makes S cyclic also.

The other direction is obvious.

Recall [Wa, p. 193] that a module M over a ring R with Jacobson radical J is homogeneously serial if for all pairs of nonzero finitely generated submodules A and B we have that A/IA ≅ B/JB. A ring is homogeneously serial if its indecomposable summands are serial. The next lemma is needed in the proof of Theorem 4.1, but it is of interest in its own right, since it shows that modules over noetherian hereditary homogeneously serial rings share many characteristics with modules over Dedekind-like rings.

Lemma 3.6. Let R be a pullback ring with hereditary noetherian homogeneously serial coordinate rings R_{1}, R_{2}, finitely generated as modules over their centers. Let M be an indecomposable R-module with a separated representation

0 ⟶ K ⟶ S ⟶ M ⟶ 0.

Write S = S_{1} ⊕ ⋯ ⊕ S_{n} as a direct sum of indecomposable separated modules. Then, for each α = 1, 2, at most one S_{i}, i = 1, …, n, has the property that A(S_{i}) is an R_{α}-module.

Proof. It is sufficient to consider the case α = 2; the case α = 1 is analogous. First of all, using Corollary 3.2 and [Wa, Lemma 1.11] we can reduce ourselves to considering the case where R/I is a division ring Δ. Let

S = S_{1} ⊕ ⋯ ⊕ S_{n},

with each S_{i} indecomposable and cyclic, generated by s_{i}. Let

K = K_{1} ⊕ ⋯ ⊕ K_{m},

with each K_{j} a simple Δ-module, generated by x_{j}. The inclusion K ↣ S represents each x_{j} as a linear combination of the s_{i}:

x_{j} ↦ (a_{1j}, b_{1j})s_{1} + ⋯ + (a_{nj}, b_{nj})s_{n},

for j = 1, …, m, where (a_{ij}, b_{ij}) ∈ R.

By [Wa, Theorem 1.14] the ideal I_{2} is principal as left and right R_{2}-module. Now consider the R_{2}-module S_{1}/I_{1}S_{1} = T. If ann(T) = 0, then A(T) = 0. Since K ⊂ A(S) this implies that (a_{i1}, b_{i1}) = 0 for all i = 1, …, m. So assume that ann(T) ≠ 0. Since R_{2} is serial there exists an integer r such that

I_{2}^{r+1} ⊊ ann(T) ⊆ I_{2}^{r}.

Let π_{2} ∈ I_{2} be a generator of I_{2} as a left R_{2}-module. Then, by [Wa, Theorem 1.14], ann(T) is principal, generated by tπ_{2}^{e} as a left R_{2}-module, for some t ∈ R_{2} \ I_{2}. Since K is a Δ-module it follows that b_{1j} = u_{j}tπ_{2}^{r} for some u_{j} ∈ R_{2} which is either zero or a unit, since I_{2} is the radical of R_{2}. Similarly, I_{1} is generated by π_{1} as a left R_{1}-module, and a_{1j} = v_{j}sπ_{2}^{f}. Similar considerations hold for the other summands S_{i}.

Arrange the coefficients in the relations for the x_{j} in a matrix ((a_{ij}, b_{ij})) of size n × m, with entries in R. We will use “admissible” elementary row and column operations over R to transform this matrix into a suitable normal form. The process is similar to part of the procedure described in [NR2] (see also [LS]). Given an entry (a_{ij}, b_{ij}) write a_{ij} = v_{ij}sπ_{2}^{f_{j}} with f_{j} maximal. Then call f_{j} the a-degree of the entry. Define the b-degree e_{j} of the entry similarly. Note that all entries of a given row have the same a- and b-degree. Now call all elementary column operations with scalars from R admissible. An elementary row operation of the form (row i)+x(row j) is admissible if either the two rows have the same a- and b-degrees or if row i has larger a-, resp. b-, degree and x is a product of (π_{1}^{f_{i}−f_{j}}, 0) with a unit of R, resp. a product of (0, π_{2}^{e_{i}−e_{j}}) with a unit. For instance, to add a multiple of the i-th row to the j-th row to eliminate the b-part of the first entry of the j-th row, in the case where both rows have the same a- and b-degrees, observe that b_{i1} = u_{i}t_{i}π_{2}^{f_{i}}, where u_{i} ∈ R_{2} is a unit. Likewise, b_{j1} = u_{j}t_{i}π_{2}^{f_{j}} with u_{j} ∈ R_{2} a unit. Choose units z_{i} = (v_{i}, u_{i}) and z_{j} = (v_{j}, u_{j}) ∈ R, multiply the i-th row with z_{j}z_{i}^{−1} and subtract it from the j-th row.

Step 1. First we diagonalize the a-part of the matrix. Permute the rows (by rearranging the indecomposable summands of S) so that they occur in order of decreasing a-degree from top to bottom. Order the rows with the same a-degree in order of decreasing b-degree. Now locate the rows with nonzero entries of the lowest occuring a-degree. Of those choose one with lowest b-degree and permute columns to move a nonzero entry in this row to the last column. We can now perform admissible elementary row and column operations to clear out all a-entries from this row and the last column. Now move to the next lowest row with nonzero a-entries, move one to the next to last column and proceed as before. Moving up the matrix in this way results in a matrix with a-part in diagonal form. Note that every column must contain an entry with nonzero a-part, otherwise one of the properties of a separated representation is violated [Le1, Prop. 2.3].

We now assign weights to the rows and columns as follows. If a column contains an entry with a-degree e, then its weight is e. (Note that there is exactly one entry with nonzero a-part in each column.) If a row contains an entry with b-degree f, then its weight is f. If it does not contain any nonzero b-entries, then its weight is infinite.

