Wei Ren

Chongqing Normal University | CNU · School of Mathematical Sciences

We prove that any faithful Frobenius functor between abelian categories preserves the Gorenstein projective dimension of objects. Consequently, it preserves and reflects Gorenstein projective objects. We give conditions on when a Frobenius functor preserves the stable categories of Gorenstein projective objects, the singularity categories and the G...

We study homotopy categories of model categories arising from a cotorsion triple, and the equivalences between corresponding stable categories. We characterize homological dimensions with respect to a cotorsion triple. Then, we lift cotorsion triple to complexes, and get the equivalence of homotopy categories of complexes via Quillen equivalence of...

Let $R\subset A$ be a Frobenius extension of rings. We prove that: (1) for any left $A$-module $M$, $_{A}M$ is Gorenstein projective (injective) if and only if the underlying left $R$-module $_{R}M$ is Gorenstein projective (injective). (2) if $\mathrm{G}\text{-}\mathrm{proj.dim}_{A}M<\infty$, then $\mathrm{G}\text{-}\mathrm{proj.dim}_{A}M = \mathr...

We prove that for a Frobenius extension, if a module over the extension ring is Gorenstein projective, then its underlying module over the the base ring is Gorenstein projective; the converse holds if the Frobenius extension is either left-Gorenstein or separable (e.g. the integral group ring extension $\mathbb{Z}\subset \mathbb{Z}G$). Moreover, fo...

Let CC be a triangulated category with a proper class EE of triangles. Asadollahi and Salarian introduced and studied EE-GGprojective and EE-GGinjective objects, and developed a relative homological algebra in CC. In this paper, we further study Gorenstein homological dimensions for triangulated categories. First, we discuss the finiteness of EE-GG...

For any group $G$ and any commutative ring $R$, the Gorenstein homological dimension ${\rm Ghd}_RG$, defined as the Gorenstein flat dimension of the trivial $RG$-module $R$, is characterized. We prove that ${\rm Ghd}_RG < \infty$ if and only if Gorenstein flat dimension of any $RG$-module is finite, whenever the supremum of flat dimension of inject...

Owing to the difference in K-theory, an example by Dugger and Shipley implies that the equivalence of stable categories of Gorenstein projective modules should not be a Quillen equivalence. We give a sufficient and necessary condition for the Frobenius pair of faithful functors between two abelian categories to be a Quillen equivalence, which is al...

We prove that any faithful Frobenius functor between abelian categories preserves the Gorenstein projective dimension of objects. Consequently, it preserves and reflects Gorenstein projective objects. We give conditions on when a Frobenius functor preserves the stable categories of Gorenstein projective objects, the singularity categories and the G...

First we study the Gorenstein cohomological dimension ${\rm Gcd}_RG$ of groups $G$ over coefficient rings $R$, under changes of groups and rings; a Gorenstein version of Serre's theorem is proved, which gives an equality between Gorenstein cohomological dimensions of a group and the subgroup with finite index; characterizations for finite groups ar...

Let $G$ be a group and $R$ a commutative ring. We define the Gorenstein homological dimension of $G$ over $R$, denoted by ${\rm Ghd}_{R}G$, as the Gorenstein flat dimension of trivial $RG$-module $R$. It is proved that for any flat extension $R\rightarrow S$ of rings, ${\rm Ghd}_SG \leq {\rm Ghd}_RG$; in particular, ${\rm Ghd}_{R}G$ is a refinement...

Owing to the difference in $K$-theory, an example by Dugger and Shipley implies that the equivalence of stable categories of Gorenstein projective modules should not be a Quillen equivalence. We give a sufficient and necessary condition such that the Frobenius pair of faithful functors between two abelian categories is a Quillen equivalence, which...

