DIRECT SUMS OF REPRESENTATIONS AS

MODULES OVER THEIR ENDOMORPHISM RINGS

Birge Huisgen-Zimmermann and Manuel Saorín

1. Introduction, terminology, and background

Our investigation into the endo-structure of infinite direct sums ⨁_{i∈I} M_{i} of indecomposable modules M_{i} – over a ring R with identity – is centered on the following question: If S = End_{R}(⨁_{i∈I} M_{i}), how much pressure, in terms of the S-structure of ⨁_{i∈I} M_{i}, is required to force the M_{i} into finitely many isomorphism classes? One of the consequences of our principal result in this direction (Theorem H of Section 4) is as follows. If all of the M_{i} are endofinite (think, for instance, of finitely generated or generic modules over an Artin algebra) and if (M_{t})_{t∈T} is a transversal of the isomorphism types of the M_{i}, then the following conditions (1)-(4) are equivalent: (1) T is finite; (2) ⨁_{i∈I} M_{i} is endo-artinian and (M_{t})_{t∈T} is left semi-T-nilpotent; (3) ⨁_{i∈I} M_{i} is endo-noetherian and (M_{t})_{t∈T} is right semi-T-nilpotent; (4) (M_{t})_{t∈T} is left and right semi-T-nilpotent and, for any cofinite subset T′ ⊆ T, the endosocle of ⨁_{t∈T′} M_{t} has finite support in T′. Here we call a family (M_{i})_{i∈I} left semi-T-nilpotent in case, for every sequence (i_{n}) of distinct indices in I, every sequence of non-isomorphisms f_{n} ∈ Hom_{R}(M_{i_{n+1}}, M_{i_{n}}) and every finitely cogenerated factor module M_{i_{1}}/X of M_{i_{1}}, the image of f_{1} ⋯ f_{n} is contained in X for n ≫ 0. The familiar dual condition of right semi-T-nilpotence, going back to Harada (see [11]), links our results to well-known theorems addressing the exchange property of direct sums.

The mentioned equivalence applies with particular strength to Artin algebras, one of the reasons for this being the following asset (Proposition L): Given any family (M_{i}) of finitely generated representations of an Artin algebra, the direct sum ⨁_{i∈I} M_{i} is endo-artinian if and only if it is Σ-algebraically compact (= Σ-pure injective).

Our study of Σ-algebraically compact and, more restrictively, endo-artinian direct sums hinges crucially on an analysis of endosocles. The importance of this invariant in measuring the supply of maps among the M_{i} emerges clearly in the following consequence of Theorem H: A finite dimensional algebra over an algebraically closed field has finite representation type if and only if the endosocles of all direct sums of indecomposable left representations of constant finite dimension have finite supports (Corollary N).

Another point of independent interest lies in the general connections between T -nilpo-tence conditions and the endo-structure of direct sums exhibited in Proposition E of Section 3: A family (M_{i})_{i∈I} of indecomposable modules is right T -nilpotent if and only if ⨁_{i∈I} M_{i}

Typeset by AMS-TEX

While carrying out this project, the first-named author was partially supported by an NSF grant, and the second-named author by grants from the DGES of Spain and the Fundación ‘Séneca’ of Murcia.

has the descending chain condition for finitely generated endo-submodules; moreover, these conditions are equivalent to the requirement that the direct sum ⨁_{i∈I} M_{i} be semi-artinian over its endomorphism ring, i.e., that all endo-factor modules have essential socles. As a consequence, each Σ-algebraically compact module M is a direct sum of indecomposables M_{i}, i ∈ I, with the property that the family (M_{i})_{i∈I} is right T -nilpotent.

We add a few comments on the background of our project for motivation. It is well known that the structure of non-finitely generated representations of R, viewed as modules over their endomorphism rings, is decisive in understanding the behavior of the finitely generated representations. We point to a few specific instances of such connections. As was first observed in [19] and [17], finite representation type of R occurs precisely when all (left) R-modules have finite lengths over their endomorphism rings. Moreover, the rings with vanishing left pure global dimension are characterized by certain endo-chain conditions satisfied by their modules. Along a different line, Crawley-Boevey proved that, for tame finite dimensional algebras over an algebraically closed field, the generic modules (that is, the infinite dimensional indecomposable endofinite modules) govern the one-parameter families of finitely generated indecomposable representations in an extremely strong sense [6, 7]. (It appears plausible that the endosocles of direct sums of generic modules should reflect domestic versus non-domestic representation type; see, e.g., Example C(3), due to Ringel [20].) There is a common skein tying the listed scenarios at least loosely together, namely the fact that Σ-algebraic compactness of an R-module is tantamount to the descending chain condition for a selection of its endo-submodules. Other results in a related vein link endo-chain conditions of infinite direct sums of finitely generated representations of Artin algebras to the existence of preprojective or preinjective partitions, the existence of almost split maps, and the structure of direct products (see, e.g., [3, 13, 8, 18, 1] and our concluding remarks). These multiple bridges between endo-chain conditions on one hand, and purely representation-theoretic assets of classes of modules on the other, motivate our present investigation.

Section 2 is devoted to exploring endosocle series of direct sums. In Section 3, we compare T-nilpotence conditions for families of modules with endo-chain conditions for their direct sums, and in Section 4, we follow with our main results, applications, and examples.

Prerequisites

Recall from [22] and [23] that a p-functor on R -Mod is a subfunctor of the forgetful functor R -Mod → ℤ -Mod which commutes with direct products; here R -Mod stands for the category of all left R-modules. Special instances of p-functors can be described as follows: A pointed matrix over R consists of a row-finite matrix A = (a_{ij})_{i∈I,j∈J} of elements in R, paired with a column index α ∈ J. Given a pointed matrix (A, α), we call the following p-functor [A, α] on R -Mod a matrix-functor: For any R-module M, the subgroup [A, α]M is defined to be the α-th projection of the solution set in M^{J} of the homogeneous system

∑_{j∈J} a_{ij}X_{j} = 0 for all i ∈ I.

In other words,

[A, α]M = {m ∈ M | ∃ a solution (m_{j}) ∈ M^{J} of the above system with m_{α} = m}.

Following [23], we call [A, α]M a matrix subgroup of M, a finite matrix subgroup in case I and J are finite. We refer the reader to [23] or [14] for more information on p-functors.

One of the most salient reasons for our present interest in p-functors lies in the following equivalent description of Σ-algebraically compact modules, i.e., of the modules M with the property that all direct sums of copies of M are algebraically compact (= pure injective): Namely, M is Σ-algebraically compact if and only if every descending chain of p-functorial subgroups of M becomes stationary, a condition which is in turn equivalent to the DCC for finite matrix subgroups of M; see [22, 9, 23, 10]. Since, clearly, every matrix subgroup of M is an endo-submodule, this is a well-known instance of a link between the R- and endo-structures of M. While it guarantees that endo-artinian modules are Σ-algebraical-ly compact, the converse fails in general; see, e.g., [5] or [15, Theorem 5]. By contrast, M is endo-noetherian if and only if M has the ascending chain condition for p-functorial subgroups, since every finitely generated endo-submodule of M is a matrix subgroup. In case M is a direct sum of finitely presented modules, the ACC for finite matrix subgroups – not equivalent to the ACC for arbitrary matrix subgroups in general – already suffices to guarantee endo-noetherianness (Observation 8 of [17]). In light of the fact that p-functors automatically commute with direct sums (as arbitrary subfunctors of the forgetful functor do), one deduces the following observation which will prove useful in connection with dualities.

Observation. For any family (M_{i})_{i∈I} of finitely presented modules, the following conditions are equivalent:

(1) ⨁_{i∈I} M_{i} is endo-noetherian.

(2) ⨁_{i∈I} M_{i} satisfies the ACC for (finite) matrix subgorups.

(3) ∏_{i∈I} M_{i} satisfies the ACC for (finite) matrix subgroups.

(4) ∏_{i∈I} M_{i} is endo-noetherian.