Now partition the matrix into blocks such that in each block all rows have the same weight and all columns have the same weight. Call a block a b-block if it contains no entries with nonzero a-part, otherwise an ab-block.

Step 2. Choose the highest horizontal strip of lowest occuring row weight. In this strip choose the leftmost nonzero block, that is, the one with the highest vertical weight, call it B.

Step 2.1. Suppose that B is a b-block. Perform admissible elementary row and column operations to bring it into the form

( \begin{matrix}... & 0 \\ 0 & 0 \end{matrix} ) ,

where the upper left part is a diagonal block. Since the vertical or the horizontal strip containing B also contains an ab-block we need to perform admissible column operations to restore the diagonal form of its a-part.

Now introduce new partitions into the matrix so that B only contains zero and diagonal blocks. For any ab-block introduce further partitions so that the upper left part is square. Proceed by using the nonzero entries in B to clear out their rows and columns of b-entries. Then restore the diagonal form of the a-part of any ab-blocks that were affected. Proceed to the horizontal strip of next lowest horizontal weight.

Step 2.2. Suppose B is an ab-block. Then we can use the nonzero b-entries in this block to clear out all b-entries in their rows and columns and split off a direct summand. This contradicts the indecomposability of the module M, so this case cannot happen.

After this procedure has been carried out the matrix has the property that each horizontal strip contains at most one nonzero a-block, and each vertical strip contains exactly one nonzero ab-block and one nonzero b-block. By permuting blocks we can transform the matrix into one of the following two types:

( \begin{matrix}ab & 0 & 0 & ⋯ & 0 & b \\ b & ab & 0 & ⋯ & 0 & 0 \\ 0 & b & ab & ⋯ & 0 & 0 \\ ⋮ & ⋮ & ... & ... & ⋮ & ⋮ \\ 0 & 0 & 0 & ... & ab & 0 \\ 0 & 0 & 0 & ⋯ & b & ab \end{matrix} );

( \begin{matrix}ab & 0 & 0 & ⋯ & 0 & 0 \\ b & ab & 0 & ⋯ & 0 & 0 \\ 0 & b & ab & ⋯ & 0 & 0 \\ ⋮ & ⋮ & ... & ... & ⋮ & ⋮ \\ 0 & 0 & 0 & ... & ab & 0 \\ 0 & 0 & 0 & ⋯ & b & ab \\ 0 & 0 & 0 & ⋯ & 0 & b \end{matrix} ) .

Observe that if any of the blocks are larger than 1 × 1, then we can permute the rows and columns to decompose the matrix into a sum of smaller matrices of the same type.

It now follows that if M is indecomposable, then this matrix must have at least n − 1 columns, that is, m ≥ n − 1. Otherwise the matrix is of the second type, and either the b-block in the last row is of size larger than 1 × 1, or there is a row of zeros. In both cases we obtain a contradiction to the indecomposability of M.

To complete the proof of the lemma, assume now that there are two summands, S_{1} and S_{2} say, such that A(S_{1}) and A(S_{2}) are R_{1}-modules. Then the b-part of the first two rows of our matrix is zero. But then there will be a column of the matrix in normal form which contains no nonzero b-entries. This contradicts the property of separated representations [Le1, Prop. 2.3] that K ∩ I_{1}S = 0. The proof of the lemma is now complete.

4. Indecomposable modules and the functor A

Theorem 4.1. Let R be a pullback ring with hereditary noetherian homogeneously serial coordinate rings R_{1}, R_{2}, such that I_{i} = J(R_{i}) and I = I_{1} × I_{2} = J(R) for i = 1, 2. Assume, furthermore, that R_{i}, i = 1, 2, is finitely generated as a module over its center.

(1) If M, M′ are indecomposable modules in Mod(R) such that A(M) ≅ A(M′) and M/A(M) ≅ M′/A(M′), then M ≅ M′.

(2) If M in Mod(R) is indecomposable, then A(M) is also indecomposable.

Proof. We first give an outline of the proof to help the reader through the maze of technical details to come. The first step in the proof is to reduce the theorem to the case of separated modules via separated representations. Now, a separated module S is described via a pullback square

S ⟶ S/I_{1}S
↓  ↓
S/I_{2}S ⟶ S/IS .

To give an isomorphism between separated modules, it is sufficient to give isomorphisms of the lower left and upper right corners of the respective defining pullback diagrams, which agree when extended to the lower right corner. Under the above assumptions on the coordinate rings of R, the previous analysis of indecomposable separated R-modules imposes certain restrictions on the coordinate modules S/I_{1}S and S/I_{2}S which will be exploited to construct the desired isomorphisms.

To prove the first assertion we may clearly assume that 0 ≠ A(M) ≠ M, and similarly for M′. Let

0 ⟶ K ⟶ S ⟶ M ⟶ 0

be a separated representation of M, and

0 ⟶ K′ ⟶ S′ ⟶ M′ ⟶ 0

a separated representation of M′. By Lemma 2.1.4, there is a separated representation

0 ⟶ K ∩ IA(S) ⟶ T ⟶ A(M) ⟶ 0

of A(M), such that K = (K ∩ IA(S))⊕L and A(S) = T ⊕L. Similarly for A(M′). Observe that K is a module over the semi-simple ring R/I, hence so is L. Let ϕ : A(M) ⟶ A(M′) be an isomorphism. We obtain a commutative diagram of short exact sequences

0 ⟶ K ⊕ L′ ⟶ T ⊕ L ⊕ L′ ⟶ A(M) ⟶ 0
≅↓  ≅↓  ^{ϕ}↓
0 ⟶ K′ ⊕ L ⟶ T′ ⊕ L′ ⊕ L ⟶ A(M′) ⟶ 0

The middle vertical map is the direct sum of an isomorphism T ⟶ T′, which is a lifting of ϕ [Le1, Theorem 2.6], and the obvious isomorphism L ⊕ L′ ⟶ L′ ⊕L. Hence, the middle vertical map is an isomorphism A(S)⊕L′ ⟶ A(S′)⊕L. This map makes the right-hand square commute, hence induces the left vertical isomorphism. It thus suffices to extend the middle isomorphism to an isomorphism μ : S ⊕ L′ ⟶ S′ ⊕ L, so that we obtain a commutative diagram

K ⊕ L′ ⟶ S ⊕ L′
≅↓  ≅↓
K′ ⊕ L ⟶ S′ ⊕ L .

