Related papers: A New Outlook on Cofiniteness

For a flat commutative $k$-algebra $A$ such that the enveloping algebra $A\otimes_k A$ is noetherian, given a finitely generated bimodule $M$, we show that the adic completion of the Hochschild cohomology module $HH^n(A/k,M)$ is naturally…

Let $R$ be a commutative noetherian ring, $\fa$ an ideal of $R$ and $M,N$ finite $R$--modules. We prove that the following statements are equivalent. \begin{enumerate} \item[(i)] $\lc^{i}_{\fa}(M,N)$ is finite for all $i< n$. \item[(ii)]…

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.…

In this paper, we show that for an $F$-pure local ring $(R,\m)$, all local cohomology modules $H_{\m}^i(R)$ have finitely many Frobenius compatible submodules. This answers positively an open question raised by F.Enescu and M.Hochster. We…

Let $(R, {\frak m})$ be a local ring, $I$ a proper ideal of $R$ and $M$ a finitely generated $R$-module of dimension $d$. We discuss the local homology modules of $H^d_I(M)$. When $M$ is Cohen-Macaulay, it is proved that $H^d_{{\frak…

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,…

Let T be a commutative Noetherian local ring of dimension at least two and R=T[x_1,...,x_n] a polynomial ring in n variables over T. Consider R as a graded ring with deg T = 0 and deg x_i = 1 for all i. Let I=R_+ and f a homogeneous…

Let $R$ be a commutative Noetherian ring, $\Phi$ a system of ideals of $R$ and $M,X$ two $R$-modules. In this paper, we study the Artinianness and cofiniteness of the module $H^i_{\Phi}(M,X)$ which is an extension of the generalized local…

Let R be a regular, local and F-finite ring defined over a field of finite characteristic. Let I be an ideal of height c with normal quotient $A=R/I$. It is shown that the local cohomology module H^c_I(R) contains a unique simple…

Let $R$ be a commutative Noetherian local ring and let $\fa$ be a proper ideal of $R$. A non-zero finitely generated $R$-module $M$ is called relative Cohen-Macaulay with respect to $\fa$ if there is precisely one non vanishing local…

Let R be a (not necessarily local) Noetherian ring and M a finitely generated R-module of finite dimension d. Let \fa be an ideal of R and \fM denote the intersection of all prime ideals \fp in Supp_RH^d_{\fa}(M). It is shown that…

Let $\fa$ be an ideal of a commutative Noetherian ring $R$ with identity and let $M$ and $N$ be two finitely generated $R$-modules. Let $t$ be a positive integer. It is shown that $\Ass_R(H_{\fa}^t(M,N))$ is contained in the union of the…

Let $(R,\frak m)$ be a commutative noetherian local ring. In this paper, we prove that if $\frak m$ is decomposable, then for any finitely generated $R$-module $M$ of infinite projective dimension $\frak m$ is a direct summand of (a direct…

Let $R$ be a noetherian commutative ring, and \[ \mathbb F: ...\rightarrow F_2\rightarrow F_1\rightarrow F_0\rightarrow 0 \] a complex of flat $R$-modules. We prove that if $\kappa(\mathfrak p)\otimes_R\mathbb F$ is acyclic for every…

Let $R = \bigoplus_{n \in \mathbb{N}_0} R_n$ be a Noetherian homogeneous ring with local base ring $(R_0, \mathfrak{m}_0)$ and let $M$ and $N$ be finitely generated graded $R$-modules. Let $i,j\in\mathbb{N}_0$. In this paper we will study…

Let R be a commutative noetherian ring. Denote by mod R the category of finitely generated R-modules. In this paper, we investigate the finiteness of the radii of resolving subcategories of mod R with respect to a fixed semidualizing…

Let $A$ be a commutative Noetherian ring containing a field of characteristic zero. Let $R= A[X_1, \ldots, X_m]$ be a polynomial ring and $A_m(A) = A \langle X_1, \ldots, X_m, \partial_1, \ldots, \partial_m \rangle$ be the $m^{th}$ Weyl…

Let $G$ be an abelian group of order $n$ and let $R$ be a commutative ring which admits a homomorphism ${\Bbb Z}[\zeta_{n}]\ra R$, where $\zeta_{n}$ is a (complex) primitive $n$-th root of unity. Given a finite $R[G\e]$-module $M$, we…

Let $R$ be a commutative noetherian ring and let $\frak a$ be an ideal of $R$. In this paper, we study a certain condition, namely $C_{\frak a}$, introduced by Aghapournahr and Melkersson, on the extension of two subcategories of…

For $K$ a field, consider a finite subgroup $G$ of $\operatorname{GL}_n(K)$ with its natural action on the polynomial ring $R:=K[x_1,\dots,x_n]$. Let $\mathfrak{n}$ denote the homogeneous maximal ideal of the ring of invariants $R^G$. We…