2. Endosocle series of direct sums of indecomposable modules

The following elementary observations will be very useful to us in the sequel.

Lemma A. Every Σ-algebraically compact R-module M is semi-artinian over its endomorphism ring S, meaning that every S-factor module of M has essential socle.

Proof. Suppose that M is a Σ-algebraically compact R-module. As we pointed out above, M then satisfies the descending chain condition for matrix subgroups which, in particular, yields the DCC for finitely generated S-submodules of M. But this latter condition clearly entails our claim: Indeed, given S-submodules U ⫋ V of M, there exists a minimal Ssubmodule W of V not contained in U, and any such choice of W gives rise to a simple submodule (W + U)/U ⊆ V/U. □

The converse of Observation A fails in general: Let R be any left perfect ring which fails to be left Σ-algebraically compact (for examples see [24]); then all left R^{op}-modules have essential socles, and therefore the regular left R-module is semi-artinian over its endomorphism ring.

Recall that the ascending socle series of an S-module M is defined as follows: soc_{0} M = 0, soc_{1} M = soc_{S} M, and, if α = β + 1 is a successor ordinal, soc_{α} M is the preimage of soc_{S}(M/ soc_{β} M) under the canonical epimorphism M → M/ soc_{β} M; for a limit ordinal α, finally, soc_{α} M is defined to be the union ⋃_{β<α} soc_{β} M. Accordingly, the socle length of M is the least ordinal μ with the property that soc_{μ+1} M = soc_{μ} M.

Given any left R-module M with endomorphism ring S, we will refer to the S-socle of M as the endosocle, denoted by endosoc M, and to the S-socle series as the endosocle series; a generic term of this series will be labeled endosoc_{α} M. The endosocle length of M, finally, is the length of its S-socle series. Whenever we speak of endosoc M as having a property (X), we are referring to the S-structure, not the R-structure.

Since indecomposable algebraically compact modules have local endomorphism rings ([15]), our first observation yields

Corollary A′. Suppose that M is a Σ-algebraically compact R-module with endosocle length μ. Then endosoc_{μ} M = M.

Hence: If the module M is also indecomposable with endomorphism ring S, it has an exhaustive, well-ordered chain of S-submodules with consecutive factors of the form (S/J(S))^{(A)}. □

In contrast to the preceding observation, direct sums of Σ-algebraically compact modules may have trivial endosocles – think of M = ⨁_{n∈ℕ} ℤ/(p^{n}), for instance, where p is prime. This can be seen as a pronounced failure of the direct sum of Σ-algebraically compact modules to inherit crucial properties implied by Σ-algebraic compactness.

Recall that any Σ-algebraically compact module M is a direct sum of submodules with local endomorphism rings. Our main interest being in families (M_{i}) of Σ-algebraically compact or, more restrictively, endo-artinian modules, most of our discussion will therefore focus on the situation where all of the M_{i} have local endomorphism rings. In this setting, we record how the endosocle of ⨁_{i∈I} M_{i} relates to the endosocles of the individual M_{i}. In case all of the M_{i} are pairwise isomorphic, the connection is straightforward, otherwise it reflects the behavior of maps among the summands M_{i} as follows.

Lemma B. Let (M_{i})_{i∈I} be a family of modules with local endomorphism rings, and S = End_{R}(⨁_{i∈I} M_{i}).

(1) First suppose that M_{i} = N for all i ∈ I. Then:

• endosoc(N^{(I)}) = (endosoc(N))^{(I)}, and the endosocle of N^{(I)} consists of a single homogeneous component.

• The endosocle of N^{(I)} is finitely generated as an endo-submodule if and only if the

same is true for the endosocle of N.

(2) In general, endosoc(⨁_{i∈I} M_{i}) = ⨁_{i∈I} B_{i}, where each B_{i} is the End_{R}(M_{i})-submodule of M_{i} consisting precisely of those elements of M_{i} which are annihilated by all nonisomorphisms in ⋃_{j∈I} Hom_{R}(M_{i}, M_{j}).

In particular: If M_{i} ≅ M̸_{j} for i ≠ j, then each B_{i} is a semisimple End_{R}(M_{i})-submodule of M_{i}, as well as a semisimple S-submodule of the direct sum ⨁_{i∈I} M_{i}.

(3) Whenever (M_{n})_{n∈ℕ} is a family of pairwise nonisomorphic modules with local endomorphism rings which permits a sequence of monomorphisms M_{1} ↪ M_{2} ↪ M_{3} ↪ ⋯, the direct sum ⨁_{n∈ℕ} M_{n} has trivial endosocle.

Proof. Set M = ⨁_{i∈I} M_{i}, and observe that each S-submodule U of M is of the form U = ⨁_{i∈I}(U ∩ M_{i}), where the intersection U ∩ M_{i} is an End_{R}(M_{i})-submodule of M_{i}. We deduce that each simple S-submodule of M can be written in the form Sx_{i} for a nonzero element x_{i} of some M_{i}.

Part (1) is immediate from these remarks.

(2) Set B_{i} = (endosoc M) ∩ M_{i}. To see that B_{i} is contained in the annihilator of the set of non-isomorphsms in ⋃_{j∈I} Hom_{R}(M_{i}, M_{j}), observe that, due to the locality of the rings End_{R}(M_{i}), each of the indicated non-isomorphisms belongs to the Jacobson radical of S.

For the other inclusion, fix i ∈ I, set I(i) = {j ∈ I | M_{j} ≅ M_{i}}, and S(i) = End_{R}(⨁_{j∈I(i)} M_{j}). If x ∈ M_{i} is annihilated by all non-isomorphisms in the above union, then x clearly belongs to the End_{R}(M_{i})-socle of M_{i}, and consequently S(i)x is contained in the S(i)-socle of ⨁_{j∈I(i)} M_{j} by part (1). Since Sx = S(i)x, this shows that Sx ⊆ endosoc M, which yields x ∈ B_{i} as required.

The final assertion under (2) is now obvious, as is (3). □

Examples C.

(1) Let R = ℤ and p a prime. The terms of the endosocle series of M = ⨁_{n∈ℕ} ℤ/(p^{n}) ⊕ ℤ(p^{∞}) are as follows: endosoc_{α} M equals the copy of ℤ/(p^{n}) inside ℤ(p^{∞}) for α < ω, and endosoc_{α} M = ℤ(p^{∞}) for α ≥ ω.

(2) Let Λ be the Kronecker algebra. If (M_{n})_{n∈ℕ} is the family of all preprojective modules in R -ind, then endosoc(⨁_{n∈ℕ} M_{n}) = 0; in fact, endosoc(⨁_{n≥m} M_{n}) = 0 for all m. On the other hand, if M′ = ⨁_{n∈ℕ} M′_{n} is the direct sum of all preinjective modules, with length(M′_{n}) = 2n − 1, the endosocle of M′ equals M′_{1} ⊕ eM′_{2}, and, for m ≥ 2, we have endosoc(⨁_{n≥m} M′_{n}) = M′_{m}.

(3) The following is due to Ringel [20]: If Λ is a string algebra (i.e., a finite dimensional monomial relation algebra which is special biserial), but fails to be of domestic representation type, then there exists a family (M_{n})_{n∈ℕ} of pairwise non-isomorphic generic Λ-modules allowing for consecutive embeddings M_{1} ↪ M_{2} ↪ …; by Lemma B(3), we infer that the endosocle of the direct sum of the M_{n}’s is zero.

We continue to assume that M = ⨁_{i∈I} M_{i}, where all M_{i} have local endomorphism rings. In exploring the endo-structure of M, a different sequence of iterated endosocles – not forming an ascending chain – is frequently more helpful than the series we just discussed. This is due to the fact that the traditional endosocle series may be infinite and still get stalled within a finite subsum of M_{i}’s (see Example C(1)). By contrast, any infinite sequence of nonzero terms of the following relative endosocle series involves infinitely many summands M_{i}. This often makes it a more effective tool in studying homomorphisms among non-isomorphic M_{i}’s.