In this corrigendum, a typo in Proposition 3.2 is fixed, and an additional assumption is added for Theorems 3.3, 3.4, and Corollary 3.5: the Frobenius extension R⊂A of rings should be split (i.e., R is a direct summand of A as an R-bimodule). Some examples of split Frobenius extensions are given, which imply that this additional assumption is not t...

Firstly, we compare the bounded derived categories with respect to the pure-exact and the usual exact structures, and describe bounded derived category by pure-projective modules, under a fairly strong assumption on the ring. Then, we study Verdier quotient of bounded pure derived category modulo the bounded homotopy category of pure-projective mod...

Firstly, we compare the bounded derived categories with respect to the pure-exact and the usual exact structures, and describe bounded derived category by pure-projective modules, under a fairly strong assumption on the ring. Then, we study Verdier quotient of bounded pure derived category modulo the bounded homotopy category of pure-projective mod...

We prove that the weak equivalences, cofibrations and fibrations in Gillespie’s flat model structure are invariant under excellent extensions of rings. As applications, the flat and cotorsion dimensions of unbounded complexes along excellent extensions are investigated, and moreover, some homological invariant properties under excellent extensions...

Let $R$ be a left-Gorenstein ring. We construct a Quillen equivalence between singular contraderived model category and singular coderived model category introduced by Becker [Adv. Math., (2014) 187-232]. As an application, we explicitly give an equivalence $\mathbf{K}_{ex}(\mathcal{P})\simeq \mathbf{K}_{ex}(\mathcal{I})$ for the homotopy categorie...

Let V,W be two classes of modules. In this paper, we introduce and study VW-Gorenstein complexes as a common generalization of W-complexes, Gorenstein projective (resp., Gorenstein injective) complexes, and GC - projective (resp., GC -injective) complexes. It is shown that under certain hypotheses a complex X is VW-Gorenstein if and only if each Xⁿ...

Let C be a triangulated category with a proper class E of triangles. We prove that there exists an Avramov–Martsinkovsky type exact sequence in C, which connects E-cohomology, E-Tate cohomology and E-Gorenstein cohomology.

In this article, we investigate the stability of Cartan-Eilenberg Gorenstein categories. To this end, we introduce and study the concept of two-degree Cartan-Eilenberg W-Gorenstein complexes. We prove that a complex C is two-degree Cartan-Eilenberg W-Gorenstein if and only if C is Cartan-Eilenberg W-Gorenstein. As applications, we show that a compl...

The notion of relative derived category with respect to a subcategory is introduced. A triangle-equivalence, which extends a theorem of Gao and Zhang [Gorenstein derived categories, \emph{J. Algebra} \textbf{323} (2010) 2041-2057] to the bounded below case, is obtained. Moreover, we interpret the relative derived functor $\mathrm{Ext}_{\mathcal{X}\...

Let C be a triangulated category and Ɛ a proper class of triangles. We show that Tate cohomology in triangulated category is balanced, i.e. there is an isomorphism (Formula presented.) for any integer i∈ℤ, where the first cohomology group is computed by complete Ɛ-projective resolution for A∈C and the second one is computed by complete Ɛ-injective...

Let R → S be a ring homomorphism. We consider the relationships of the Gorenstein dimensions of an R-complex X (possibly unbounded) with those of the S-complexes RHomR (S, X) and [Inline formula]. More generally, the Gorenstein injective dimension of RHomR (U, X) is considered where U is an S-complex with finite projective dimension. As an applicat...

We extend the cotorsion dimension of R-modules to unbounded R-complexes by applying the flat model structure on Ch(R) proposed by J. Gillespie. This is not natural because there has been no sufficiently general result available for the existence of proper “cotorsion” resolutions of unbounded complexes, for which one would be able to define the deri...

We introduce the relative derived categories with respect to Ding modules, and give the relation with derived and Gorenstein derived categories. Especially, a triangle-equivalence of Ding derived categories over Ding-Chen rings and some applications are obtained. Generalized Tate cohomology with respect to Ding projective and Ding injective modules...