From this diagram we then obtain an induced isomorphism of quotients M ≅ M′. If we drop the assumption that S, resp. S′, is a separated representation of M, resp. M′, then we can assume that L, resp. L′, is a summand of S, resp. S′. The property of S and S′ being separated representations is not needed for the rest of the argument. Thus we may replace S by S ⊕ L′ and S′ by S′ ⊕ L and assume that A(S) ≅ A(S′).

Let S = ⨁_{k=1}^{n} S_{k}, where each S_{k} is separated and indecomposable. We group the summands of S as follows:

A = ⨁{S_{k}|0 ≠ A(S_{k}) = S_{k}};
U_{1} = ⨁{S_{k}|0 ≠ A(S_{k}) = I_{1}S_{k} ≅ I_{1}(S_{k}/I_{2}S_{k})};
U_{2} = ⨁{S_{k}|0 ≠ A(S_{k}) = I_{2}S_{k} ≅ I_{2}(S_{k}/I_{1}S_{k})}.

Note that, unless M is separated, A(S_{k}) ≠ 0 for all k, since M is indecomposable and the image of K in S_{k} is contained in A(S_{k}). By Lemma 2.2 and Lemma 3.3, we have S = A ⊕ U_{1} ⊕ U_{2}. There is a similar decomposition of S′. Furthermore, A(S) = ⨁_{k} A(S_{k}), with each A(S_{k}) indecomposable; similarly for A(S′). Now,

U_{1}/A(U_{1}) ⊕ U_{2}/A(U_{2}) ≅ S/A(S) ≅ M/A(M)

≅ M′/A(M′) ≅ S′/A(S′) ≅ U′_{1}/A(U′_{1}) ⊕ U′_{2}/A(U′_{2}).

Since U_{1}/A(U_{1}) = U_{1}/I_{1}U_{1} is an R_{2}-module, and U_{2}/A(U_{2}) = U_{2}/I_{2}U_{2} is an R_{1}- module – similarly for U′_{1}, U′_{2} – it follows that U_{i}/A(U_{i}) ≅ U′_{i}/A(U′_{i}), i = 1, 2.

Observe that I_{1}U_{1}, I_{1}U′_{1} are R_{1}-modules, and I_{2}U_{2}, I_{2}U′_{2} are R_{2}-modules. Furthermore, we can decompose A as A = A_{1} ⊕ A_{2} ⊕ A_{3}, where A_{i} consists of those summands that are R_{i}-modules, for i = 1, 2, but not R_{j}-modules for j ≠ i, and A_{3} is the sum of the remaining summands of A, including those that are R/I-modules.

The isomorphism ϕ can be written as a (3 × 3)-matrix of homomorphisms. Since none of the off-diagonal entries can be isomorphisms, the diagonal entries

\begin{matrix}ϕ_{1} : A(U_{1} ⊕ A_{1}) = I_{1}U_{1} ⊕ A_{1} & ⟶ I_{1}U′_{1} ⊕ A′_{1} = A(U′_{1} ⊕ A′_{1}), \\ ϕ_{2} : A(U_{2} ⊕ A_{2}) = I_{2}U_{2} ⊕ A_{2} & ⟶ I_{2}U′_{2} ⊕ A′_{2} = A(U′_{2} ⊕ A′_{2}), \\ ϕ_{3} : A_{3} & ⟶ A′_{3} \end{matrix}

have to be isomorphisms. Now observe that in the separated representations above we have L ⊂ K ⊂ A_{3} and L′ ⊂ K′ ⊂ A′_{3}. We may replace the off-diagonal entries of ϕ by zero, to obtain another isomorphism A(M) ≅ A(M′) which induces an isomorphism K ⊕ L′ ≅ K′ ⊕ L as above. Consequently, it suffices to extend this new isomorphism, which contains no cross terms. We may now assume that ϕ = ϕ_{1} ⊕ ϕ_{2} ⊕ ϕ_{3}.

Since S is separated, it is described via the pullback square

S ⟶ S/I_{1}S
↓  ↓
S/I_{2}S ⟶ S/IS ,

where

S/IS = U_{1}/IU_{1} ⊕ U_{2}/IU_{2} ⊕ A/IA,
S/I_{i}S = U_{1}/I_{i}U_{1} ⊕ U_{2}/I_{i}U_{2} ⊕ A_{1}/I_{i}A_{1} ⊕ A_{2}/I_{i}A_{2} ⊕ A_{3}/I_{i}A_{3},

for i = 1, 2.