As usual, we start with endosoc′_{0} M = 0; moreover, we set I_{0} = ∅. Next, we set endosoc′_{1} M = endosoc M, and denote by I_{1} the support of endosoc′_{1} M in I. Assume that the terms endosoc′_{β} M have already been defined for all ordinal numbers β < α. In case α is a limit ordinal, we set endosoc′_{α} M = 0. If, on the other hand, α is a successor ordinal, say α = β +1, we let I_{β} be the support of ∑_{γ≤β} endosoc′_{γ} M, and define endosoc′_{α} M to be the endosocle of the trimmed direct sum ⨁_{I\I_{β}} M_{i}. In particular, we see that I_{α} is defined for each ordinal α and equals the support of ∑_{β≤α} endosoc′_{β} M = ⨁_{β≤α} endosoc′_{β} M. Finally, we refer to the least ordinal number μ such that endosoc′_{μ+1} M = 0 (or, equivalently, the least ordinal number μ with I_{μ+1} = I_{μ}) as the relative endosocle length of M.

Clearly, the isomorphism types of the R-submodules endosoc′_{α} M are isomorphism invariants of M, and the sum ∑_{α≤μ} endosoc′_{α} M is direct by construction. Moreover, we remark that the relative endosocle length of M is always smaller than or equal to the traditional endosocle length. Here is an example in which strict inequality occurs: If M is the ℤ-module (ℤ/(p^{n}))^{(ℕ)} ⊕ ℤ(p^{∞}) for a prime p, then the relative endosocle length of M is 2, whereas the traditional endosocle length is ω + n.

The second of the following observations will be used repeatedly in the next section. Both are straightforward from the definitions.

Lemma D. Let M = ⨁_{i∈I} M_{i}, where each M_{i} has local endomorphism ring. Moreover, suppose that each subsum of M_{i}’s has essential endosocle (keep in mind that these hypotheses are satisfied if M is Σ-algebraically compact). Then:

(1) I_{μ} = I, where μ is the relative endosocle length of M.

(2) Suppose U is an endo-submodule of M which is not contained in ⨁_{i∈I_{α}} M_{i}. Then (J(S)U)∩endosoc′_{β+1} M is nonzero for all β < α; in particular, (J(S)U)∩endosoc′_{α} M ≠ 0 if α is a successor ordinal. □

3. T-nilpotence versus endo-chain conditions

Harada [11] called a family (M_{i})_{i∈I} of indecomposable R-modules semi-T-nilpotent in case, for any sequence (i_{n})_{n∈ℕ} of distinct indices in I, any family of non-isomorphisms f_{n} ∈ Hom_{R}(M_{i_{n}}, M_{i_{n+1}}), and any finitely generated R-submodule X of M_{i_{1}}, there exists n_{0} ∈ ℕ with

f_{n_{0}}f_{n_{0}−1} ⋯ f_{1}(X) = 0.

Obviously, this is a condition which can be tested ‘pointwise’, i.e., the family (M_{i}) satisfies it precisely when, for each family of indices and non-isomorphisms f_{n} as above and any element x ∈ M_{i_{1}}, there exists a number n_{0} with f_{n_{0}}f_{n_{0}−1} ⋯ f_{1}(x) = 0. To make room for a dual concept, we will refer to the described condition as right semi-T-nilpotence. The following twin concept of left semi-T-nilpotence of a family (M_{i})_{i∈I} of indecomposables requires that, for any sequence (i_{n})_{n∈ℕ} of distinct indices in I, any family of nonisomorphisms f_{n} ∈ Hom_{R}(M_{i_{n+1}}, M_{i_{n}}), and any finitely cogenerated factor module M_{i_{1}}/X of the R-module M_{i_{1}}, there exists a natural number n_{0} such that

Im(f_{1}f_{2} ⋯ f_{n_{0}}) ⊆ X.

Left semi-T-nilpotence clearly implies the following elementwise condition: For each family of indices and homomorphisms f_{n} as specified above, and for each nonzero y ∈ M_{i_{1}}, there exists n_{0} ∈ ℕ such that y ∉ Im(f_{1}f_{2} ⋯ f_{n_{0}}). While the converse fails in general, it does hold if all of the M_{i}’s are finitely cogenerated – so, in particular, if they have finite length – in which case either condition is equivalent to the requirement that the family of homomorphisms among the M_{i}’s be artinian in the sense of Auslander (see below).

The stronger conditions of right/left T-nilpotence, finally, call for the same conclusions as the corresponding semi-T-nilpotence conditions, but on waiving the premise that the families of indices considered be free of repetitions.

We point out that, for the special case of finitely generated indecomposable modules M_{i} over an Artin algebra, Auslander introduced alternate terms for these nilpotence conditions (see [2]). He labeled a family of morphisms among such modules noetherian (resp., artinian) in case, for every countable sequence of non-isomorphisms (f_{n}) selected within the family – consecutively composable in the appropriate sense – there exists an index n_{0} with f_{n_{0}} ⋯ f_{1} = 0 (resp., f_{1} ⋯ f_{n_{0}} = 0). So, in this situation, the family (M_{i}) is right semi-T-nilpotent if and only if it is right T-nilpotent, if and only if the family of homomorphisms among the M_{i} is noetherian. (We alert the reader to the fact that this condition automatically comes with endo-artinianness of ⨁_{i∈I} M_{i} – see Proposition E below – but may fail for endo-noetherian direct sums ⨁_{i∈I} M_{i}.)

The second of the following known characterizations of right semi-T-nilpotent families of modules will be used freely in the sequel, while the third will enter into an application of our main theorem. The implications ‘(1) ⟺ (2) ⟹ (3)’ are due to Harada ([11], and the remaining one was filled in by W. Zimmermann and the firstnamed author [16]. Recall that a module M is said to have the exchange property in case, for every equality of the form M′ ⊕ A = ⨁_{l∈L} A_{l} with M′ ≅ M, there exist submodules B_{l} ⊆ A_{l} such that M′ ⊕ A = M′ ⊕ ⨁_{l∈L} B_{l}.

Known facts. Suppose that all M_{i} have local endomorphism rings. Then the following conditions are equivalent:

(1) The family (M_{i})_{i∈I} is right semi-T-nilpotent.

(2) The Jacobson radical of End_{R}(⨁_{i∈I} M_{i}) coincides with the set of those endomorphisms f which have the property that all compositions pr_{j} ·f · in_{i} are non-isomorphisms (here pr_{j} and in_{i} denote the canonical projections and injections, respectively).

(3) The direct sum ⨁_{i∈I} M_{i} has the exchange property.

We start by relating right semi-T-nilpotence to the endo-semi-artinian condition we encountered in Lemma A. These connections will not only feed into the proof of our main result, but are of interest as supplements. Part (3) of the following proposition is a variant of [17, Proposition 4].

Proposition E. Suppose that (M_{i})_{i∈I} is a family of indecomposable left R-modules, and set M = ⨁_{i∈I} M_{i}.

(1) If (M_{i}) is a right T-nilpotent family, then all of the M_{i} have local endomorphism rings.

(2) The following conditions are equivalent:

(a) The family (M_{i}) is right T-nilpotent.

(b) The direct sum M is semi-artinian over its endomorphism ring.

(c) M satisfies the DCC for cyclic endo-submodules (or, equivalently, the DCC for

finitely generated endo-submodules). As a consequence, every Σ-algebraically compact R-module M is a direct sum ranging

over a right T-nilpotent family of indecomposable modules.

(3) The following condition is sufficient for right semi-T-nilpotence of the family (M_{i}): Each M_{i} has local endomorphism ring, and every descending chain of matrix functors on R -Mod becomes uniformly stationary on almost all of the M_{i}’s.

Proof. Set S = End_{R}(M).

(1) Suppose that M_{i} belongs to a right T-nilpotent family, and let f ∈ End_{R}(M_{i}) be a non-isomorphism. Then f fails to be a monomorphism, for, given any element x ∈ M, there exists m ∈ ℕ with f^{m}(x) = 0. To see that, for any g ∈ End_{R}(M_{i}), the map id_{M_{i}} −gf is an isomorphism, it thus suffices to check injectivity. To that end, we observe that x = gf(x) implies x = (gf)^{n}(x) for all n ∈ ℕ, and hence x = 0 by our T-nilpotence hypothesis.

