Related papers: Rigid modules and homological dimensions
Our purpose in this work is multifold. First, we provide general criteria for the finiteness of the projective and injective dimensions of a finite module $M$ over a (commutative) Noetherian ring $R$. Second, in the other direction, we…
Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings. As opposed to their classical…
We explore the implications of the finiteness of homological dimensions for Ext modules, focusing on projective dimension, injective dimension, and their Gorenstein counterpart. In this direction, we establish several finiteness criteria…
We prove rigidity type results on the vanishing of stable (co)homology for modules of finite complete intersection dimension, results which generalize and improve upon known results. We also introduce a notion of pre-rigidity, which…
We obtain various characterizations of commutative Noetherian local rings $(R, \fm)$ in terms of homological dimensions of certain finitely generated modules. For example, we establish that $R$ is Gorenstein if the Gorenstein injective…
We study homological properties of test modules that are, in principle, modules that detect finite homological dimensions. The main outcome of our results is a generalization of a classical theorem of Auslander and Bridger: we prove that,…
In this paper we study homological dimensions of finitely generated modules over commutative Noetherian local rings, called reducing homological dimensions. We obtain new characterizations of Gorenstein and complete intersection local rings…
In this paper, we first study the Gorenstein projective/flat dimension of complexes of modules. The relation between the Gorenstein projective/flat dimension for complexes and that for modules are investigated. Then we study Tate, stable…
We investigate modules for which vanishing of Tor-modules implies finiteness of homological dimensions (e.g., projective dimension and G-dimension). In particular, we answer a question of O. Celikbas and Sather-Wagstaff about ascent…
Starting from the notion of totally reflexive modules, we survey the theory of Gorenstein homological dimensions for modules over commutative rings. The account includes the theory's connections with relative homological algebra and with…
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…
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…
A central problem in the theory of Gorenstein dimensions over commutative noetherian rings is to find resolution-free characterizations of the modules for which these invariants are finite. Over local rings, this problem was recently solved…
It is proved that if one of the finite modules M and N, over a local ring R, has reducible complexity and has finite Gorenstein dimension then the depth formula holds, provided TorR_i(M,N) = 0 for i>>0. We also study the vanishing of…
For finitely generated modules $M$ and $N $ over a commutative Noetherian local ring $R$, we give various sufficient criteria for detecting freeness of $M$ or $N$ via vanishing of some finitely many Ext modules $\textrm{Ext}^i_R(M,N)$ and…
We prove versions of results of Foxby and Holm about modules of finite (Gorenstein) injective dimension and finite (Gorenstein) projective dimension with respect to a semidualizing module. We also verify special cases of a question of…
We study rings which have Noetherian cohomology under the action of a ring of cohomology operators. The main result is a criterion for a complex of modules over such a ring to have finite injective dimension. This criterion generalizes, by…
A new homological dimension is introduced to measure the quality of resolutions of `singular' finite dimensional algebras (of infinite global dimension) by `regular' ones (of finite global dimension). Upper bounds are established in terms…
We study the vanishing of (co)homology along ring homomorphisms for modules that admit certain filtrations, and generalize a theorem of O. Celikbas-Takahashi. Our work produces new classes of rigid and test modules, in particular over local…
In this paper, we explore the implications of the finiteness of complete intersection dimensions for RHom complexes and Ext modules. We prove various stability results and criteria for detecting finite complete intersection homological…