Syzygy modules for Noetherian rings
Let \(\Lambda\) be a two-sided Noetherian ring and \(\text{mod }\Lambda\) be the category of finitely generated \(\Lambda\)-modules. For each integer \(k>0\) we denote by \(\Omega^k(\text{mod }\Lambda)\) the full subcategory of \(\text{mod }\Lambda\) whose objects are the \(C\) in \(\text{mod }\Lambda\) which are the \(k\)th syzygies of modules in \(\text{mod }\Lambda\). While the projective resolutions of modules have been studied extensively, little is known about the subcategory \(\Omega^k(\text{mod }\Lambda)\) of \(\text{mod }\Lambda\). One of the few general properties known about these subcategories is that they are covariantly finite in \(\text{mod }\Lambda\). While in general covariantly finite subcategories of \(\text{mod }\Lambda\) have been proven to be of interest in various situations, they are particularly interesting when they are closed under module extension. The main theme of the paper under review is to describe when the subcategories \(\Omega^k(\text{mod }\Lambda)\) of \(\text{mod }\Lambda\) are extension closed. The main result is Theorem 0.1, which is too technical to be stated here. Interested readers are referred to the paper itself and the bibliography listed there, in particularly the following related papers by the same authors [Adv. Math. 86, No. 1, 111-152 (1991; Zbl 0774.16006); Proc. Tsukuba Workshop on Representations of Algebras, Lond. Math. Soc. Lect. Note Ser. 168, 1-42 (1992; Zbl 0774.16005); J. Pure Appl. Algebra 92, No. 1, 1-27 (1994; Zbl 0803.16016)].
categories of finitely generated modules
scientific article; zbMATH DE number 923677