(2) Start by recalling that the descending chain condition for cyclic submodules is equivalent to that for finitely generated submodules, due to Bjork [4]. The canonical embeddings and projections coming with the direct sum ⨁_{i∈I} M_{i} will be denoted by in_{i} and pr_{i}, respectively.

To back up ‘(a) ⟹ (c)’, suppose (a) holds.

First we observe that, for any element m = (m_{i})_{i∈I} ∈ M, the cyclic S-module Sm/J(S)m is semisimple. Indeed, right semi-T-nilpotence of (M_{i}) guarantees that the Jacobson radical J(S) of S contains all endomorphisms f with the property that the compositions pr_{i} f in_{j} are non-isomorphisms for arbitrary i, j ∈ I. We will make the assumption that M_{i} ≅ M̸_{j} whenever i ≠ j; this does not affect the generality of our argument, for every endo-submodule U of a direct sum of powers ⨁_{i∈I} M_{i}^{(L_{i})} is of the form ⨁_{i∈I} U_{i}^{(L_{i})} for a suitable endo-submodule U_{i} of M_{i}. Consequently,

Sm/J(S)m = ⨁_{finite} End_{R}(M_{i})m_{i}/J(End_{R}(M_{i}))m_{i},

where each of the summands on the right is an S-module, the S-structure of which coincides with the pertinent End_{R}(M_{i})/J(End_{R}(M_{i}))-structure.

To the contrary of our claim assume that M contains a strictly descending chain Sx_{1} ⫌ Sx_{2} ⫌ Sx_{3} ⫌ ⋯ of cyclic S-submodules. Now the induced sequence in the semisimple factor module Sx_{1}/J(S)x_{1} does become stationary, yielding an integer m such that Sx_{m} ⊆ Sx_{m+1} + J(S)x_{1}. By possibly dropping some terms from our descending chain, we may assume that Sx_{2} ⊆ Sx_{3}+J(S)x_{1}. We apply the same argument to Sx_{2}/J(S)x_{2} and iterate, whence an obvious induction permits us to thin out our original sequence of cyclics so as to assure Sx_{n} ⊆ Sx_{n+1} + J(S)x_{n−1} for all n ≥ 2. For each n ≥ 2 we thus obtain an equality x_{n} = f_{n+1}(x_{n+1}) + g_{n−1}(x_{n−1}) with f_{n+1} ∈ S and g_{n−1} ∈ J(S). By substituting x_{2} = f_{3}(x_{3})+g_{1}(x_{1}) into the equality x_{3} = f_{4}(x_{4})+g_{2}(x_{2}), we deduce that (1−g_{2}f_{3})(x_{3}) = f_{4}(x_{4}) + g_{2}g_{1}(x_{1}), and multiplication by the inverse of 1 − g_{2}f_{3} yields an equality of the form x_{3} = f′_{4}(x_{4}) + h_{1}(x_{1}) with h_{1} ∈ J(S). In a next step, we insert this expression for x_{3} into x_{4} = f_{5}(x_{5}) + g_{3}(x_{3}), thus obtaining an element h_{2} ∈ J(S) with the property that x_{4} = f′_{5}(x_{5}) + h_{2}h_{1}(x_{1}) for a suitable f′_{5} ∈ S. Again we iterate, and arrive at a sequence of elements h_{1}, h_{2}, h_{3}, ⋯ ∈ J(S) satisfying x_{n} = f′_{n+1}(x_{n+1}) + h_{n−2} ⋯ h_{2} h_{1}(x_{1}) for suitable choices of f′_{n} ∈ S.

T-nilpotence of the family (M_{i})_{i∈I}, combined with Koenig’s graph theorem, will now provide us with a natural number N such that h_{N} ⋯ h_{2} h_{1}(x_{1}) = 0. For, if we had h_{n} ⋯ h_{2} h_{1}(x_{1}) ≠ 0 for all n, the following graph G would have paths of arbitrary length: the roots of G are the nonzero components x_{1i} ∈ M_{i} of the element x_{1} – finite in number, say x_{1,i(1)}, …, x_{1,i(m)}, and the edge paths of length l in G correspond to the nonzero evaluations

(pr_{j(l)} h_{n(l)} in_{j(l−1)})(pr_{j(l−1)} h_{n(l−1)} in_{j(l−2)}) ⋯ (pr_{j(1)} h_{n(1)} in_{j(0)})(x_{1,j(0)}),

for suitable distinct j(s) ∈ I such that j(0) ∈ {i(1), …, i(m)}. Koenig’s graph theorem would then imply the existence of an infinite path. But this is incompatible with our T-nilpotence condition, since each of the maps pr_{j(s)} h_{n(s)} in_{j(s−1)} : M_{j(s−1)} → M_{j(s)} is a non-isomorphism by our choice of the h_{n}.

Thus, we obtain a number N as described and infer x_{N+2} ∈ Sx_{N+3}, contrary to our assumption that the chain of Sx_{n} be strictly descending.

To justify the implication ‘(c) ⟹ (b)’, carry over the argument proving Lemma A. For ‘(b) ⟹ (a)’, see [21, Chapter VIII].

(3) Suppose that (i_{n})_{n∈ℕ} is a sequence of distinct elements of I, and f_{i_{n}} : M_{i_{n}} → M_{i_{n+1}} a non-isomorphism for n ∈ ℕ. Given x ∈ M_{i_{1}}, consider the following chain of principal S-submodules of M:

Sf_{i_{1}}x ⊇ Sf_{i_{2}}f_{i_{1}}x ⊇ Sf_{i_{3}}f_{i_{2}}f_{i_{1}}x ⊇ · · · .

All terms of this chain are matrix subgroups of the R-module M (see Section 1), say Sf_{i_{n}} ⋯ f_{i_{1}}x = P_{n}(M), where P_{n} is a matrix functor. Since the class of matrix functors on R -Mod is closed under intersections, we can normalize to the situation where P_{1} ⊇ P_{2} ⊇ P_{3} ⊇ · · ·. Assuming that this chain becomes uniformly stationary on amost all of the M_{i}’s, we obtain a natural number N, together with a finite subset I_{0} ⊆ I, such that P_{n}(⨁_{i∈I\I_{0}} M_{i}) = P_{N}(⨁_{i∈I\I_{0}} M_{i}) for all n ≥ N. Since the i_{n} are pairwise different, we can adjust N upwards, if necessary, so as to guarantee that, moreover, f_{i_{n}} ⋯ f_{i_{1}}x ∈ ⨁_{i∈I\I_{0}} M_{i} for all n ≥ N. Denoting the R-endomorphism ring of ⨁_{i∈I\I_{0}} M_{i} by S′, we infer that the chain

S′f_{i_{n}} ⋯ f_{i_{N+1}}(f_{i_{N}} f_{i_{N−1}} ⋯ f_{i_{1}}x), n ≥ N

is stationary. We will deduce that f_{i_{N}} ⋯ f_{i_{1}}x = 0. Indeed, our setup yields a map g ∈ S′ with gf_{i_{N+1}}f_{i_{N}} ⋯ f_{i_{1}}x = f_{i_{N}} ⋯ f_{i_{1}}x. Clearly, we may assume that g is a map from M_{i_{N+2}} to M_{i_{N+1}}. If f_{i_{N}} ⋯ f_{i_{1}}x were nonzero, Nakayama’s Lemma would exclude gf_{i_{N+1}} from the Jacobson radical of End_{R}(M_{i_{N}}), and locality of this endomorphism ring would guarantee gf_{i_{N+1}} to be a unit of End_{R}(M_{i_{N}}). But this is impossible, since M_{i_{N+2}} is indecomposable and f_{i_{N+1}} a non-isomorphism. Hence f_{i_{N}} ⋯ f_{i_{1}}x = 0 as claimed, and the argument is complete. □

Part (2) of the preceding proposition can actually be de-specified from the situation of direct sums regarded as modules over their endomorphism rings: If S is any ring and M a left S-module, then the DCC for finitely generated submodules forces M to be semiartinian; the latter condition in turn implies that, given any m ∈ M and any sequence (s_{1}, s_{2}, …) in J(S), we have s_{n} ⋯ s_{1}m = 0 for n ≫ 0. Moreover, these three conditions are equivalent in case Sm/J(S)m is semisimple for every element m ∈ M.

