Recollements of Singularity Categories and Monomorphism Categories

Pin, Liu; Lu, Ming

doi:10.1080/00927872.2014.894048

Cited by 18 publications

(17 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…P. Zhang proved that if Λ is Gorenstein and A M is projective, then GprojΛ admits a recollement relative to GprojA and GprojB. The author together with P. Liu [25] generalized this to consider the singularity categories, proved that D sg (Λ) admits a recollement relative to D sg (A) and D sg (B) if proj. dim A M < ∞ and proj.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Gorenstein defect categories of triangular matrix algebras

2017

Journal of Algebra

Self Cite

View full text Add to dashboard Cite

Abstract. We apply the technique of recollement to study the Gorenstein defect categories of triangular matrix algebras. First, we construct a left recollement of Gorenstein defect categories for a triangular matrix algebra under some conditions, using it, we give a categorical interpretation of the Gorenstein properties of the triangular matrix algebra obtained by X-W. Chen, B. L. Xiong and P. Zhang respectively. Second, under some additional conditions, a recollement of Gorenstein defect categories for a triangular matrix algebra is constructed. As an application, for a special kind of triangular matrix algebras, which are called simple gluing algebras, we describe their singularity categories and Gorenstein defect categories.

show abstract

Section: Introductionmentioning

confidence: 99%

“…dim M B < ∞ for any Artin rings A and B. In [25], the machinery of localization to the three categories in a recollement is introduced, and some sufficient conditions for the quotient categories to form a new recollement are found. We refer the reader to [24] for localization theory of triangulated categories.…”

Section: Introductionmentioning

confidence: 99%

Gorenstein defect categories of triangular matrix algebras

2017

Journal of Algebra

Self Cite

View full text Add to dashboard Cite

show abstract

“…As a corollary of Theorem 4.16 in [20], we get the following theorem: Theorem 4.19. Let X be a finite-dimensional Noetherian separated scheme over a field k. Then there exists a commutative diagram of recollements:…”

Section: Lemma 418mentioning

confidence: 91%

Derived Equivalences and Recollements of Differential Graded Algebras and Schemes

Chen

2016

Algebra Colloq.

Self Cite

View full text Add to dashboard Cite

In this paper, we first prove for two differential graded algebras (DGAs) A, B which are derived equivalent to k-algebras Λ, Γ, respectively, that D(A⊗ k B) ≃ D(Λ⊗ k Γ).Secondly, for two quasi-compact and separated schemes X, Y and two algebras A, B over k which satisfy D(Qcoh(X)) ≃ D(A) and). Finally, we prove that if X is a quasi-compact and separated scheme over k, then D(Qcoh(X × P 1 )) admits a recollement relative to D(Qcoh(X)), and we describe 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].for tensor products of algebras. Precisely, given a commutative ring R, let Λ be an R-algebra with a tilting complex P * whose endomorphism algebra is isomorphic to Γ and let Λ 0 be an R-algebra with a tilting complex Q * whose endomorphism algebra is isomorphic to Γ 0 . IfBeilinson [1] and Bernstein and Gelfand's [3] used derived categories to establish a beautiful relation between coherent sheaves on projective spaces and representations of certain finite-dimensional algebras. Their constructions had numerous generalizations (see [8,15]). It need very strong conditions for a variety or scheme to be derived equivalent to an algebra. But in [5], Bondal and Van den Bergh showed that a quasi-compact and separated scheme can be derived equivalent to a differential graded algebra (abbreviated below to DGA). Furthermore, the derived categories of DGAs, or more generally, DG categories, provide a conceptual framework for tilting theory (see [16,17]). So we first consider how derived equivalences occur for tensor products of DGAs which generalizes Rickard's result. We get Q)). Subsequently, we prove the following theorem. Proposition 1.1. Let A and B be two DG k-algebras. If P and Q are compact generators of D(A) and D(B), respectively, then P ⊗ k Q is a compact generator of D(A ⊗ k B). In particular, D(A ⊗ k B) ≃ D(HomTheorem 1.2. Let A and B be two DG k-algebras, and Λ and Γ be two k-algebras. If D(A) ≃ D(Λ) and D(B) ≃ D(Γ), then D(A ⊗Corresponding to every scheme X, there are two very important abelian categories, Coh(X) of coherent sheaves and Qcoh(X) of quasi-coherent sheaves, and then their derived categories. Parallel to the tensor products of algebras or DGAs, we consider the products of schemes and get: Theorem 1.3. Let X, Y be two quasi-compact and separated schemes over a field k and A, B two algebras also over k.Besides derived equivalences, recollements are also important in the study of algebraic geometry and representation theory. A recollement of triangulated categories is a diagram

show abstract

“…=Im F, and they called it the Gorenstein defect category of A-Mod: Nowadays, singularity category and related topic has been studied by many authors, see for example [13,14,15,27,28,29,33,35].…”

Section: Appendix a Triangle-equivalences Associative To Gorenstein Projective Dimension For Complexesmentioning

confidence: 99%

“…where these six functors are the derived versions of those as defined in Lemma 4.1 (1). Liu and Lu [28] show that if pd R M < 1, then (1.1) In order to solve this question, we first give an explicit description for Gorenstein projective T-modules over the triangular matrix ring T. We get the following: We note that the above theorem generalizes Theorem 3.5 in [19] to a more general case, whereas the authors therein assume the ring R to be left Gorenstein regular.…”

Section: Introductionmentioning

confidence: 99%

Gorenstein projective modules and recollements over triangular matrix rings

Zheng

et al. 2020

Communications in Algebra

View full text Add to dashboard Cite

R-S-bimodule. We describe Gorenstein projective modules over T. In particular, we refine a result of Enochs, Cort es-Izurdiaga, and Torrecillas [Gorenstein conditions over triangular matrix rings, J. Pure Appl. Algebra 218 (2014), no. 8, 1544-1554]. Also, we consider when the recollement of D b T-Mod ð Þrestricts to a recollement of its subcategory D b T-Mod ð Þ fgp consisting of complexes with finite Gorenstein projective dimension. As applications, we obtain recollements of the stable category T-GProj and recollements of the Gorenstein defect category D def T-Mod ð Þ : ARTICLE HISTORY

show abstract

Recollements of Singularity Categories and Monomorphism Categories

Abstract: We generalize results on existence of recollement situations of singularity categories of lower triangular Gorenstein algebras and stable monomorphism categories of CohenMacaulay modules.

Cited by 18 publications

References 27 publications

Gorenstein defect categories of triangular matrix algebras

Gorenstein defect categories of triangular matrix algebras

Derived Equivalences and Recollements of Differential Graded Algebras and Schemes

Gorenstein projective modules and recollements over triangular matrix rings

Contact Info

Product

Resources

About