Since M/A(M) ≅ M′/A(M′), it follows that U_{1} = 0 if and only if U′_{1} = 0. From Lemma 3.6 it then follows that if U_{1} ≠ 0, then A′_{1} = 0. Hence the isomorphism ϕ_{1} cannot have any components mapping from I_{1}U_{1} to A′_{1}. By Lemma 3.1, U_{1}/A(U_{1}) = U_{1}/I_{1}U_{1} is a projective R_{2}–module, and U_{2}/A(U_{2}) = U_{2}/I_{2}U_{2} is a projective R_{1}–module. Since I_{1}U_{1} = I_{1}(U_{1}/I_{2}U_{1}) and U_{1}/I_{2}U_{1}, as well as A_{1}, are artinian R_{1}–modules, the isomorphism

ϕ_{1} : I_{1}U_{1} ⊕ A_{1} ⟶ I_{1}U′_{1} ⊕ A′_{1}

lifts to an isomorphism

α_{1} : U_{1}/I_{2}U_{1} ⊕ A_{1} ⟶ U′_{1}/I_{2}U′_{1} ⊕ A′_{1},

by Lemma 3.4. Passing to quotients, α_{1} induces an isomorphism

γ_{1} : U_{1}/IU_{1} ⊕ A_{1}/IA_{1} ⟶ U′_{1}/IU′_{1} ⊕ A′_{1}/IA′_{1}

between summands of S/IS and S′/IS′. Since A_{1} is an R_{1}–module, it follows that A_{1}/IA_{1} = A_{1}/I_{1}A_{1}. Also, the R_{2}–module U_{1}/I_{1}U_{1} is a projective cover of U_{1}/IU_{1}. Therefore, the isomorphism γ_{1} lifts to an isomorphism

β_{1} : U_{1}/I_{1}U_{1} ⊕ A_{1}/I_{1}A_{1} ⟶ U′_{1}/I_{1}U′_{1} ⊕ A_{1}/I_{1}A′_{1}

of summands of S/I_{1}S and S′/I_{1}S′.

Similarly, the isomorphism

ϕ_{2} : I_{2}U_{2} ⊕ A_{2} ⟶ I_{2}U′_{2} ⊕ A′_{2}

induces an isomorphism

β_{2} : U_{2}/I_{1}U_{2} ⊕ A_{2} ⟶ U′_{2}/I_{1}U′_{2} ⊕ A′_{2}

between summands of S/I_{1}S and S′/I_{1}S′. This, in turn, induces an isomorphism

γ_{2} : U_{2}/IU_{2} ⊕ A_{2}/IA_{2} ⟶ U′_{2}/IU′_{2} ⊕ A′_{2}/IA′_{2}

between summands of S/IS and S′/IS′. As before, the isomorphism γ_{2} lifts to an isomorphism

α_{2} : U_{2}/I_{2}U_{2} ⊕ A_{2}/I_{2}A_{2} ⟶ U′_{2}/I_{2}U′_{2} ⊕ A′_{2}/I_{2}A′_{2}

between summands of S/I_{2}S and S′/I_{2}S′.

In summary, we have constructed isomorphisms

α = α_{1} ⊕ α_{2} ⊕ ϕ_{3} : S/I_{2}S ⟶ S′/I_{2}S′,

and

β = β_{1} ⊕ β_{2} ⊕ ϕ_{3} : S/I_{1}S ⟶ S′/I_{1}S′,

both induced by ϕ : A(S) ⟶ A(S′), which induce the same isomorphism γ_{1} ⊕ γ_{2} ⊕ ϕ_{3} : S/IS ⟶ S′/IS′. (By abuse of notation, ϕ_{3} denotes three different isomorphisms.) It follows now from the universal property of pullback diagrams that α and β induce an isomorphism S ⟶ S′ which is an extension of ϕ, as desired. This completes the proof of statement (1).

To prove the second part, let M be an indecomposable R–module. Let

0 ⟶ K ⟶ S ⟶ M ⟶ 0

be a separated representation of M. Assume that A(M) = A_{1} ⊕ A_{2}. Then, by Lemma 2.1.4, there is a short exact sequence

0 ⟶ K = K_{1} ⊕ K_{2} ⟶ A(S) = B_{1} ⊕ B_{2} ⟶ A(M) = A_{1} ⊕ A_{2} ⟶ 0,

which is an extension of a direct sum of separated representations of A_{1} and A_{2}. As before, S = ⨁_{k=1}^{n} S_{k} is a direct sum of indecomposable separated modules S_{k}.

Then A(S) = ⨁_{k} A(S_{k}), with each A(S_{k}) indecomposable, by Lemma 3.3. It now follows from the Krull–Schmidt Theorem for artinian modules that

B_{i} ≅ C_{i} = ⨁_{k∈E_{i}} A(S_{k})

for i = 1, 2 and some partition {E_{1}, E_{2}} of {1, …, n}. Let ϕ be an automorphism of A(S) such that ϕ(B_{i}) = C_{i}. Define

M′ = S/ (ϕ(K_{1}) ⊕ ϕ(K_{2})) = D_{1}/ϕ(K_{1}) ⊕ D_{2}/ϕ(K_{2}),

with D_{i} = ⨁_{k∈E_{i}} S_{k}. Note that this makes sense, since ϕ(K_{i}) ⊂ C_{i} ⊂ D_{i}. Then

ϕ : A(M) = A(S)/K ⟶ A(M′) = A(S)/ (ϕ(K_{1}) ⊕ ϕ(K_{2}))

is an isomorphism and M/A(M) ≅ S/A(S) ≅ M′/A(M′). It now follows from the first part that M ≅ M′. (Note that, even though M′ is not assumed to be indecomposable, we have that A(S_{k}) ≠ 0 for all k, so the proof of the first part still applies.) But M was assumed to be indecomposable, hence D_{i}/ϕ(K_{i}) = 0 for some i, and so C_{i}/ϕ(K_{i}) ≅ A_{i} = B_{i}/K_{i} = 0. This shows that A(M) is indecomposable, completing the proof of the theorem.

