Classifying thick subcategories of the stable category of Cohen–Macaulay modules

Abstract

Various classification theorems of thick subcategories of a triangulated category have been obtained in many areas of mathematics. In this paper, as a higher-dimensional version of the classification theorem of thick subcategories of the stable category of finitely generated representations of a finite p-group due to Benson, Carlson and Rickard, we consider classifying thick subcategories of the stable category of Cohen-Macaulay modules over a Gorenstein local ring. The main result of this paper yields a complete classification of the thick subcategories of the stable category of Cohen-Macaulay modules over a local hypersurface in terms of specialization-closed subsets of the prime ideal spectrum of the ring which are contained in its singular locus.

Generators and Dimensions of Derived Categories of Modules

“…Hence the completion R of R also has an isolated singularity. Then it follows from [36,Corollary 2.9] that CM R = G holds. Corollary 5.3 and Lemma 5.4 yield CM R = G n for some n ≥ 1.…”

Singularity categories and singular equivalences for resolving subcategories

“…In particular, if R is a commutative Gorenstein local ring and X is a quasi-resolving subcategory of mod R containing CM(R), then X and CM(R) are singularly equivalent. If R is moreover a complete intersection, then X is singularly equivalent to X ∩ CM(R) for all resolving subcategories X of mod R. Combining this with a classification of resolving subcategories given in [50] yields that if R is an isolated hypersurface singularity, then X is singularly equivalent to either CM(R) or the zero category 0, so there are only at most two singular equivalence classes.…”

“…This notion has been introduced by Auslander and Bridger [7] to prove that the category of totally reflexive modules is a resolving subcategory of the category mod R of finitely generated modules over a noetherian ring R. The category CM(R) of maximal Cohen-Macaulay modules over a Cohen-Macaulay local ring R is also a resolving subcategory of mod R, and there are many other important subcategories known to be resolving. The studies of resolving subcategories have been done widely so far; see [2,15,16,28,29,40,47,[49][50][51][52]56] for example. For a resolving subcategory X of an abelian category A, let X = X / proj A be the stable category of X .…”

Derived Witt group formalism

“…There is a list of examples of resolving subcategories of Mod(A) in [16] which can be adapted to subcategories of Coh(X). The one that is of our particular interest is the one corresponding to the notion of semi-dualizing A-modules ω.…”

“…More recently, there has been considerable amount of activities (e.g. [16,17]) on resolving subcategories of the category Mod(A) of finitely generated modules over noetherian commutative rings A. Many of these are directed toward the classification of such resolving subcategories under various conditions, which is encompassed by similar classification of variety of types of subcategories of the module categories.…”