4. Main results and examples

Theorem H below will zero in on our primary concern: Namely to significantly weaken conditions on the endo-structure of ⨁_{i∈I} M_{i} which are known to guarantee that the M_{i} fall into finitely many isomorphism classes. An instance of such a condition is endofiniteness of ⨁_{i∈I} M_{i}; see [7]. Our result also provides background for the following well-known characterization of the Artin algebras Λ having finite representation type, in terms of morphisms among the indecomposable finitely generated left Λ-modules, due to Auslander ([2]): Namely, Λ has finite representation type precisely when every countable family of non-isomorphisms is both artinian and noetherian in the sense given at the beginning of Section 2. Related results for families of finitely generated indecomposable modules were recently obtained by Dung [8, Theorem 3.3 and Corollary 3.12]. We, too, are particularly interested in the situation where all of the M_{i}’s have finite length over R – next to the case where the M_{i}’s are generic – and will address it in subsequent corollaries and examples.

Lemma F. Let (M_{i})_{i∈I} be a right semi-T-nilpotent family of pairwise non-isomorphic modules with local endomorphism rings. If, for each cofinite subset I′ of I, the direct sum ⨁_{i∈I′} M_{i} is finitely generated over its endomorphism ring, then I is finite.

Proof. Suppose that the endo-condition of our claim is satisfied. For any subset I′ ⊆ I, we denote by pr_{I′} the canonical projection M → ⨁_{i∈I′} M_{i} along ⨁_{i∈I\I′} M_{i}, preserving the convention pr_{i} = pr_{{i}} for i ∈ I. Moreover, we set S = End_{R}(⨁_{i∈I} M_{i}). Keeping in mind that every S-submodule U of M is of the form ⨁_{i∈I}(M_{i} ∩ U), where each M_{i} ∩ U is an End_{R}(M_{i}) submodule of M_{i}, we find that, given any S-generating set G of such a module U, the collection {pr_{i}(x) | i ∈ I, x ∈ G} of projections again belongs to U (and thus generates U).

By hypothesis, we can therefore pick a finite family of elements (x_{l_{1}})_{l_{1}∈L_{1}} in ⋃_{i∈I} M_{i} with the property that M = ∑_{l_{1}∈L_{1}} Sx_{l_{1}}; by K_{1} ⊆ I we denote the (finite) union of the supports of the x_{l_{1}}. Setting S_{1} = End_{R}(⨁_{i∈I\K_{1}} M_{i}) and applying our hypothesis to the pared-down direct sum, we see that each of the modules S_{1} pr_{I\K_{1}}(Sx_{l_{1}}) can be written as a finite sum, S_{1} pr_{I\K_{1}}(Sx_{l_{1}}) = ∑_{l_{2}∈L_{2}} S_{1}f_{l_{2}l_{1}}(x_{l_{1}}), where L_{2} is a finite index set disjoint from L_{1} and the f_{l_{2}l_{1}} are homomorphisms from suitable summands M_{i} of ⨁_{k∈K_{1}} M_{k} to summands M_{j} of ⨁_{k∈I\K_{1}} M_{k}; here the sources and targets of the f_{l_{2}l_{1}} may appear repeatedly. By allowing zero maps among the f_{l_{1}l_{2}}, we can make the same index set L_{2} work for all l_{1} ∈ L_{1} simultaneously. We thus obtain

M = ⨁_{k∈K_{1}} M_{k} + ∑_{l_{1}∈L_{1}} S_{1} pr_{I\K_{1}}(Sx_{l_{1}})

= ⨁_{k∈K_{1}} M_{k} + ∑_{l_{1}∈L_{1},l_{2}∈L_{2}} S_{1}f_{l_{2}l_{1}}(x_{l_{1}}).

Next we let K_{2} ⊆ I be a finite set containing K_{1} and the supports of the elements f_{l_{2}l_{1}}(x_{l_{1}}) for all choices of l_{1}, l_{2}. Now we set S_{2} = End_{R}(⨁_{i∈I\K_{2}} M_{i}) and, using the same guidelines as for the choice of L_{1} and the f_{l_{2}l_{1}}’s, we choose a finite index set L_{3}, disjoint from L_{1} ∪ L_{2}, and homomorphisms f_{l_{3}l_{2}}, each from a suitable summand M_{i} of ⨁_{k∈K_{2}} M_{k} to a summand M_{j} of ⨁_{k∈I\K_{2}} M_{k}, such that

S_{2} pr_{I\K_{2}}(S_{1}f_{l_{2}l_{1}}(x_{l_{1}})) = ∑_{l_{3}∈L_{3}} S_{2}f_{l_{3}l_{2}}f_{l_{2}l_{1}}(x_{l_{1}}),

for all l_{1} ∈ L_{1} and l_{2} ∈ L_{2}. This yields the equality

M = ⨁_{k∈K_{2}} M_{k} + ∑_{l_{1}∈L_{1},l_{2}∈L_{2},l_{3}∈L_{3}} S_{2}f_{l_{3}l_{2}}f_{l_{2}l_{1}}(x_{l_{1}}).

We repeat the above procedure to inductively obtain pairwise disjoint finite index sets L_{1}, L_{2}, L_{3}, …, finite subsets K_{1} ⊆ K_{2} ⊆ K_{3} ⊆ ⋯ of I, and homomorphisms f_{l_{n+1}l_{n}} – for l_{n} ∈ L_{n} and l_{n+1} ∈ L_{n+1} – from summands M_{i} of ⨁_{k∈K_{n}} M_{k} to summands M_{j} of ⨁_{i∈K_{n+1}\K_{n}} M_{i}, respectively, such that for each n ∈ ℕ,

M = ⨁_{k∈K_{n}} M_{k} + ∑_{l_{m}∈L_{m},1≤m≤n} S_{n}f_{l_{n+1}l_{n}} ⋯ f_{l_{2}l_{1}}(x_{l_{1}}),

where S_{n} = End_{R}(⨁_{i∈I\K_{n}} M_{i}).

Let G be the graph having as vertices the indices in ⋃_{i∈ℕ} K_{i} and as edge paths those concatenations f_{l_{n+1}l_{n}} ⋯ f_{l_{2}l_{1}} which have the property that f_{l_{n+1}l_{n}} ⋯ f_{l_{2}l_{1}}(x_{l_{1}}) ≠ 0. Since all of the maps f_{l_{i+1}l_{i}} are non-isomorphisms by construction, our T-nilpotence hypothesis excludes the occurrence of an infinite path. Therefore, König’s Graph Theorem guarantees the existence of an upper bound on the lengths of paths in G – N say – which shows that M = ⨁_{k∈K_{N}} M_{k}. But this says that I = K_{N} is finite as required. □

We now come to our key lemma. In spite of the dual formats of Lemmas F and G, the latter cannot be obtained by dualizing the preceding argument.

Lemma G. Let (M_{i})_{i∈I} be a left semi-T-nilpotent family of pairwise non-isomorphic modules with local endomorphism rings. If, for each cofinite subset I′ of I, the direct sum ⨁_{i∈I′} M_{i} is finitely cogenerated over its endomorphism ring, then I is finite.