Corollary 4.2. Let R be a pullback ring, such that each coordinate ring R_{i}, i = 1, 2, is a finite product of maximal orders over commutative complete discrete valuation domains, and let I_{i} = J(R_{i}) be the Jacobson radical of R_{i}. Then

(1) if M in Mod(R) is indecomposable, then A(M) is indecomposable;

(2) if M, M′ in Mod(R) are indecomposable, A(M) ≅ A(M′), and M/A(M) ≅ M′/A(M′), then M ≅ M′.

Proof. Such a ring is noetherian and homogeneously serial [Wa, Thm. 6.8], thus Theorem 4.1 applies.

5. Reduction to the case of F -algebras

We now relate R-modules to modules over the associated graded ring G(R) of R with respect to I ⊂ R. Let R be a noetherian ring and I a two-sided ideal of R. Let G(R) denote the associated graded ring, that is,

G(R) = ⨁_{i=0}^{∞} I^{i}/I^{i+1},

where I^{0} = R. There is a functor

G : Mod(R) ⟶ Mod (G(R)) ,

defined by G(M) = ⨁_{i} I^{i}M/I^{i+1}M, for a finitely generated R-module M.

Theorem 5.1. Let R be a noetherian ring with Jacobson radical I, such that R/I is (semi-simple) artinian. Assume furthermore that R is finitely generated as a module over its center. Then the functor G induces a bijection on isomorphism classes of objects, which preserves and reflects indecomposables.

Proof. First of all, it is straightforward to see that G preserves finite direct sums. Next we need to show that, if G(M) ≅ G(N) in Mod ((G(R)), then M ≅ N in Mod(R). First observe that ⋂_{n} I^{n}M = 0 for all R-modules M. This can be seen as follows. Since R is finitely generated over its center, R is right and left fully bounded [GW, Prop. 8.1], hence satisfies the second layer condition. Therefore, we have that ⋂_{n} I^{n} = 0 [GW, Thm. 12.8]. Using Nakayama’s Lemma it now follows that ⋂_{n} I^{n}M = 0.

Since R is noetherian, the category of finitely generated G(R)-modules is equivalent to the category Mod_{gr} (G(R)) of ℕ-graded finitely generated G(R)-modules [NV, Lemma 3.3.2]. In this category, the morphism groups are as follows. Let \underline{M} = ⨁_{i∈ℕ} M_{i}, \underline{N} = ⨁_{i∈ℕ} N_{i} be graded G(R)-modules, and let Hom(\underline{M}, \underline{N})_{p} be the abelian group of degree p homomorphisms, p ≥ 0. Here, f ∈ Hom(\underline{M}, \underline{N})_{p} if f : M_{i} ⟶ M_{i+p}. Then

Hom(\underline{M}, \underline{N}) = ⨁_{p} Hom(\underline{M}, \underline{N})_{p}.

Therefore, we may work in the category Mod_{gr} (G(R)) instead of Mod (G(R)). So suppose we are given an isomorphism

φ : ⨁_{i} I^{i}M/I^{i+1}M ⟶ ⨁_{i} I^{i}N/I^{i+1}N

in Mod_{gr} (G(R)). Then φ = ⨁_{p} φ_{p}, where

φ_{p} ∈ Hom (⨁_{i} I^{i}M/I^{i+1}M, ⨁_{i} I^{i}N/I^{i+1}N)_{p} .

Since φ is an isomorphism,

φ_{0} : I^{i}M/I^{i+1}M ⟶ I^{i}N/I^{i+1}N

must also be an isomorphism for all i ∈ ℕ. (Observe that there are no maps of negative degrees.) Let x_{1}, …, x_{n} be a set of generators for M, such that x̄_{1}, …, x̄_{n} are a minimal set of generators for M/IM. Let ȳ_{1} = φ_{0}(x̄_{1}), …, ȳ_{n} = φ_{0}(x̄_{n}) be the corresponding set of generators for N/IN. Then we can lift these to a set of generators y_{1}, …, y_{n} of N, since I is the Jacobson radical. Now define

f : M ⟶ N

by f(x_{i}) = y_{i}, for i = 1, …, n. To see that f is well-defined, suppose that x = ∑ r_{i}x_{i} = 0 in M. Since

φ_{0}(x̄) = 0 = ∑ r̄_{i}ȳ_{i} = ∑ \overline{r_{i}y_{i}},

it follows that y = ∑ r_{i}y_{i} ∈ IN. But, since φ_{0} is a graded homomorphism, we have

y + I^{2}N = ∑(r_{i}y_{i} + I^{2}N) = ∑ φ_{0}(r_{i}x_{i} + I^{2}M) = φ_{0}(x + I^{2}M) = φ_{0}(0) = 0.

This shows that y ∈ I^{2}N. Inductively, we can push y farther down the filtration. Since ⋂_{i} I^{i}N = 0, this implies that y = 0. This shows that the map f is welldefined, and R-linear by definition.

To see that f is one-to-one, suppose that f(x) = 0 for some x ∈ M. Then φ_{0}(x̄) = 0, hence x ∈ IM. Similarly, φ_{0}(x + I^{2}M) = f(x) + I^{2}N = 0, hence x ∈ I^{2}M. Since ⋂_{i} I^{i}M = 0, we can see inductively, as above, that x = 0.

Let y ∈ N. Then y + IN = φ_{0}(x + IM) for some x ∈ M. Thus, f(x) − y ∈ IN. Therefore, f(x) − y + I^{2}N = φ_{0}(x′ + I^{2}M) for some x′ ∈ M. So f(x) − f(x′) − y ∈ I^{2}N. Again, since ⋂_{i} I^{i}N = 0, we can see inductively that y is in the image of f, so that f is onto, hence an isomorphism. Thus, we have proven that the functor G reflects isomorphisms.

