Indecomposable Modules

If Λ \Lambda is an artin algebra there is a partition of ind ⁡ Λ \operatorname {ind} \Lambda , the category of indecomposable finitely generated Λ \Lambda -modules, ind ⁡ Λ = ∪ i ⩾ 0 P _ _ i \operatorname {ind} \Lambda = { \cup _{i \geqslant 0}}{\underline {\underline {P}}_i} , called the preprojective partition. We show that P _ _ i \underline {\underline {P}}_i can be easily constructed for hereditary artin algebras, if P _ _ i − 1 \underline {\underline {P}}_{i - 1} is known: A A is in P _ _ i \underline {\underline {P}}_i if and only if A A is not in P _ _ i − 1 \underline {\underline {P}}_{i - 1} and there is an irreducible map B → A B \to A , where B B is in P _ _ i − 1 \underline {\underline {P}}_{i - 1} .

4 - Indecomposable Modules

An Artin algebra Λ is said to be of finite Cohen–Macaulay type if, up to isomorphism, there are only finitely many indecomposable modules in G(Λ), the full subcategory of modΛ consisting of all Gorenstein projective (right) Λ-modules. In this paper, we study 1-Gorenstein algebras of finite Cohen–Macaulay type through mod(G(Λ)), the category of finitely presented G(Λ)-modules. Some applications will be provided. In particular, a necessary and sufficient condition is given for T3(Λ), the 3 by 3 lower triangular matrices over Λ, to be of finite Cohen–Macaulay type. Finally, the structure of almost split sequences will be described explicitly in a special subcategory of mod(G(Λ)), denoted by ϑ−1(G(Λ)). If Λ is self-injective, ϑ−1(G(Λ))=mod(G(Λ)).

In 1968–1969, A. V. Jategaonkar published his famous constructions of left but not right noetherian rings that provided counterexamples to several important conjectures of that era. These examples, and others like them, seemed to indicate that, in general, the task of completely understanding the structure of indecomposable injective modules over one-sided noetherian rings was hopeless. In this paper I show how to deduce by natural methods, directly from the known description of these rings and their properties, explicit computational descriptions of the indecomposable injective left modules over Jategaonkar's rings. I use these explicit descriptions to answer some simple structural questions about the indecomposables. The structure of indecomposable injective modules over a commutative noetherian ring has been well understood since the work of Matlis[10] in the 1950's. There is a natural correspondence between the indecomposable injectives and the localizations of the ring, and the structure of ...

This paper is devoted to proving the following result: An artin algebra is of infinite representation type if and only if there is no bound on the lengths of the endomorphism rings of finitely generated indecomposable modules, where the endomorphism rings are viewed as modules over themselves.

Decompositions in categories of highest weight modules

Let R be a ring, see below for other notation. The functor categories (mod-R, Ab) and ((R-mod)op, Ab) have received considerable attention since the 1960s. The first of these has achieved prominence in the model theory of modules and most particularly in the investigation of the representation theory of Artinian algebras. Both [11, Chapter 12] and [8] contain accounts of the use (mod-R, Ab) may be put to in the model theoretic setting, and Auslander's review, [1], details the application of (mod-R, Ab) to the study of Artinian algebras. The category ((R-mod)op, Ab) has been less fully exploited. Much work, however, has been devoted to the study of the transpose functor between R-mod and mod-R. Warfield's paper, [13], describes this for semiperfect rings, and this duality is an essential component in the construction of almost split sequences over Artinian algebras, see [4]. In comparison, the general case has been neglected. This paper seeks to remedy this situation, giving a concrete description of the resulting equivalence between (mod-R, Ab) and ((R-mod)op, Ab) for an arbitrary ring R.

Invariants and constructions of separable equivalences

The aim of this paper is to study the modules M with a finite direct sum decomposition M = M 1 ⊕ … ⊕ Mn such that all the endomorphism rings End(Mi ) are homogeneous semilocal. In particular, we find complete results about the uniqueness of such decompositions (Krull-Schmidt Theorem). *Partially supported by Ministero dell'Università e della Ricerca Scientifica e Tecnologica (Italy) through the project “Teoria degli anelli, moduli e gruppi abeliani: metodi omologici, topologici e categoriali.”

On subcategories closed under predecessors and the representation dimension

Some modules with nearly prescribed endomorphism rings

In this note, we prove that if Λ \Lambda is an Artin algebra with a simple module S S of finite projective dimension, then the finiteness of the finitistic dimension of Λ \Lambda implies that of ( 1 − e ) Λ ( 1 − e ) (1-e)\Lambda (1-e) where e e is the primitive idempotent supporting S S . We derive some consequences of this. In particular, we recover a result of Green-Solberg-Psaroudakis: if Λ \Lambda is the quotient of a path algebra by an admissible ideal I I whose defining relations do not involve a certain arrow α \alpha , then the finitistic dimension of Λ \Lambda is finite if and only if the finitistic dimension of Λ / Λ α Λ \Lambda /\Lambda \alpha \Lambda is finite.

Twisted generalized Weyl algebras (TGWAs) A(R,σ,t) are defined over a base ring R by parameters σ and t, where σ is an n-tuple of automorphisms, and t is an n-tuple of elements in the center of R. We show that, for fixed R and σ, there is a natural algebra map \(A(R,\sigma ,tt^{\prime })\to A(R,\sigma ,t)\otimes _{R} A(R,\sigma ,t^{\prime })\). This gives a tensor product operation on modules, inducing a ring structure on the direct sum (over all t) of the Grothendieck groups of the categories of weight modules for A(R,σ,t). We give presentations of these Grothendieck rings for n = 1,2, when \(R=\mathbb {C}[z]\). As a consequence, for n = 1, any indecomposable module for a TGWA can be written as a tensor product of indecomposable modules over the usual Weyl algebra. In particular, any finite-dimensional simple module over \(\mathfrak {sl}_{2}\) is a tensor product of two Weyl algebra modules.

Poor modules with no proper poor direct summands

Ehresmann semigroups whose categories are EI and their representation theory