Proof. Again suppose that the endo-condition of our claim is satisfied. Set M = ⨁_{i∈I} M_{i} and S = End_{R}(M). Assume that I is infinite. In order to deduce that the family (M_{i})_{i∈I} then fails to be left semi-T-nilpotent, we consider the relative endosocle series (endosoc′_{α} M)_{α}, as introduced at the end of Section 2. As before, we denote the support of ⨁_{β≤α} endosoc′_{β} M by I_{α}. By hypothesis, endosoc M is finitely generated, whence the support I_{1} is finite by Lemma B(2). An obvious induction, based on the hypothesis that the endosocle of any cofinite subsum ⨁_{i∈I′} M_{i} has finite support in I′, thus yields finiteness of I_{k} for all k ∈ ℕ.

Our hypothesis moreover guarantees that each of the relative endosocles endosoc′_{k+1} M is essential in the corresponding direct sum ⨁_{i∈I\I_{k}} M_{i} viewed as a module over its endomorphism ring S_{k}. Since, under the canonical embedding of S_{k} into S, we have J(S_{k}) ⊆ J(S), we deduce from Lemma D(2) that, for any nonzero S_{k}-submodule U of ⨁_{i∈I\I_{k}} M_{i}, the intersection (J(S)U) ∩ endosoc′_{k} M is nonzero. For each k ∈ ℕ, we now consider a descending chain A_{k1} ⊇ A_{k2} ⊇ A_{k3} ⊇ … of nonzero S_{k}-submodules of endosoc′_{k} M, where the A_{ki} are inductively defined as follows: A_{k1} = (J(S) endosoc′_{k+1} M) ∩ endosoc′_{k} M for k ≥ 1, and

A_{k,i+1} = (J(S)A_{k+1,i}) ∩ endosoc′_{k} M.

Being contained in the finitely generated semisimple S_{k−1}-module endosoc′_{k} M, the chain (A_{ki})_{i≥1} becomes stationary. Furthermore, by Observations D(2), it consists of nonzero terms. Therefore it converges to a nonzero S_{k−1}-submodule C_{k} ⊆ endosoc′_{k} M. From the definition of the A_{ki}, we moreover derive C_{k} = (J(S)C_{k+1}) ∩ endosoc′_{k} M for all k ∈ ℕ, which yields a sequence of inclusions

C_{1} ⊆ J(S)C_{2} ⊆ J(S)^{2}C_{3} ⊆ · · · .

Since C_{1} is an S-module, we can pick i_{1} ∈ I_{1} so that M_{i_{1}} ∩C_{1} contains a nonzero element x. The above string of equations then yields disjoint finite sets L_{n} for n ∈ ℕ, together with families of homomorphisms (f_{l_{n}})_{l_{n}∈L_{n}}, each of the form f_{l_{n}} = pr_{v(l_{n})} ·f_{l_{n}} · in_{u(l_{n})} ∈ Hom_{R}(M_{u(l_{n})}, M_{v(l_{n})}) for suitable indices u(l_{n}) ∈ I_{n+1} \ I_{n}, v(l_{n}) ∈ I_{n}, and v(l_{1}) = i_{1} for all l_{1} ∈ L_{1}, such that

x ∈ Im( ∑_{l_{1}∈L_{1},...,l_{n}∈L_{n}} f_{l_{1}}f_{l_{2}} ⋯ f_{l_{n}})

for each n. Here all of the f_{l_{n}}’s are non-isomorphisms since the set of u(l_{k})’s is distinct from that of the v(l_{k})’s for each k; however, we permit repetitions u(l_{k}) = u(l′_{k}) and v(l_{k}) = v(l′_{k}) for l_{k} ≠ l′_{k}. Next, we let Y ⊂ M_{i_{1}} be a maximal R-submodule of M_{i_{1}} with x ∉ Y, and consider the tree having as root the index i_{1} and as branches all those concatenations f_{l_{1}}f_{l_{2}} ⋯ f_{l_{n}} the image of which is not contained in Y. By construction, there is no upper bound on the lengths of the edge paths in this graph, and therefore König’s graph theorem guarantees the existence of a path of infinite length. But, since M_{i_{1}}/Y is a finitely cogenerated factor module of M_{i_{1}}, this means that the family (M_{i}) fails to be left semi-T-nilpotent. Thus, our hypothesis to the contrary ensures finiteness of I. □

This smooths the road to

Theorem H. Let (M_{i})_{i∈I} be a family of indecomposable R-modules and (M_{t})_{t∈T} a transversal of the isomorphism classes of the M_{i}.

(I) If all of the M_{i} have local endomorphism rings, the following statements are equivalent:

(1) ⨁_{i∈I} M_{i} is endo-noetherian, and (M_{t})_{t∈T} is right semi-T-nilpotent.

(2) T is finite, and each M_{t} is endo-noetherian. (II) The following statements are equivalent:

(1) ⨁_{i∈I} M_{i} is endo-artinian, and (M_{t})_{t∈T} is left semi-T-nilpotent.

(2) T is finite, and each M_{t} is endo-artinian. (III) Suppose that (M_{t})_{t∈T} is a left semi-T-nilpotent and right T-nilpotent family sat-

isfying the following additional finiteness condition: (•) For any cofinite subset T′ ⊆ T, the endosocle of ⨁_{t∈T′} M_{t} is finitely generated.

Then T is finite.

Proof. Start by observing that, in each of the first two parts, the implication ‘(2) ⟹ (1)’ is trivial; Indeed, it suffices to note that both endo-noetherianness and endo-artinianness are inherited by arbitrary powers and finite direct sums. Moreover, note that, in proving the remaining implications, it is innocuous to assume that I = T, since the conditions involved are stable under passage from ⨁_{i∈I} M_{i} to ⨁_{t∈T} M_{t}. The implications ‘(1) ⟹ (2)’ under (I) and (II) now follow from Lemma F and Lemma G, respectively; note that (II.1) guarantees all of the M_{i} to have local endomorphsim rings, while this is a blanket hypothesis for (I).

In order to prove (III), adopt all of the listed hypotheses. By part (1) of Proposition E, the M_{t} again have local endomorphism rings. Moreover, for each cofinite subset T′ ⊆ T, the direct sum ⨁_{t∈T′} M_{t} is finitely cogenerated over its endomorphism ring: Indeed, in view of Proposition E(2), right T-nilpotence of (M_{t})_{t∈T′} ensures that this direct sum has essential endosocle and, by condition (•), this endosocle is finitely generated. Thus, once more, Lemma G yields finiteness of T. □

By combining part (III) of Theorem H and Proposition E(2), we immediately obtain

Corollary I. If (M_{i})_{i∈I} is a family of indecomposable R-modules and (M_{t})_{t∈T} a transversal of the isomorphism classes of the M_{i}, then the following statements are equivalent:

(1) ⨁_{i∈I} M_{i} is Σ-algebraically compact, (M_{t})_{t∈T} left semi-T-nilpotent, and (•) for each cofinite subset T′ ⊆ T, the endosocle of ⨁_{t∈T′} M_{t} is finitely generated.

(2) T is finite, and each M_{i} is Σ-algebraically compact. □

Remarks. The semi-T-nilpotence conditions in Theorem H and Corollary I are not redundant, not even when all of the M_{i} are finitely generated over a finite dimensional algebra – see Examples O below.

As for condition (•) in Theorem H(III) and Corollary I: Dropping it means jeopardizing the equivalence in either case, as illustrated by Example J(a) below. In case the individual modules M_{i} have endosocles of finite lengths, condition (•) can clearly be weakened to the requirement that, for each cofinite subset T′ ⊆ T, the endosocle of ⨁_{t∈T′} M_{t} have finite support. We do not know whether condition (•) can always be relaxed in this fashion, but point out that finite generation of endosoc(⨁_{t∈T′} M_{t}) is not automatic in the presence of the remaining hypotheses for a fixed cofinite subset T′ of T. Indeed, we provide a Σ-algebraically compact direct sum ⨁_{n∈ℕ} M_{n}, ranging over a left and right T-nilpotent family (M_{n})_{n∈ℕ} of pairwise non-isomorphic indecomposable modules, which has a nonfinitely generated endosocle with singleton support (Example J(b)).

