Quillen equivalences for stable categories

Abstract

For an abelian category A we investigate when the stable categories GProj(A) and GInj(A) are triangulated equivalent. To this end, we realize these stable categories as homotopy categories of certain (non-trivial) model categories and give conditions on A that ensure the existence of a Quillen equivalence between the model categories in question. We also study when such a Quillen equivalence transfers from A to categories naturally associated to A, such as Ch(A), the category of chain complexes in A, or Rep(Q, A), the category A-valued representations of a quiver Q. de Ciencia y Tecnología de la Región de Murcia in the framework of III PCTRM 2011-2014 . 1 In the important special case where A is a quasi-Frobenius ring, for example, if A = kG is the group algebra of a finite group G with coefficients in a field k, the category MCM(A) is just the category mod(A) of all finitely generated A-modules.2 The singularity category D sg (A) is defined to be the Verdier quotient D b (A)/D b perf (A) of the bounded derived category D b (A), whose objets are complexes of A-modules with bounded and finitely generated homology, by the subcategory D b perf (A), whose objects are isomorphic (in D b (A)) to a perfect complex, that is, to a bounded complex of finitely generated projective A-modules. The name singularity category and the symbol D sg (A) seem to be the popular choices nowadays, however, in the work of Buchweitz [8, Def. 1.2.2], this category is called the stabilized derived category and denoted by D b (A), and in the work of Orlov [31], it is called the triangulated category of singularities and denoted by D sg (A).

Frobenius functors, stable equivalences and $K$-theory of Gorenstein projective modules

“…It is well-known that GP(A) is a Frobenius category where the projective-injective objects are precisely the projective objects in A; see for example [10,Proposition 2.2]. Thus, the stable category GP(A) is triangulated.…”

“…We use GP <∞ (A) and P <∞ (A) to denote the subcategory of objects in A with finite Gorenstein projective dimension and finite projective dimension, respectively. Note that GP <∞ (A) is a weakly idempotent complete exact category [10,Lemma 3.3], and GP(A) ∩ P <∞ (A) = P(A) holds by [16,Proposition 10.2.3].…”

“…Proposition 2.1. ( [10,Theorem 3.7]) On the category GP <∞ (A), there is a model structure (GP(A), P <∞ (A), GP <∞ (A)).…”

“…are also known as modules of G-dimension zero, totally reflexive modules and maximal Cohen-Macaulay modules in different literatures. By [10,Theorem 3.7], for any ring there always exists a Gorenstein projective model structure on the weakly idempotent complete exact category consisted by modules of finite Gorenstein projective dimension.…”

Quillen equivalences inducing Grothendieck duality for unbounded chain complexes of sheaves

“…Other authors have studied when the stable category of Gorenstein projectives is equivalent to the stable category of Gorenstein injectives. For example, some Quillen equivalences in this direction were found in [DEH18]. Also, it is shown in [ZH17] that, for a Noetherian ring R with a dualizing complex, the stable category of Gorenstein projective R-modules is equivalent to the stable category of Gorenstein injective R-modules.…”