Related papers: On Gerko's Strongly Tor-independent Modules

Let $R$ be a Cohen-Macaulay local ring of dimension one with a canonical module $\rm{K_R}$. Let $I$ be a faithful ideal of $R$. We explore the problem of when $I \otimes_RI^{\vee}$ is torsionfree, where $I^{\vee} = \operatorname{Hom_R(I,…

Let $R$ be a Cohen-Macaulay local ring and let $M$ and $N$ be non-zero finitely generated $R$-modules. We investigate necessary conditions for the depth formula $\depth(M)+\depth(N)=\depth(R)+\depth(M\otimes_{R}N)$ to hold. We show that,…

Let $R$ be a finitely generated positively graded algebra over a Noetherian local ring $B$, and $\mathfrak{m} = [R]_+$ be the graded irrelevant ideal of $R$. We provide a local criterion characterizing the $B$-freeness of all the local…

For finitely generated modules $M$ and $N$ over a Gorenstein local ring $R$, one has $depth M + depth N= depth(M\otimes N) +depth R$, i.e., the depth formula holds, if $M$ and $N$ are Tor-independent and Tate homology…

Let $(A,\mathfrak{m})$ be a Henselian Cohen-Macaulay local ring and let CM(A) be the category of maximal Cohen-Macaulay $A$-modules. We construct $T \colon CM(A)\times CM(A) \rightarrow mod(A)$, a subfunctor of $Ext^1_A(-, -)$ and use it to…

We answer a question of Celikbas, Dao, and Takahashi by establishing the following characterization of Gorenstein rings: a commutative noetherian local ring $(R,\mathfrak m)$ is Gorenstein if and only if it admits an integrally closed…

Let $\mathfrak{a}$ be an ideal of local ring $(R,\mathfrak{m})$ and $M$ a finitely generated $R$-module and $n\in\Bbb N$. It is shown that some results concerning cominimaxness of formal local cohomology modules.

One of the most stunning results in the representation theory of Cohen-Macaulay rings is Auslander's well known theorem which states a CM local ring of finite CM type can have at most an isolated singularity. There have been some…

Let $R$ be a commutative Noetherian ring and $M$ a finitely generated $R$-module. We show in this paper that, for an integer $t$, if the local cohomology module $H^{i}_\mathfrak{a}(M)$ with respect to an ideal $\frak a$ is finitely…

We consider the class $\mathfrak M$ of $\bf R$--modules where $\bf R$ is an associative ring. Let $A$ be a module over a group ring $\bf R$$G$ where $G$ is a group and let $\mathfrak L(G)$ be a set of all proper subgroups of $G$ such that…

Let $(R,\mathfrak m)$ be a Noetherian local ring, $I$ an ideal of $R$ and $M$ a weakly finite or a coatomic $R$-module of dimension $n$. In this article, we resolve the Artinianness and non-Artinianness of top local cohomology modules,…

Let $A$ be a noetherian ring, $\fa$ an ideal of $A$, and $M$ an $A$--module. Some uniform theorems on the artinianness of certain local cohomology modules are proven in a general situation. They generalize and imply previous results about…

Let R be a commutative Noetherian local ring. We show that R is Gorenstein if and only if every finitely generated R-module can be embedded in a finitely generated R-module of finite projective dimension. This extends a result of Auslander…

Let $(R,\mathfrak{m})$ be a regular local ring containing a field $K$. Let $I$ be a Cohen-Macaulay ideal of height $g$. If $\text{char } K = p > 0$ then by a result of Peskine and Szpiro the local cohomology modules $H^i_I(R)$ vanish for $i…

Let $(Q,\mathfrak{n})$ be a regular local ring of dimension $c \geq 2$ with algebraically closed residue field $k = Q/\mathfrak{n}$. Let $f_1, f_2, \ldots f_{c-1}, g$ be a regular sequence in $Q$ such that $ f_i \in \mathfrak{n}^2$ for all…

Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. An $R$-module $M$ is said to be a uniformly $S$-Artinian ($u$-$S$-Artinian for abbreviation) module if there is $s\in S$ such that any descending chain of…

Let $R$ be a local complete intersection ring and let $M$ and $N$ be nonzero finitely generated $R$-modules. We employ Auslander's transpose in the study of the vanishing of Tor and obtain useful bounds for the depth of the tensor product…

Let $(R, \mathfrak{m}, k)$ be a commutative Noetherian local ring. We study the suitable chains of semidualizing $R$-modules. We prove that when $R$ is Artinian, the existence of a suitable chain of semidualizing modules of length…

Let $R$ be a complete local Gorenstein ring of dimension one, with maximal ideal m. We show that if $M$ is a Cohen-Macaulay $R$-module which begins an AR-sequence, then this sequence is produced by a particular endomorphism of m…

Let $(A,\mathfrak{m})$ be a Noetherian local ring, $M$ a finite $A$-module and $x_1,...,x_n\in \m$ such that $\lambda (M/\x M)$ is finite. Serre proved that all partial Euler characteristics of $M$ with respect to $\x$ is non-negative. This…