Approximations and almost split sequences in homologically finite subcategories
The importance of almost split sequences (also called Auslander-Reiten sequences) is well-known in the representation theory of Artin algebras. The general theory of almost split sequences was established in a series of joint papers of Auslander and Reiten in the seventies. This general theory was further developed by \textit{M. Auslander} and \textit{S. O. Smalø} [in J. Algebra 69, 426-454 (1981; Zbl 0457.16017)]. There are still many questions dealing with almost split sequences in a subcategory. Among others Auslander and Smalø proposed the following two questions: (1) Given a full subcategory \(\mathcal X\) of the module category of finitely generated modules over an Artin algebra. Describe the Ext-projective and Ext-injective objects in \(\mathcal X\). In particular, is there only a finite number of isomorphism classes of indecomposable objects which are Ext-projective or Ext-injective? Also, need these numbers be the same when finite? (2) Suppose \(0\to A\to B\to C\to 0\) is an almost split sequence in the subcategory \(\mathcal X\). Is there a method of constructing \(A\) from \(C\) and vice versa similar to the construction given by the functors \(DTr\) and \(TrD\)?, where \(D\) is the usual duality and \(Tr\) stands for the transpose with respect to the given Artin algebra. In the paper under review problem (2) is solved in terms of minimal approximations, \(DTr\), \(TrD\) and the functor \(\text{Ext}^1\). To question (1), the author has also partial answers which characterize the Ext-projective and Ext-injective objects. He also gives a necessary and sufficient condition for the finiteness of the number of non-isomorphic indecomposable Ext-projective modules in a certain subcategory in terms of approximations. Concerning the two questions, some special attention is paid also to the subcategories closed under submodules.
categories of finitely generated modules
scientific article; zbMATH DE number 1106636