Examples J. (a) The direct sum M = ⨁_{p prime} ℤ(p^{∞}) is a Σ-algebraically compact ℤmodule having as endomorphism ring the direct product of the rings of p-adic integers, where p runs through all primes. Moreover, the family (ℤ(p^{∞}))_{p prime} is clearly left semi-T-nilpotent and right T-nilpotent. However, the endosocle of M coincides with the ℤ-socle, and thus has infinite support. This shows that the endosocle condition in (III) is crucial.

(b) Set R = K[X_{n} | n ∈ ℕ]/(X_{n} | n ∈ ℕ)^{3}, where K is a field, and let x_{n} be the residue class of X_{n} in R. Define M_{0} = (x_{n} | n ∈ ℕ)/(x_{n}^{2} | n ∈ ℕ) and M_{n} = R/(x_{n}, x_{j}x_{k} | j, k ∈ ℕ) for n ∈ ℕ. One readily checks that Hom_{R}(M_{i}, M_{j}) = Hom_{R}(M_{i}, JM_{j}) for i ≠ j, i ≥ 0, j ≥ 1, and that the endomorphism rings of the M_{i} are local with vanishing radical squares. In particular, the family (M_{i})_{i≥0} is left and right T-nilpotent. Noting that the M_{i} are modules over R/J^{2} and using [23, Satz 6.5] for instance, one moreover checks that ⨁_{i≥0} M_{i} is Σ-algebraically compact. We further obesrve that, for n ∈ ℕ, the module M_{n} embeds into M_{0} via \overline{1} ↦ \overline{x_{n}}. Hence the endosocle of ⨁_{i≥0} M_{i} coincides with the endosocle of M_{0}, that is, its support is a singleton. On the other hand, it fails to be finitely generated.

Prime targets for applications of the following consequence of Theorem H are families of generic or finitely generated modules over Artin algebras. In light of the subsequent proposition addressing families of finitely generated modules over Artin algebras, the picture can be sharpened in that situation.

Corollary K. For any family (M_{i})_{i∈I} of indecomposable endofinite modules the following statements are equivalent:

(1) The number of isomorphism types of M_{i} is finite.

(2) ⨁_{i∈I} M_{i} is endo-noetherian, and the family (M_{i}) is right semi-T-nilpotent.

(3) ⨁_{i∈I} M_{i} is endo-artinian, and the family (M_{i}) is left semi-T-nilpotent.

(4) If (M_{t})_{t∈T} is a transversal of the isomorphism types of the M_{i}, then (M_{t}) is left and right semi-T-nilpotent and, for any cofinite subset T′ ⊆ T, the endosocle of ⨁_{t∈T′} M_{t} has finite support in T.

Proof. In view of Theorem H, only the implication ‘(4) ⟹ (1)’ requires some justification: Since the M_{i} are endofinite, each singleton family {M_{i}} is right T-nilpotent by Proposition E(2), whence right T-nilpotence of the family (M_{i}) follows from right semi-T-nilpotence. □

The next result exhibits a simplification of the picture for Artin algebras. Namely, a direct sum ⨁_{i∈I} M_{i} of finitely generated modules M_{i} is Σ-algebraically compact precisely when it is endo-artinian. In other words, for Artin algebras, Proposition L provides an ‘endo-artinian counterpart’ to the final observation of Section 1: This observation remains true on replacing ‘endo-noetherian’ by ‘endo-artinian’, and substituting the ACC for (finite) matrix subgroups by the corresponding DCC.

Throughout our excursion to the narrowed setting, Λ will be an Artin algebra with center C, and D : Λ -mod → mod- Λ will denote the standard duality, i.e., D = Hom_{C}(−, E), where E is the C-injective envelope of C/J(C). Note that D induces a Morita duality on C -mod as well; this obvious fact is useful, because endo-submodules of Λ-modules are of course C-submodules.

Proposition L. For any family (M_{i})_{i∈I} of finitely generated left Λ-modules, the following conditions are equivalent:

(1) ⨁_{i∈I} M_{i} is Σ-algebraically compact.

(2) ⨁_{i∈I} M_{i} is endo-artinian.

(3) ⨁_{i∈I} D(M_{i}) is endo-noetherian.

Proof. To verify ‘(1) ⟹ (3)’, we adopt (1). This means that M = ⨁_{i∈I} M_{i} satisfies the DCC for finite matrix subgroups, whence, by Proposition 3 of [17], the dual D(M) = ∏_{i∈I} D(M_{i}) has the ACC for finite matrix subgroups. In view of the final observation of Section 1, this entails (3).

To prove ‘(3) ⟹ (2)’, we assume that ⨁_{i∈I} D(M_{i}) is endo-noetherian. We begin by noting that a C-submodule U of ⨁_{i∈I} M_{i} is an endo-submodule of ⨁_{i∈I} M_{i} if and only if U = ⨁_{i∈I} U_{i}, where the C-submodules U_{i} = M_{i} ∩ U of the M_{i} have the property that, for all i, j ∈ I and all f ∈ Hom_{Λ}(M_{i}, M_{j}), the image f(U_{i}) is contained in U_{j}. Now suppose that U is an endo-submodule of ⨁_{i∈I} M_{i}, and, for i ∈ I, let ι_{i} : U_{i} ↪ M_{i} be the canonical embedding. Moreover, let V_{i} ⊆ D(M_{i}) denote the kernel of the dual map D(ι_{i}); i.e., V_{i} = {ρ ∈ Hom_{K}(M_{i}, K) | ρ|_{U_{i}} = 0}. We claim that V = ⨁_{i∈I} V_{i} is an endo-submodule of ⨁_{i∈I} D(M_{i}). Clearly, V is a C-submodule, whence it suffices to check stability under homomorphisms g ∈ Hom_{Λ}(D(M_{i}), D(M_{j})). Suppose g = D(f) with f ∈ Hom_{Λ}(M_{j}, M_{i}). Dualizing the fact that f · ι_{j} = ι_{i} · (f|_{U_{j}}), we obtain, for each ρ ∈ V_{i}, the equality g(ρ)|_{U_{j}} = (ρ · f)|_{U_{j}} = (ρ|_{U_{i}}) · (f|_{U_{j}}) = 0; in other words, g(ρ) belongs to V_{j} as required.

Let U_{1} ⊇ U_{2} ⊇ U_{3} ⊇ ⋯ be a descending chain of endo-submodules of ⨁_{i∈I} M_{i}, and label the embedding U_{k+1} ↪ U_{k} by ϕ_{k}. Extending the above notation, we moreover write the embedding U_{ki} = M_{i} ∩ U_{k} ↪ M_{i} as ι_{ki}, and the kernel of the C-module epimorphism D(ι_{ki}) as V_{ki}. The preceding paragraph shows that each of the direct sums V_{k} = ⨁_{i∈I} V_{ki} is in fact an endo-submodule of ⨁_{i∈I} D(M_{i}). Clearly, the V_{k} form an ascending chain which, by our hypothesis, becomes stationary. This means that the ascending chains V_{1i} ⊂ V_{2i} ⊂ ⋯ become uniformly stationary on I, and, in view of the exact sequences

0 → V_{ki} → M_{i} → D(U_{ki}) → 0,

we see that it is again uniformly on I that the lengths of the C-modules in the sequences D(U_{1i}) → D(U_{2i}) → ⋯ induced by the epimorphisms D(ϕ_{k}|_{U_{ki}}) converge to a constant. Since the lengths of the C-modules appearing along the chains U_{1i} ⊇ U_{2i} ⊇ ⋯ for i ∈ I converge in tandem, they once more do so uniformly on I. This proves our initial descending chain (U_{k})_{k∈ℕ} to be eventually stationary.

In view of the characterization of Σ-algebraic compactness quoted in Section 1, the implication ‘(2) ⟹ (1)’ is clear. □

The next corollary results from a combination of points K and L with the Harada-Sai Lemma [12].

Corollary M. Suppose Λ is an Artin algebra and (M_{i})_{i∈I} a family of left Λ-modules having uniformly bounded (finite) composition lengths. Moreover, let (M′_{t})_{t∈T} be a transversal of the isomorphism types of the indecomposable direct summands of the M_{i}. Then the following conditions are equivalent:

(1) T is finite.

