Indecomposable Cohen-Macaulay modules and irreducible maps

Abstract

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Annihilation of Cohomology and Strong Generation of Module Categories

“…The identification of the singular locus with the closed subset defined by the cohomology annihilator is related to results of Wang [31,32], that in turn extend work of Dieterich [9], Popescu and Roczen [26], and Yoshino [34,35] stemming from Brauer-Thrall conjectures for maximal Cohen-Macaulay modules over Cohen-Macaulay rings; see the comments preceding Theorems 5.3 and 5.4.…”

“…In this section we explain how certain ideas introduced by Noether [24], and developed by Auslander and Goldman [2], and also Scheja and Storch [29], can be used to find cohomology annihilators, especially those that are non-zerodivisors. The results presented are inspired by, and extend to not necessarily commutative rings, those of Wang [31,32]; see also [26,35]. The novelty, if any, is that some arguments are simpler, and we contend more transparent; confer the proofs of Lemma 3.2 below with that of [31,Proposition 5.9].…”

“…The central theme of this article is that the two topics that make up its title are intimately related. Inklings of this can be found in the literature, both on annihilators of cohomology, notably work of Popescu and Roczen [26] from 1990, and on generators for module categories that is of more recent vintage; principally the articles of Dao and Takahashi [8], and Aihara and Takahashi [1]. We make precise the close link between the two topics, by introducing and developing appropriate notions and constructions, and use it to obtain more comprehensive results than are currently available in either one.…”

Indecomposable generalized Cohen-Macaulay modules

“…This proposition follows from our next lemma which is stronger as is needed for the proof of 3.4 but suits better for induction. of [25]. This fact would simplify the presentation of [25].…”

“…This technique works very well for the category of maximal CohenMacaulay modules over a Cohen-Macaulay local ring A with isolated singularity (see, [9,24,25,32]), the proof using the following facts:…”