Related papers: On vanishing of certain Ext modules

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…

Let (R,m) be a complete Noetherian local ring and let M be a finite R--module of positive Krull dimension n. It is shown that any subset T of Assh_R(M) can be expressed as the set of attached primes of the top local cohomology module…

Let $\frak a$ denote an ideal in a commutative Noetherian ring $R$ and $M$ a finitely generated $R$-module. In this paper, we introduce the concept of the cohomological dimension filtration $\mathscr{M} =\{M_i\}_{i=0}^c$, where $ c={\rm cd}…

Let R be a commutative noetherian ring. In this paper, we study, for the singularity category of R, the vanishing of the complexity $\delta_t(X,Y)$ in the sense of Dimitrov, Haiden, Katzarkov and Kontsevich. We prove that the set of real…

We construct a class of Gorenstein local rings $R$ which admit minimal complete $R$-free resolutions $\bd C$ such that the sequence $\{\rank_R C_i\}$ is constant for $i< 0$, and grows exponentially for all $i>0$. Over these rings we show…

Assume $R$ is a local Cohen-Macaulay ring. It is shown that $\Ass_R (H^l_I(R))$ is finite for any ideal $I$ and any integer $l$ provided $\Ass_R (H^2_{(x,y)}(R))$ is finite for any $x,y\in R$ and $\Ass_R (H^3_{(x_1,x_2,y)}(R))$ is finite…

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…

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring, and suppose $R$ is Cohen-Macaulay with canonical module $\omega_R$. We develop new tools for analyzing questions involving annihilators of several homologically defined objects.…

Let $(R,\m)$ be a Noetherian local ring. Consider the notion of homological dimension of a module, denoted H-dim, for H= Reg, CI, CI$_*$, G, G$^*$ or CM. We prove that, if for a finite $R$-module $M$ of positive depth, $\Hd_R({\m}^iM)$ is…

Let R be a commutative noetherian ring. We prove that if R is either an equidimensional finitely generated algebra over a perfect field, or an equidimensional equicharacteristic complete local ring with a perfect residue field, then the…

Let $(R, \frak m)$ be a Noetherian local ring. This paper deals with the annihilator of Artinian local cohomology modules $H^i_{\frak m}(M)$ in the relation with the structure of the base ring $R$, for non negative integers $i$ and finitely…

Let $(R,\mathfrak{m})$ be a local Noetherian ring with residue field $k$. While much is known about the generating sets of reductions of ideals of $R$ if $k$ is infinite, the case in which $k$ is finite is less well understood. We…

It is proved that a module $M$ over a Noetherian ring $R$ of positive characteristic $p$ has finite flat dimension if there exists an integer $t\ge 0$ such that $\operatorname{Tor}_i^R(M, {}^{f^{e}}\!R)=0$ for $t\le i\le t+\dim R$ and…

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…

Let $(R,\frak{m})$ be a Noetherian local ring, $I$ an ideal of $R$ and $N$ a finitely generated $R$-module. Let $k{\ge}-1$ be an integer and $ r=\depth_k(I,N)$ the length of a maximal $N$-sequence in dimension $>k$ in $I$ defined by M.…

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring which contains a regular sequence $ \underline{x} = x_1,\ldots,x_d \in \mathfrak{m} \smallsetminus \mathfrak{m}^2 $ such that $ \mathfrak{m}^3 \subseteq (\underline{x}) $. Let $…

Motivated by work of C. U. Jensen, R.-O. Buchweitz, and H. Flenner, we prove the following result. Let $R$ be a commutative Noetherian ring and $a$ an ideal in the Jacobson radical of $R$. Let $\hat{R}^a$ be the $a$-adic completion of $R$.…

The paper investigates the non-vanishing of $H^1(E(n))$, where $E$ is a (normalized) rank two vector bundle over any smooth irreducible threefold $X$ of degree $d$ such that $Pic(X) \cong \ZZ$. If $\epsilon$ is the integer defined by the…

Let $(R,\fm)$ be a local ring, and let $C$ be a semidualizing complex. We establish the equality $r_R(Z) = \nu(\Ext^{g-\inf C}_R(Z,C))\mu^{\depth C}_R(\mathfrak{m}, C)$ for a homologically finite and bounded complex $Z$ with finite…

In this paper, we investigate the maximal Cohen-Macaulay property of tensor products of modules, and then give criteria for projectivity of modules in terms of vanishing of Ext modules. One of the applications shows that the…