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Nanqing Ding

Publications and source records attributed to Nanqing Ding.

6 recordsLinked to original sources
arXiv2022

The singularity category of an exact category applied to characterize Gorenstein schemes

We study singularity categories of exact categories with a focus on those associated to a complete hereditary cotorsion pair. As an application we identify a non-affine analogue of the singularity category of a Gorenstein local ring; with this Buchweitz's classic equivalence of three categories over Gorenstein local rings has been generalized to schemes, a project started by Murfet and Salarian more than ten years ago. As another application we use the framework to characterize rings of finite finitistic dimension.

math.KT↗
arXiv2021

A new characterization of silting subcategories in the stable category of a Frobenius extriangulated category

We give a new characterization of silting subcategories in the stable category of a Frobenius extriangulated category, generalizing the result of Di et al. (J. Algebra 525 (2019) 42-63) about the Auslander-Reiten type correspondence for silting subcategories over triangulated categories. More specifically, for any Frobenius extriangulated category $\mathcal{C}$, we establish a bijective correspondence between silting subcategories of the stable category $\underline{\mathcal{C}}$ and certain covariantly finite subcategories of $\mathcal{C}$. As a consequence, a characterization of silting subcategories in the stable category of a Frobenius exact category is given. This result is applied to homotopy categories over abelian categories with enough projectives, derived categories over Grothendieck categories with enough projectives as well as to the stable category of Gorenstein projective modules over a ring $R$.

math.RA↗
arXiv2020

Tate-Vogel and relative cohomologies of complexes with respect to cotorsion pairs

We study Tate-Vogel and relative cohomologies of complexes by applying the model structure induced by a complete hereditary cotorsion pair ($\A$, $\B$) of modules. We show first that the class of complexes admitting a complete $\A$ resolution is exactly the class of complexes with finite Gorenstein $\A$ dimension. This lets us give general techniques for computing Tate-Vogel cohomoloies of complexes with finite Gorenstein $\A$ dimension. As a consequence, relative cohomology groups for complexes with finite Gorenstein $\A$ dimension are investigated. Finally, the relationships between Gorenstein $\A$ dimensions and $\A$ dimensions for complexes are given.

math.RA↗
arXiv2020

Auslander-Buchweitz Approximation Theory for Extriangulated Categories

Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. In this paper, we introduce and develop an analogous theory of Auslander-Buchweitz approximations for extriangulated categories. We establish the existence of precovers pand preenvelopesq and obtain characterizations of relative homological dimensions, which are based on certain subcategories under finiteness of resolutions. Finally, we give a description of cotorsion pairs on extriangulated categories under some conditions, and provide a characterization of silting subcategories on stable categories. Keywords: Extriangulated category; Homological dimension; Cogenerator; Cotorsion pair.

math.CT↗
arXiv2019

Recollements associated to cotorsion pairs over upper triangular matrix rings

Let $A$, $B$ be two rings and $T=\left(\begin{smallmatrix} A & M 0 & B \end{smallmatrix}\right)$ with $M$ an $A$-$B$-bimodule. Given two complete hereditary cotorsion pairs $(\mathcal{A}_{A},\mathcal{B}_{A})$ and $(\mathcal{C}_{B},\mathcal{D}_{B})$ in $A$-Mod and $B$-Mod respectively. We define two cotorsion pairs $(Φ(\mathcal{A}_{A},\mathcal{C}_{B}), \mathrm{Rep}(\mathcal{B}_{A},\mathcal{D}_{B}))$ and $(\mathrm{Rep}(\mathcal{A}_{A},\mathcal{C}_{B}), Ψ(\mathcal{B}_{A},\mathcal{D}_{B}))$ in $T$-Mod and show that both of these cotorsion pairs are complete and hereditary. Given two cofibrantly generated model structures $\mathcal{M}_{A}$ and $\mathcal{M}_{B}$ on $A$-Mod and $B$-Mod respectively. Using the result above, we investigate when there exist a cofibrantly generated model structure $\mathcal{M}_{T}$ on $T$-Mod and a recollement of $\mathrm{Ho}(\mathcal{M}_{T})$ relative to $\mathrm{Ho}(\mathcal{M}_{A})$ and $\mathrm{Ho}(\mathcal{M}_{B})$. Finally, some applications are given in Gorenstein homological algebra.

math.CT↗