Related papers: On pure derived and pure singularity categories

This paper is a sequel to "T-structures and twisted complexes on derived injectives" by the same author with W. Lowen and M. Van den Bergh. We define a dg-category of unbounded twisted complexes on a dg-category, which is particularly…

We consider the derived category of permutation modules for a finite group, in positive characteristic. We stratify this tensor triangulated category using Brauer quotients. We describe the spectrum of its compact objects, by reducing the…

Let $(S, \n)$ be a commutative noetherian local ring and $\omega\in\n$ be non-zerodivisor. This paper deals with the behavior of the category $\mon(\omega, \cp)$ consisting of all monomorphisms between finitely generated projective…

In this paper, we characterize several properties of commutative notherian local rings in terms of the left perpendicular category of the category of finitely generated modules of finite projective dimension. As an application we prove that…

It is now very known how the subprojectivity of modules provides a fruitful new unified framework of the classical projectivity and flatness. In this paper, we extend this fact to the category of complexes by generalizing and unifying…

We study the canonical orbit category of the bounded derived category of finite dimensional representations of the quiver of type $D_{\infty}$. We prove that this orbit category is a cluster category, that is, its cluster-tilting…

We describe all possible universal localisations of a hereditary ring in terms of suitable full subcategories of the category of finitely presented modules. For these universal localisations we then identify the category of finitely…

We classify the module categories over the double (possibly twisted) of a finite group.

Let $R$ be a ring. An $R$-module $M$ is said to be an absolutely $w$-pure module if and only if $\Ext^1_R(F,M)$ is a GV-torsion module for any finitely presented module $F$. In this paper, we introduce and study the concept of…

We study the properties of the relative derived category $D_{\mathscr{C}}^{b}$($\mathscr{A}$) of an abelian category $\mathscr{A}$ relative to a full and additive subcategory $\mathscr{C}$. In particular, when $\mathscr{A}=A{\text -}\mod$…

Let $\mathscr{A}$ be a small abelian category. For a closed subbifunctor $F$ of $\Ext_{\mathscr{A}}^{1}(-,-)$, Buan has generalized the construction of the Verdier's quotient category to get a relative derived category, where he localized…

Buchweitz related the singularity category of a (strongly) Gorenstein ring and the stable category of maximal Cohen-Macaulay modules by a triangle equivalence. We phrase his result in a relative categorical setting based on N-complexes…

We analyze Auslander-Reiten components for the bounded derived category of a finite-dimensional algebra. We classify derived categories whose Auslander-Reiten quiver has either a finite stable component or a stable component with finite…

We compare closed and rigid monoidal categories. Closedness is defined by the tensor product having a right adjoint: the internal hom functor. Rigidity, on the other hand, generalises the duality of finite-dimensional vector spaces. In the…

Our aim is to give a fairly complete account on the construction of compatible model structures on exact categories and symmetric monoidal exact categories, in some cases generalizing previously known results. We describe the close…

In this note, we extend the quasi-projective dimension of finite (that is, finitely generated) modules to homologically finite complexes, and we investigate some of homological properties of this dimension.

Originally a technical tool, the derived category of coherent sheaves over an algebraic variety has become over the last twenty years an important invariant in the birational study of algebraic varieties. Problems of birational invariance…

For any ring R we construct two triangulated categories, each admitting a functor from R-modules that sends projective and injective modules to 0. When R is a quasi-Frobenius or Gorenstein ring, these triangulated categories agree with each…

Let $\Mod \CS$ denote the category of $\CS$-modules, where $\CS$ is a small category. In the first part of this paper, we provide a version of Rickard's theorem on derived equivalence of rings for $\Mod \CS$. This will have several…

We give a detailed proof of the following fundamental result: the singularity category of a ring is triangle equivalent to the stabilization of its stable module category. The result yields singular equivalences between rings of different…