Now let N = ⨁_{i} N_{i} be a graded module over G(R). We need to show that N = G(M) for some R-module M. Let x_{1}, …, x_{m} be a set of generators for the R/I-module N_{0}. Since (I/I^{2})N_{i} = N_{i+1}, we get that N_{i} has generators y_{1}, …, y_{m} as an I^{i}/I^{i+1}-module. Let F be a free R-module with generators z_{1}, …, z_{m}, and let the submodule K of F be generated by elements ∑ r_{j}z_{j} such that ∑ r̄_{j}y_{j} = 0 is a relation in N_{i} for some i. Define M = F/K. Then it is straightforward to see that G(M) ≅ N.

Now it is easy to see that the functor G reflects indecomposables. Suppose that G(M) ≅ N_{1} ⊕ N_{2}. Then

G(M) ≅ N_{1} ⊕ N_{2} ≅ G(M_{1}) ⊕ G(M_{2}) ≅ G(M_{1} ⊕ M_{2}),

so that M ≅ M_{1} ⊕ M_{2}, since G reflects isomorphisms. This completes the proof.

Remarks. 1. Define a quotient category MOD(R) of Mod(R) to have the same objects as Mod(R) but morphism sets Hom_{R}(M, N)/Hom_{R}(M, IN). It is straightforward to see that MOD(R) is an additive category with finite direct sums. One observes that the canonical quotient functor Mod(R) ⟶ MOD(R) induces a bijection on objects. The functor G induces a functor of quotient categories MOD(R) ⟶ MOD (G(R)), which can be shown to be a category equivalence. (For details, see [AL].)

2. The preceding theorem, together with Theorem 4.1, implies that, if R is a pullback ring with noetherian hereditary serial coordinate rings, finitely generated over their centers, then the classification of indecomposable R-modules is reduced to the classification of indecomposable artinian G(R)-modules.

Proposition 5.2. Let R be a pullback ring of R_{1}, R_{2}. If J_{i} is the Jacobson radical of R_{i}, i = 1, 2, then assume that the Jacobson radical of R is J = J_{1} × J_{2}. Then G(R), formed with respect to J, is also a pullback ring, the Jacobson radical J (G(R)) is equal to G(J), and G(R)/J (G(R)) ≅ F. Moreover, if R_{i} is noetherian serial, then so is G(R_{i}).

Proof. Using the universal property of the following pullback diagram, it is straightforward to check that G(R) is isomorphic to the ring A in the pullback diagram

A ⟶ G(R_{1})
↓  ↓
G(R_{2}) ⟶ F .

The remaining assertions of the proposition follow immediately.

Corollary 5.3. Let R be a pullback ring with semiperfect coordinate rings R_{1}, R_{2}, finitely generated over their centers. Then

(1) G(R) is semiperfect;

(2) if F is a matrix ring over a field k, and each R_{i} is hereditary noetherian serial with A(R_{i}) = 0, then G(R) is Morita equivalent to the localization of the k-algebra k[x, y]/⟨xy⟩ at the maximal ideal ⟨x, y⟩.

Proof. 1. Since R is semiperfect, a primitive idempotent f of F lifts to a primitive idempotent e of R. Thus, Re is an indecomposable summand of R. By Theorem 5.1, G(Re) is an indecomposable summand of G(R). It now follows that G(Re) = G(R)e′ for some idempotent e′ ∈ G(R), such that e′ maps to f ∈ F. Thus, G(R) is semiperfect.

2. By Corollary 3.2, G(R) is Morita equivalent to a basic ring Λ = eG(R)e, which is a pullback of basic rings Λ_{i} = e_{i}G(R_{i})e_{i} over fF f. Since F is a matrix ring over a field k, fF f = k.

Now (identifying corresponding idempotents in R and G(R)), we obtain

Λ = ⨁_{m} e(J^{m}/J^{m+1})e,

with e(R/J)e = fF f, and

e(J^{m}/J^{m+1})e = (e_{1}J_{1}^{m}e_{1})/(e_{1}J_{1}^{m+1}e_{1}) ⊕ (e_{2}J_{2}^{m}e_{2})/(e_{2}J_{2}^{m+1}e_{2}).

Furthermore, each Λ_{i} is a noetherian serial ring which is indecomposable. Therefore, it is serial as a left Λ_{i}-module. Thus, J(Λ_{i}) = e_{i}J_{i}e_{i} = Λ_{i}x_{i}, for some x_{i} ∈ Λ_{i}. Furthermore,

J(Λ_{i})^{m}/J(Λ_{i})^{m+1} = (e_{i}J_{i}^{m}e_{i}/e_{i}J_{i}^{m+1}e_{i})

is simple, hence isomorphic to k. It follows that J(Λ_{i})^{m} = Λ_{i}x_{i}^{m}. We now have

Λ = k ⊕ (Λ_{1}x_{1}/Λ_{1}x_{1}^{2} ⊕ Λ_{2}x_{2}/Λ_{2}x_{2}^{2}) ⊕ (Λ_{1}x_{1}^{2}/Λ_{1}x_{1}^{3} ⊕ Λ_{2}x_{2}^{2}/Λ_{2}x_{2}^{3}) ⊕ ⋯
≅ k[x_{1}, x_{2}]_{⟨x_{1},x_{2}⟩}/⟨x_{1}x_{2}⟩,

as desired.

