Singularity categories and singular equivalences for resolving subcategories

Abstract

Abstract. Let X be a resolving subcategory of an abelian category. In this paper we investigate the singularity category Dsg(X ) = D b (mod X )/K b (proj(mod X )) of the stable category X of X . We consider when the singularity category is triangle equivalent to the stable category of Gorenstein projective objects, and when the stable categories of two resolving subcategories have triangle equivalent singularity categories. Applying this to the module category of a Gorenstein ring, we prove that the complete intersections over which the stable categories of resolving subcategories have trivial singularity categories are the simple hypersurface singularities of type (A 1 ). We also generalize several results of Yoshino on totally reflexive modules.

Gorenstein Homological Aspects of Monomorphism Categories via Morita Rings

“…above is a consequence of Theorem 5.8 which provides sufficient conditions on a subcategory B of an exact category A with enough projectives such that mod-B is a Gorenstein abelian category. It should be noted that this result generalizes, and is inspired by, a result of Matsui and Takahashi [32]. Conventions and Notation.…”

“…In this context, the fourth main result of this paper is Theorem B (ii), see Corollary 5.12, which shows that the category of coherent functors over C is a Gorenstein abelian category in the sense of [12]. Finally, using a result of Beligiannis [8] we realize the singularity category [32] of mod-C as the stable category of Cohen-Macaulay objects over mod-C . Theorem B.…”

“…We refer to Theorem 5.5 for its proof and to Definition 5.2 for the precise notion of Gorenstein subcategories in the setting of exact categories. Moreover, inspired by recent work of Matsui and Takahashi [32] we continue our study on the exact subcategory C of mono(Λ) by considering the category of coherent functors mod-C over the stable category C of C . Also, we define the subcategory Ω n (C ) of C consisting of all nth syzygies of objects in C (subsection 5.2).…”

“…We refer to [2,3] for more details on coherent functors. Then, Matsui and Takahashi [32] considered the Verdier quotient D sg (mod-A ) := D b (mod-A )/K b (proj(mod-A )) and call it the singularity category of mod-A . We remark that this triangulated category is included in the general framework of the stabilization of an abelian or exact category studied by Beligiannis [8].…”

On the Recollements of Functor Categories

“…The singularity category in this generality was considered by Beligiannis, see for instance [11,Corollary 3.9], see also [51,Sec.5], [16] and [40].…”