(2) ⨁_{i∈I} M_{i} is endo-noetherian.

(3) ⨁_{i∈I} M_{i} is endo-artinian.

(4) ⨁_{i∈I} M_{i} is Σ-algebraically compact.

(5) For any cofinite subset T′ ⊆ T, the endosocle of ⨁_{t∈T′} M′_{t} has finite support.

Proof. It suffices to observe that, due to the Harada-Sai Lemma, our blanket hyposthesis forces the family (M′_{t}) to be both left and right T-nilpotent. □

The final corollary in this series highlights the role played by the endosocles of direct sums of indecomposable representations. It is an immediate consequence of Corollary M and the (confirmed) second Brauer-Thrall conjecture.

Corollary N. For any finite dimensional algebra Λ over an algebraically closed field, the following conditions are equivalent:

(1) Λ has finite representation type.

(2) For every family (M_{i})_{i∈I} of pairwise non-isomorphic indecomposable modules of fixed finite dimension, the endosocle of ⨁_{i∈I} M_{i} has finite support. □

The following examples attest to the fact that none of the T-nilpotence conditions in Theorem H and Corollaries I, K can be dropped without penalty.

Examples O. Let Λ = KΓ be the Kronecker algebra over a field K, where Γ is the quiver

\xymatrix{1 \ar[r]^{α} \ar[r]_{β} & 2}

(1) Moreover, let (M_{n})_{n∈ℕ} be the family of all preprojective left Λ-modules, i.e., M_{1} = Λe_{2}/J(Λ)e_{2}, and for n ≥ 2, M_{n} is the string module with graph

[figure: 1; α; β; 1; α; β; ⋯; 1; α; β; 2; 2; 2; ⋯; 2; 2]

having n − 1 summands S_{1} in its top. We claim that the direct sum ⨁_{n∈ℕ} M_{n} is endonoetherian.

To verify our claim, we set M = ⨁_{n∈ℕ} M_{n}, S = End_{R}(M), and, given K-subspaces X ⊆ Y of M, we write [Y : X] for the codimension of X in Y. The dimension of any subspace X ⊆ M_{n} is clearly determined by the two codimensions [e_{1}M_{n} : e_{1}X] and [soc_{Λ} M_{n} : X∩soc_{Λ} M_{n}]. Let U be an S-submodule of M. In view of the equalities Hom_{Λ}(M_{n}, M_{i}) = 0 for n > i and the two obvious linearly independent embeddings of M_{n} into M_{n+1}, it is straightforward to verify that [e_{1}M_{n} : e_{1}(U ∩ M_{n})] ≥ [e_{1}M_{n+1} : e_{1}(U ∩ M_{n+1})] provided that e_{1}(U ∩ M_{n}) is nonzero, and [soc_{Λ} M_{n} : U ∩ soc_{Λ} M_{n}] ≥ [soc_{Λ} M_{n+1} : U ∩ soc_{Λ} M_{n+1}] in case U ∩ soc_{Λ} M_{n} is nonzero. Now consider an ascending chain U_{1} ⊆ U_{2} ⊆ … of Ssubmodules of M, and suppose that, for l large enough, U_{l} has non-trivial intersection with some e_{1}M_{i} and soc_{Λ} M_{j} (the remaining cases are treated similarly). Fix such an index l. For k ≥ 1, set c(k) = min{[e_{1}M_{n} : e_{1}(U_{k} ∩ M_{n})] + [soc_{Λ} M_{n} : U_{k} ∩ soc_{Λ} M_{n}] | n ∈ ℕ}, and note that c(l) ≥ c(l + 1) ≥ . . .. Therefore, we can find m ≥ l such that c(m) is minimal, say c(m) = [e_{1}M_{n} : e_{1}(U_{m} ∩ M_{n})] + [soc_{Λ} M_{n} : U_{m} ∩ soc_{Λ} M_{n}] for n ≥ n_{0}. For i ≥ m, the intersections M_{n} ∩ U_{i} can then differ from M_{n} ∩ U_{m} only when n < n_{0}, which shows that the chain of U_{i}’s becomes stationary.

(2) Let Λ still be the Kronecker algebra and (N_{n})_{n∈ℕ} the family of all preinjective left Λ-modules. We claim that ⨁_{n∈ℕ} N_{n} is endo-artinian, and so, in particular, Σ-algebra-ically compact. Indeed, since the preprojective left and right Λ-modules have identical structures (the two simple modules simply swap roles), the direct sum ⨁_{n∈ℕ} D(N_{n}) is endo-noetherian by part (1). Therefore Proposition L yields our claim.

At this point, we return to an arbitrary base ring. In light of our remarks concerning the exchange property of direct sums (in Section 3), we finally obtain the following consequence of Theorem H.

Corollary P. An endo-noetherian direct sum ⨁_{i∈I} M_{i} of indecomposable modules M_{i} (over any ring R) has the exchange property if and only if each M_{i} has local endomorphsim ring and the M_{i} fall into finitely many isomorphism classes. □

Viewing the Kronecker examples in light of this fact, we see that the direct sum of the preprojective modules over the Kronecker algebra fails to have the exchange property. On the other hand, the direct sum ranging over the preinjective modules does enjoy this property, as do all algebraically compact modules (see [16]).

Variants of the ACC and DCC for endo-submodules have made numerous appearances in the representation theory of Artin algebras. Our concluding remarks, relating ascending endo-chain conditions of direct sums to preprojective partitions and left almost split morphisms, are merely the ‘tip of the iceberg’. (Naturally, certain relaxed descending endo-chain conditions are linked up with preinjective partitions and the existence of right almost split maps in a dual fashion.) Remarks.

Due to [3] and [13], respectively, we have the following connections between weakened endo-noetherian conditions for direct sums ⨁_{i∈I} M_{i} of finitely generated indecomposable modules on one hand and the existence of preprojective partitions (resp. strong preprojective partitions) on the other: Suppose that (M_{i}) is a family of indecomposable modules, and (M_{t})_{t∈T} a transversal of its isomorphism types. Recall that a preprojective partition of (M_{i})_{i∈I} is a partition

{M_{t} | t ∈ T } = ⨆_{n≤ω} C_{n}

such that, for all n < ω, the set C_{n} is a minimal finite generating set for ⨆_{n≤m≤ω} C_{m}. A strong preprojective partition extends this principle to arbitrary ordinals, i.e., it is a partition {M_{t} | t ∈ T } = ⨆_{α≤γ} C_{α}, where γ is an ordinal number and each C_{α} is a minimal finite generating set for ⨆_{α≤β≤γ} C_{β}. The existence of a preprojective partition of (M_{i}) is equivalent to the following endo-condition for ⨁_{i∈I} M_{i}: For each cofinite subset T_{1} ⊆ T, there exists a cofinite subset T_{2} ⊆ T_{1} such that

⨁_{t∈T_{1}} M_{t} ⊆ (End_{R}(⨁_{t∈T} M_{t})) · ( ⨁_{t∈T_{1}\T_{2}} M_{t}).

Moreover, the condition that ⨁_{i∈I} M_{i} be endo-noetherian implies the existence of strong preprojective partitions for arbitrary subfamilies of (M_{i})_{i∈I}.

This last implication was recently refined by Dung who proved that the conditions below are subject to ‘(a) ⟹ (b) ⟹ (c)’:

(a) Hom_{R}(M_{k}, ⨁_{i∈I} M_{i}) is noetherian as a left module over End_{R}(⨁_{i∈I} M_{i}) for all k ∈ I.

(b) For every subfamily C of (M_{i}), the subcategory add C of R -mod has left almost split morphisms.

(c) Every subfamily of (M_{i}) has a strong preprojective partition.

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Department of Mathematics, University of California, Santa Barbara, CA 93106, USA

   E-mail address: birge@math.ucsb.edu

Departamento de Mátematicas, Universidad de Murcia, 30100 Espinardo-MU, Spain

   E-mail address: msaorinc@fcu.um.es