Related papers: Relative global dimensions and stable homotopy cat…
Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $\xi$ of $\mathbb{E}$-triangles. The authors introduced and studied $\xi$-$\mathcal{G}$projective and $\xi$-$\mathcal{G}$injective in \cite{HZZ}.…
Let \fa be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Let \cd_{\fa}(M,N) denote the supremum of the i's such that H^i_{\fa}(M,N)\neq 0. First, by using the theory of Gorenstein homological…
We introduce the notion of relative singularity category with respect to any self-orthogonal subcategory $\omega$ of an abelian category. We introduce the Frobenius category of $\omega$-Cohen-Macaulay objects, and under some reasonable…
A semi-dualizing module over a commutative noetherian ring A is a finitely generated module C with RHom_A(C,C) \simeq A in the derived category D(A). We show how each such module gives rise to three new homological dimensions which we call…
Let $R$ be a ring with identity and $\C(R)$ denote the category of complexes of $R$-modules. In this paper we study the homotopy categories arising from projective (resp. injective) complexes as well as Gorenstein projective (resp.…
We define, via Gorenstein homomorphisms, a class of local rings over which there exist non-trivial totally reflexive modules. We also provide a general construction of such rings, which indicates their abundance.
In this paper we develop the homological properties of the $(\mathcal{L}, \mathcal{A})$-Gorenstein flat $R$-modules $\mathcal{GF}_{(\mathcal{F}(R), \mathcal{A})}$ proposed by Gillespie. Where the class $\mathcal{A} \subseteq \mathrm{Mod}…
We introduce a refinement of the Gorenstein flat dimension for complexes over an associative ring--the Gorenstein flat-cotorsion dimension--and prove that it, unlike the Gorenstein flat dimension, behaves as one expects of a homological…
We expand on two existing characterizations of rings of Gorenstein (weak) global dimension zero and give two new characterizations of rings of finite Gorenstein (weak) global dimension. We also include the answer to a question of Y.~Xiang…
We introduce the notion of balanced pair of additive subcategories in an abelian category. We give sufficient conditions under which the balanced pair of subcategories gives rise to equivalent homotopy categories of complexes. As an…
For a Cohen-Macaulay ring $R$, we exhibit the equivalence of the bounded derived categories of certain resolving subcategories, which, amongst other results, yields an equivalence of the bounded derived category of finite length and finite…
We investigate the relationship between smoothness and the relative global dimension of a ring extension. We prove that a smooth commutative algebra $A$ over $B$ has finite relative global dimension $\text{gdim}(A,B)$. Conversely, under a…
We present and study the concept of $m$-periodic Gorenstein objects relative to a pair $(\mathcal{A,B})$ of classes of objects in an abelian category, as a generalization of $m$-strongly Gorenstein projective modules over associative rings.…
It is well-known that for a large class of local rings of positive characteristic, including complete intersection rings, the Frobenius endomorphism can be used as a test for finite projective dimension. In this paper, we exploit this…
In this paper, the notion of strongly G_C-projective and injective modules is introduced, where C is a semidualizing module. Using these modules we can obtain a new characterization of G_C-projective and injective modules, similar to the…
We show that an iteration of the procedure used to define the Gorenstein projective modules over a commutative ring $R$ yields exactly the Gorenstein projective modules. Specifically, given an exact sequence of Gorenstein projective…
The aim of this article is to consider the spectral sequences induced by tensor-hom adjunction, and provide a number of new results. Let $R$ be a commutative Noetherian local ring of dimension $d$. In the 1st part, it is proved that $R$ is…
We describe the endomorphism rings of maximal rigid objects in the cluster categories of tubes. Moreover, we show that they are gentle and have Gorenstein dimension 1. We analyse their representation theory and prove that they are of finite…
Let $G$ be a finite group acting on a ring $R$ and $H$ a subgroup of $G$. In this paper we compare some homological dimensions over the skew group rings $RG$ and $RH$. Moreover, under the assumption that $RG$ is a separable extension over…
In this paper we obtain a description of the Grothendieck group of complex vector bundles over the classifying space of a p-local finite group in terms of representation rings of subgroups of its Sylow. We also prove a stable elements…