Singularity Categories with Respect to Ding Projective Modules

Abstract We introduce the singularity category with respect to Ding projective modules, Db dpsg（R）, as the Verdier quotient of Ding derived category Db DP（R） by triangulated subcategory Kb（DP）, and give some triangle equivalences. Assume DP is precovering. We show that Db DP（R） ≌K-,dpb（DP） and Dbpsg（R） ≌ DbDdefect（R）. We prove that each R-module is of finite Ding projective dimension if and only if Dbdpsg（R） = 0.

Eventually homological isomorphisms and Gorenstein projective modules

We prove that a certain eventually homological isomorphism between module categories induces triangle equivalences between their singularity categories,Gorenstein defect categories and stable categories of Gorenstein projective modules.Furthermore,we show that the Auslander-Reiten conjecture and the Gorenstein symmetry conjecture can be reduced by eventually homological isomorphisms.Applying these results to arrow removal and vertex removal,we describe the Gorenstein projective modules over some non-monomial algebras and verify the Auslander-Reiten conjecture for certain algebras.

Finiteness Conditions and Relative Singularity Categories

We introduce the n-pure projective(resp.,injective)dimension of complexes in n-pure derived categories,and give some criteria for computing these dimensions in terms of the n-pure projective(resp.,injective)resolutions(resp.,coresolutions)and n-pure derived functors.As a consequence,we get some equivalent characterizations for the finiteness of n-pure global dimension of rings.Finally,we study Verdier quotient of bounded n-pure derived category modulo the bounded homotopy category of n-pure projective modules,which is called an n-pure singularity category since it can reflect the finiteness of n-pure global dimension of rings.

Derived Equivalences and Recollements of Differential Graded Algebras and Schemes

In this paper, we first prove for two differential graded algebras （DGAs） A, B which are derived equivalent to k-algebras A, F, respectively, that ：D（Ak B） ≈D（Ak Г）. In particular, Hp^b（Ak B） ≈ Hb（proj-A k Г）. Secondly, for two quasi-compact and sepa- rated schemes X, Y and two algebras A, B over k which satisfy ：D（Qcoh（X）） ≈：D（A） and ：D（Qcoh（Y）） ≈D（B）, we show that ：D（Qcoh（X × Y）） ≈ 79（AB） and ：Db（Coh（X × Y）） ≈Db（mod-（A B））. Finally, we prove that if X is a quasi-compact and separated scheme over k, then ：D（Qcoh（X ~ pl）） admits a recollement relative to D（Qcoh（X））, and we de- scribe the functors in the recollement explicitly. This recollement induces a recollement of bounded derived categories of coherent sheaves and a recollement of singularity categories. When the scheme X is derived equivalent to a DGA or algebra, then the recollement which we get corresponds to the recollement of DGAs in [14] or the recollement of upper triangular algebras in [7].

When the Schur functor induces a triangle-equivalence between Gorenstein defect categories

Let R be an Artin algebra and e be an idempotent of R. Assume that Tor_(i)^(eRe)(Re, G) = 0 for any G ∈ GprojeRe and i sufficiently large. Necessary and sufficient conditions are given for the Schur functor S_(e) to induce a triangle-equivalence ■. Combining this with a result of Psaroudakis et al.(2014),we provide necessary and sufficient conditions for the singular equivalence ■ to restrict to a triangle-equivalence ■. Applying these to the triangular matrix algebra ■,corresponding results between candidate categories of T and A(resp. B) are obtained. As a consequence,we infer Gorensteinness and CM(Cohen-Macaulay)-freeness of T from those of A(resp. B). Some concrete examples are given to indicate that one can realize the Gorenstein defect category of a triangular matrix algebra as the singularity category of one of its corner algebras.

Applications of balanced pairs

Let(X, Y) be a balanced pair in an abelian category. We first introduce the notion of cotorsion pairs relative to(X, Y), and then give some equivalent characterizations when a relative cotorsion pair is hereditary or perfect. We prove that if the X-resolution dimension of Y(resp. Y-coresolution dimension of X)is finite, then the bounded homotopy category of Y(resp. X) is contained in that of X(resp. Y). As a consequence, we get that the right X-singularity category coincides with the left Y-singularity category if the X-resolution dimension of Y and the Y-coresolution dimension of X are finite.

Piecewise Hereditary Triangular Matrix Algebras

For any positive integer N,we clearly describe all finite-dimensional algebras A such that the upper triangular matrix algebras TN(A)are piecewise hereditary.Consequently,we describe all finite-dimensional algebras A such that their derived categories of N-complexes are triangulated equivalent to derived categories of hereditary abelian categories,and we describe the tensor algebras A⊗K[X]/(X^(N))for which their singularity categories are triangulated orbit categories of the derived categories of hereditary abelian categories.