Related papers: Gorenstein-projective and semi-Gorenstein-projecti…
We prove that for a Frobenius extension, a module over the extension ring is Gorenstein projective if and only if its underlying module over the base ring is Gorenstein projective. For a separable Frobenius extension between Artin algebras,…
This thesis is comprised of three chapters. The first chapter deals with bounded complexes of Gorenstein projective and Gorenstein injective modules. Deploying methods of relative homological algebra, we approximate such complexes with…
In this paper, it is proved that a commutative noetherian local ring admitting a finitely generated module of finite projective and injective dimensions with respect to a semidualizing module is Gorenstein. This result recovers a celebrated…
Given a Frobenius category $\mathcal{F}$ satisfying certain finiteness conditions, we consider the localization of its Hall algebra $\mathcal{H(F)}$ at the classes of all projective-injective objects. We call it the {\it "semi-derived Hall…
We consider a finite acyclic quiver $\mathcal{Q}$ and a quasi-Frobenius ring $R$. We endow the category of quiver representations over $R$ with a model structure, whose homotopy category is equivalent to the stable category of…
The principle "Every result in classical homological algebra should have a counterpart in Gorenstein homological algebra" is given in [3]. There is a remarkable body of evidence supporting this claim (cf. [2] and [3]). Perhaps one of the…
Let $A$ and $B$ be rings, $U$ a $(B,A)$-bimodule and $T=\begin{pmatrix} A&0\\U&B \end{pmatrix}$ the triangular matrix ring. In this paper, several notions in relative Gorenstein algebra over a triangular matrix ring are investigated. We…
We call a finite-dimensional K-algebra A geometrically irreducible if for all d, all connected components of the affine scheme of d-dimensional A-modules are irreducible. We show that the geometrically irreducible algebras without loops…
A well-known conjecture says that every one-relator group is coherent. We state and partly prove an analogous statement for graded associative algebras. In particular, we show that every Gorenstein algebra $A$ of global dimension 2 is…
In this paper, we study a particular case of Gorenstein projective, injective, and flat modules, which we call, respectively, strongly Gorenstein projective, injective, and flat modules. These last three modules give us a new…
This paper generalize the idea of the authors in \cite{Bennis and Mahdou1}. Namely, we define and study a particular case of modules with Gorenstein projective, injective, and flat dimension less or equal than $n\geq 0$, which we call,…
We give an example of a finite dimensional algebra with infinite delooping level, based on an example of a semi-Gorenstein-projective module due to Ringel and Zhang.
Let $A$ be a finite-dimensional local algebra over an algebraically closed field, let $J$ be the radical of $A.$ The modules we are interested in are the finitely generated left $A$-modules. Projective modules are always reflexive, and an…
We generalize the monomorphism category from quiver (with monomial relations) to arbitrary finite dimensional algebras by a homological definition. Given two finite dimension algebras $A$ and $B$, we use the special monomorphism category…
Invariants with respect to recollements of the stable category of Gorenstein projective A-modules over an algebra A and stable equivalences are investigated. Specifically, the Gorenstein rigidity dimension is introduced. It is shown that…
Let $A$ and $B$ be rings, $U$ a $(B, A)$-bimodule and $T=\left(\begin{smallmatrix} A & 0 \\ U & B \\\end{smallmatrix}\right)$ be the triangular matrix ring. In this paper, we characterize the Gorenstein homological dimensions of modules…
Let $V$ be a finite-dimensional positively-graded vector space. Let $b \in V \otimes V$ be a homogeneous element whose rank is $\text{dim}(V)$. Let $A=TV/(b)$, the quotient of the tensor algebra $TV$ modulo the 2-sided ideal generated by…
In the paper, we focus on the silting properties and the combinatorial properties of silting and Gorenstein, which is called Gorenstein silting, where the main tools used are recollements of module categories and tensor products. For a ring…
We explore questions of projectivity and tensor products of modules for finite dimensional Hopf algebras. We construct many classes of examples in which tensor powers of nonprojective modules are projective and tensor products of modules in…
In this paper, we examine the class of cofibrant modules over a group algebra $kG$, that were defined by Benson in [2]. We show that this class is always the left-hand side of a complete hereditary cotorsion pair in the category of…