Related papers: Eventually homological isomorphisms and Gorenstein…

Let $\Lambda$ be a left and right noetherian ring. First, for $m,n\in\mathbb{N}\cup\{\infty\}$, we give equivalent conditions for a given $\Lambda$-module to be $n$-torsionfree and have $m$-torsionfree transpose. Using them, we investigate…

We investigate the problem when the tensor functor by a bimodule yields a singular equivalence. It turns out that this problem is equivalent to the one when the Hom functor given by the same bimodule induces a triangle equivalence between…

For a tensor ring $T_R(M)$, under certain conditions, we characterize the Gorenstein projective modules over $T_R(M)$, and prove that a $T_R(M)$-module $(X,u)$ is Gorenstein projective if and only if $u$ is monomorphic and ${\rm coker}(u)$…

We show that recollements of module categories give rise to homomorphisms between the associated Hochschild cohomology algebras which preserve the strict Gerstenhaber structure, i.e., the cup product, the graded Lie bracket and the squaring…

We associate to a localizable module a left retraction of algebras; it is a homological ring epimorphism that preserves singularity categories. We study the behavior of left retractions with respect to Gorenstein homological properties (for…

Let a be an ideal of a commutative Noetherian ring R with identity. We study finitely generated R-modules M whose a-finiteness and a-cohomological dimensions are equal. In particular, we examine relative analogues of quasi-Buchsbaum,…

It is proved that for a commutative noetherian ring with dualizing complex the homotopy category of projective modules is equivalent, as a triangulated category, to the homotopy category of injective modules. Restricted to compact objects,…

In this paper we show that Gorensteinness, singularity categories and the finite generation condition Fg for the Hochschild cohomology are invariants under the arrow removal operation for a finite dimensional algebra.

For a finite dimensional algebra $\Lambda$ and a non-negative integer $n$, we characterize when the set $\tilt_n\Lambda$ of additive equivalence classes of tilting modules with projective dimension at most $n$ has a minimal (or…

Hochster's Monomial Conjecture and Canonical Element Conjecture date back some thirty resp. twenty years. They concern all noetherian commutative local rings, and were proved by their originator right away in equal characteristic. Thanks to…

We study criteria for a ring - or more generally, for a small category - to be Gorenstein and for a module over it to be of finite projective dimension. The goal is to unify the universal coefficient theorems found in the literature and to…

Given an artin algebra $\Lambda$ with an idempotent element $a$ we compare the algebras $\Lambda$ and $a\Lambda a$ with respect to Gorensteinness, singularity categories and the finite generation condition Fg for the Hochschild cohomology.…

We consider the equivalence from the stable module category to a subcategory $\mathcal{L}_A$ of the homotopy category constructed by Kato. This equivalence induces a correspondence between distinguished triangles in the homotopy category…

Let $\mathscr{A}$ be an abelian category having enough projective and injective objects, and let $\mathscr{T}$ be an additive subcategory of $\mathscr{A}$ closed under direct summands. A known assertion is that in a short exact sequence in…

We give sufficient conditions which ensure that a functor of finite length from an additive category to finite-dimensional vector spaces has a projective resolution whose terms are finitely generated. For polynomial functors, we study also…

In this paper we investigate homologically finite-dimensional objects in the derived category of a given small dg-enhanced triangulated category. Using these we define reflexivity, hfd-closedness, and the Gorenstein property for…

Gorenstein homological algebra is a kind of relative homological algebra which has been developed to a high level since more than four decades. In this report we review the basic theory of Gorenstein homological algebra of artin algebras.…

We study the projective dimension of finitely generated modules over cluster-tilted algebras End(T) where T is a cluster-tilting object in a cluster category C. It is well-known that all End(T)-modules are of the form Hom(T,M) for some…

In this article we investigate the relations between the Gorenstein projective dimensions of $\Lambda$-modules and their socles for minimal n-Auslander-Gorenstein algebras $\Lambda$ in the sense of Iyama and Solberg \cite{IS}. First we give…

We study and relate categories of modules, comodules and contramodules over a representation of a small category taking values in (co)algebras, in a manner similar to modules over a ringed space. As a result, we obtain a categorical…