6. Classification of Indecomposables

In this section we carry out the classification scheme for indecomposable modules for the case where R is a pullback ring of hereditary noetherian serial rings, finitely generated over their centers, and F is a matrix ring over a field k.

We first need to prove a result about modules over the ring Λ above. An indecomposable artinian Λ-module is also an indecomposable module over the artin k-algebra

Λ_{pq} = k[x, y]_{⟨x,y⟩}/⟨xy, x^{p+1} − y^{q+1}⟩,

for some positive integers p, q. The classification of finitely generated indecomposable modules over Λ_{pq} is surveyed in [BS, pp. 112–113]. There are two types, string modules and band modules, corresponding to the deleted cycle and block cycle indecomposables, respectively, in [Le2]. We will call an indecomposable artinian Λ-module a string or band module, if it is a string, respectively band, module over Λ_{pq} for some p, q. We first recall the definition of string and band modules in the form given in [AL, Sect. 4], since it is needed in the proof of Lemma 6.4 below.

Observe that Λ_{pq} can be described as follows: It is the k-algebra with generators a, b and relations ab = 0 = ba, a^{p+1} −b^{q+1} = 0. We now define two types of modules over Λ_{pq}. Observe that Λ_{pq} has {1, a^{i}, b^{j} | 1 ≤ i ≤ p + 1, 1 ≤ j ≤ q} as a k-basis.

Define a word to be a (non-empty) symbol

W = a^{i_{1}}b^{j_{1}}a^{i_{2}}b^{j_{2}} … a^{i_{n}}b^{j_{n}} ,

made up of elements in the above basis of Λ_{pq}, such that j_{1}, …, j_{n−1}, i_{2}, …, i_{n} ≥ 1. (In [BS], this is called a proper word.) We set a^{0} = 1 = b^{0}. A word W is called an irreducible band word if i_{1}, j_{n} ≥ 1, and it is not a power W = (W′)^{m} of another word W′. Note that a cyclic permutation of the constituent pairs a^{i}b^{j} in an irreducible band word results in another irreducible band word.

Definition 6.1. [BS, 112]

(1) Let W be a word (not necessarily irreducible) as above. Define the associated string module M(W) to be the Λ_{pq}-module with generators u_{1}, …, u_{n} and relations

a^{i_{1}}u_{1} = 0,  b^{j_{k}}u_{k} = a^{i_{k+1}}u_{k+1} (k = 1, …, n − 1),  b^{j_{n}}u_{n} = 0 .

(2) Let W be an irreducible band word, and let φ be an indecomposable automorphism of a finite dimensional k-vector space V. Define the band module M(W, V, φ) to be the Λ_{pq}-module generated by n vector subspaces u_{1} ⊗ V, …, u_{n} ⊗ V, subject to the relations

\begin{aligned}b^{j_{k}}u_{k} ⊗ v & = a^{i_{k+1}}u_{k+1} ⊗ v for k = 1, …, n − 1, v ∈ V; \\ b^{j_{n}}u_{n} ⊗ v & = a^{i_{1}}u_{1} ⊗ φ(v), v ∈ V . \end{aligned}

Note that both M(W) and M(W, V, ϕ) are artinian Λ-modules.

Definition 6.2. Let W be a word, as in Definition 6.1.1, and let n_{1}, n_{2} be positive integers. Define the associated non-artinian string module M(W, n_{1}, n_{2}) to be the Λ-module with generators z_{1}, z_{2}, v_{1}, …, v_{n}, such that z_{1} or z_{2} may be zero, but not both. The relations are given by:

b^{n_{2}}z_{1} = a^{i_{1}}v_{1}, b^{j_{k}}v_{k} = a^{i_{k+1}}v_{k+1}, b^{j_{n}}v_{n} = a^{n_{1}}z_{2}.

Theorem 6.3. ( [BS, p. 112]) Artinian string and band modules account for all finitely generated indecomposable Λ_{pq}-modules.

Note that a non-artinian string module M(W, n_{1}, n_{2}) is indecomposable, since A(M(W, n_{1}, n_{2})) = M(W) is an (artinian) string module, which is indecomposable.

Lemma 6.4. Let Λ = k[x_{1}, x_{2}]_{⟨x_{1},x_{2}⟩}/⟨x_{1}x_{2}⟩. There exists a finitely generated indecomposable Λ-module M such that A(M) ≅ A and M/A(M) ≅ L if and only if

(1) A ≅ M(W) is a string module with i_{1} = 1 or j_{n} = 1 or both;

(2) L = L_{1} × L_{2} is a finitely generated non-zero Λ_{1} × Λ_{2}-module, with each L_{i} indecomposable projective or zero (where Λ_{1} = Λ/⟨b⟩, and Λ_{2} = Λ/⟨a⟩);

(3) L_{1} ≠ 0 if and only if i_{1} = 1, and L_{2} = 0 if and only if j_{n} ≠ 1. In this case, M is a string module with z_{i} = 0 if and only if L_{i} = 0.

Proof. First suppose that M is a finitely generated non-artinian indecomposable Λ-module, with a separated representation

0 ⟶ K ⟶ S ⟶ M ⟶ 0.

Decompose S as a direct sum of indecomposable separated modules:

S = S_{1} ⊕ ⋯ ⊕ S_{n}.

Then some S_{m} is non-artinian. By Lemma 3.3, A(S_{m}) is an indecomposable (hence cyclic) summand of A(S) with generator z, such that either (x_{1}, 0)z = 0 and A(S_{m}) ⊂ I_{2}S_{m}, or (0, x_{2})z = 0 and A(S_{m}) ⊂ I_{1}S_{m}.

