Recollement of homotopy categories and Cohen-Macaulay modules

Abstract

We study the homotopy category of unbounded complexes with bounded homologies and its quotient category by the homotopy category of bounded complexes. We show the existence of a recollement of the above quotient category and it has the homotopy category of acyclic complexes as a triangulated subcategory. In the case of the homotopy category of finitely generated projective modules over an Iwanaga-Gorenstein ring, we show that the above quotient category are triangle equivalent to the stable module category of Cohen-Macaulay T 2 (R)-modules.1991 Mathematics Subject Classification. 18E30, 18G35, 16G99.

On the Recollements of Functor Categories

“…D b sg (S), that are defined in Theorem 3.4.3, resp. Theorem 3.3.1, to have a recollement It is easy to check that the inclusion functor ı :Gp-S −→ D b sg (S), sends stable tstructures (U , V) to (U , V ) and (V, W) to (V , W ).Hence by Corollary 1.13 of[28], the inclusion functor ı is a morphism between these recollements, i.e. we have the following commutative diagramRemark 3.4.7 Let u be a minimal object of S. Then a similar argument as in Theorem 3.4.3 works to yield the following recollement of stable categories 3.5 Gorenstein Defect Level Let A be an abelian category with enough projective objects.…”

“…Then there is the following recollement of derived categories of N -complexes Proof By Theorem 4.6, we have the following recollement Now result follows from Corollary 4.15 of [29].Remark 5.3.2 LetA be a Gorenstein ring. It is proved in[28, Theorem 4.8] that there is an equivalence between a quotient of homotopy categories and the stable category of Gorenstein projective modules. More precisely, they established an equivalenceGp-T 2 (A) K ∞,b (prj-A)/K b (prj-A)of triangulated categories, where K ∞,b (prj-A) denotes the full subcategory of K(prj-A) consisting of homologically bounded complexes.…”

Recollements of Singularity Categories and Monomorphism Categories

“…It is clearly an interesting idea to investigate recollement situations related to these triangulated categories. Existence of recollement has been studied recently for the stable category of the Gorensteinprojective modules over a Gorenstein triangular matrix algebra [15,32] and for the stable category of vector bundles [10].…”

n -representation-finite algebras and twisted fractionally Calabi-Yau algebras

“…The derived categories of finite-dimensional nonsemisimple algebras are never CY, but they are often fractionally CY. Some recent meetings [3,4,39,41] as well as recent results including [16,21,25,27,32,33,[35][36][37][38]42] suggest that the fractionally CY property becomes more and more important in representation theory, singularity theory, commutative and non-commutative algebraic geometry. The aim of this paper is to apply the fractionally CY property in the study of n-representation-finite algebras defined below.…”