By Lemma 2.1, there is a separated representation

0 ⟶ K′ ⟶ T ⟶ A(M) ⟶ 0,

with K = K′ ⊕ L, and A(S) = T ⊕ L for some k-vector space L. Since K ∩ I_{i}S = 0 for i = 1, 2, it follows that A(S_{m}) is a summand of T. Moreover, A(M) is indecomposable by Theorem 4.1.2. Then A(M) is a module over Λ/ann (A(M)), which, in turn, is a quotient of Λ_{pq} for suitable p and q. Thus, A(M) is an indecomposable Λ_{pq}-module, hence is either a string or a band module, by Theorem 6.3.

We will now derive a contradiction from the assumption that A(M) is a band module. As a band module over Λ_{pq}, A(M) has a minimal presentation

0 ⟶ K′′ ⟶ A ⟶ A(M) ⟶ 0,

as given in Definition 6.1.2, where a and b represent the images of (x_{1}, 0), respectively (0, x_{2}), in Λ_{pq}. That is, A(M) is generated by vector subspaces u_{i} ⊗ V, for i = 1, …, n, V a finite-dimensional k-vector space, satisfying the above relations.

Now let A′ be the Λ-module generated by vector subspaces u_{i} ⊗ V. Then A′ is a separated Λ-module, since each summand Λu_{i} ⊗ v_{j} is separated (and indecomposable), for v_{1}, …, v_{m} a k-basis of V. Then clearly A′/ann (A(M)) A′ ≅ A as Λ_{pq}-modules. The composite A′ ⟶ A ⟶ A(M) is a surjection of the separated module A′ onto A(M). The universal property of a separated representation now gives a commutative diagram

0 ⟶ K′ ⟶ T ⟶ A(M) ⟶ 0
↓  ^{ϕ}↓  ^{=}↓
0 ⟶ K′′ ⟶ A′ ⟶^{π} A(M) ⟶ 0,

where the middle and left vertical maps are one-to-one. If x ∈ T and ϕ(x) ∈ ann (A(M)) A′, then πϕ(x) = 0, hence x ∈ K′. But K′ ∩ A(S_{m}) ⊂ K′ ∩ I_{i}T = 0, hence A(S_{m}) ∩ ann (A(M)) T = 0. Upon passing to quotients, we therefore obtain a homomorphism

T/ann (A(M)) T ⟶ A′/ann (A(M)) A′ ≅ A

which is one-to-one on the (non-zero) summand A(S_{m})/ann (A(M)) A(S_{m}). Thus, A contains a summand C such that either aC = 0 or bC = 0. We may assume that this summand is indecomposable. Then, by the Krull-Schmidt Theorem, C ≅ Λ_{pq}u_{i} ⊗ v_{j} for some i, j. But this is a contradiction since in a band module such a relation can only occur if a or b is raised to an exponent of at least 2.

So now assume that A(M) is a string module. The preceding argument shows that there are at most two non-artinian summands S_{m} in the separated representation of M. This follows because, for the artinian string module A(M), the only relations of the form aC = 0 or bC = 0 for an indecomposable summand C in the minimal presentation of A(M) are the two relations a^{i_{1}}C = 0 and b^{j_{n}}C = 0. This proves Part 1.

To prove Part 2, observe that, given M, the proof of Part 1 shows that

S/A(S) ≅ M/A(M) ≅ L_{1} ⊕ L_{2},

with each L_{i} an indecomposable projective R_{i}-module, such that L_{1} ≠ 0 if and only if i_{1} ≠ 0, and L_{2} ≠ 0 if and only if j_{n} ≠ 0, as claimed.

In the other direction, given A and L as specified, it is straightforward to construct a non-artinian string module M, via generators and relations as in Definition 6.2, such that A(M) ≅ A and M/A(M) ≅ L.

Theorem 6.5. Let R be a pullback ring of two hereditary noetherian serial rings R_{1}, R_{2} over a matrix ring over a field k. Assume that each R_{i} is finitely generated over its center and that A(R_{i}) = 0. Then the procedure 6.6 below gives a complete classification of all indecomposable finitely generated R-modules.

Procedure 6.6.

1. Two indecomposable R-modules M and M′ are isomorphic if and only if the indecomposable G(R)-modules G(M) and G(M′) are isomorphic (Theorem 5.1), where G(R) is formed with respect to the Jacobson radical of R.

2. The ring G(R) is Morita equivalent to Λ = k[x, y]_{⟨x,y⟩}/⟨xy⟩ (Corollary 5.3).

3. The isomorphism class of an indecomposable Λ-module is determined by the isomorphism class of the artinian indecomposable A(M) and the isomorphism class of M/A(M) (Theorem 4.1).

4. An indecomposable artinian Λ-module is either a string or a band module over a quotient Λ_{pq} = Λ/⟨x^{p+1} − y^{q+1}⟩, which is a finite-dimensional k-algebra.

5. An indecomposable non-artinian Λ-module is isomorphic to Λ, Λ/⟨x⟩, Λ/⟨y⟩, or a non-artinian string module (Lemma 6.4).

Example. If the coordinate rings of the pullback ring R are finite products of maximal orders over semi-local principal ideal domains, such that I = J(R), then the indecomposable R-modules can be classified by applying Procedure 6.6 to each indecomposable component of R. For instance, if R_{1} and R_{2} are discrete valuation rings with a common residue field, then we obtain the classification in [Le2].

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David M. Arnold, Department of Mathematics, Baylor University, Waco, TX 76798-7328

E-mail address: arnoldd@baylor.edu

Reinhard C. Laubenbacher, Department of Mathematics, New Mexico State University, Las Cruces, NM 88003

E-mail address: reinhard@nmsu.edu