Related papers: Cohen-Macaulay approximations and the $\text{SC}_r…

First we study the Gorenstein cohomological dimension ${\rm Gcd}_RG$ of groups $G$ over coefficient rings $R$, under changes of groups and rings; a characterization for finiteness of ${\rm Gcd}_RG$ is given. Some results in literature…

In this paper, we establish the global analogues of some dualities and equivalences in local algebra by developing the theory of relative Cohen-Macaulay modules. Let R be a commutative Noetherian ring (not necessarily local) with identity…

We let R be a one-dimensional graded complete intersection, satisfying certain degree conditions which are satisfied whenever R is a numerical semigroup ring of embedding dimension at least three. We show that a graded maximal…

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring, $\mathfrak{a}$ be a proper ideal of $R$ and $M$ be an $R$-complex in $\mathrm{D}(R)$. We prove that if $M\in\mathrm{D}^f_\sqsubset(R)$ (respectively,…

Let $R$ be a Noetherian ring and let $C$ be a semidualizing $R$-module. In this paper, by using relative homological dimensions with respect to $C$, we impose various conditions on $C$ to be dualizing. First, we show that $C$ is dualizing…

Over $d$-dimensional Cohen-Macaulay rings with a canonical module, $d$-cotilting classes containing the maximal and balanced big Cohen-Macaulay modules are classified. Particular emphasis is paid to the direct limit closure of the balanced…

Let $(R, \frak m)$ denote a local Cohen-Macaulay ring and $I$ a non-nilpotent ideal of $R$. The purpose of this article is to investigate Faltings' finiteness dimension $f_I(R)$ and equidimensionalness of certain homomorphic image of $R$.…

Let $(R,\m)$ be a Noetherian local ring and $M$ a finitely generated $R$-module. Following I. G. Macdonald \cite{Mac}, the set of all attached primes of the Artinian local cohomology module $H^i_{\m}(M)$ is denoted by $\Att_R(H^i_{\m}(M))$.…

Let $S$ be a module finite birational extension of a $1$-dimensional local Cohen--Macaulay ring $R$. When is the conductor of $S$ in $R$ a stable ideal? If $R$ is also generically Gorenstein, then we show that the conductor of $S$ in $R$ is…

The purpose of this paper is to introduce new invariants of Cohen-Macaulay local rings. Our focus is the class of Cohen-Macaulay local rings that admit a canonical ideal. Attached to each such ring R with a canonical ideal C, there are…

Let $(A,\mathfrak{m})$ be a hypersurface ring with dimension $d$, and $M$ a MCM $A-$module with red$(M)\leq 2$ and $\mu(M)=2$ or $3$ then we have proved that depth $G(M)\geq d-\mu(M)+1$. If $e(A)=3$ and $\mu(M)=4$ then in this case we have…

An ideal I of a local Cohen-Macaulay ring R is called a cohomologically complete intersection if H^i_I(R) = 0 for all i \neq c = height(I). Here H^i_I(R), i \in Z denotes the local cohomology of R with respect to I. For instance, a…

Let $(R, \mathfrak{m} )$ be a Noetherian local ring, $M$ a finitely generated $R$-module of dimension $d$. Set $\mathfrak{a}(M):=\mathfrak{a}_0(M)\cdots \mathfrak{a}_{d-1}(M)$, where $\mathfrak{a}_i(M):={\rm Ann}_RH^i_{\mathfrak{m}}(M)$ for…

Idealization of a module $K$ over a commutative ring $S$ produces a ring having $K$ as an ideal, all of whose elements are nilpotent. We develop a method that under suitable field-theoretic conditions produces from an $S$-module $K$ and…

We describe "quasi canonical modules" for modular invariant rings $R$ of finite group actions on factorial Gorenstein domains. From this we derive a general "quasi Gorenstein criterion" in terms of certain 1-cocycles. This generalizes a…

Let $(R,\fm)$ be commutative Noetherian local ring. It is shown that $R$ is Cohen--Macaulay ring if there exists a Cohen--Macaulay finite (i.e. finitely generated) $R$--module with finite upper Gorenstein dimension. In addition, we show…

Let $R$ be a commutative noetherian local ring, and let $M$ be a finitely generated $R$-module. Inspired by works of Vasconcelos and Briggs on characterization of complete intersection local rings through the homological properties of the…

Let $M$ be a finite module over a commutative noetherian ring $R$. For ideals $\fa$ and $\fb$ of $R$, the relations between cohomological dimensions of $M$ with respect to $\fa, \fb$, $\fa\cap\fb$ and $\fa+ \fb$ are studied. When $R$ is…

Every quotient R/I of a semigroup ring R by a radical monomial ideal I has a unique minimal injective-like resolution by direct sums of quotients of R modulo prime monomial ideals. The quotient R/I is Cohen-Macaulay if and only if every…

A finitely generated module $M$ over a commutative Noetherian ring $R$ is called an $I$-Cohen Macaulay module, if \[ \grade(I,M) + \dim(M/IM)= \dim(M), \] where $I$ is a proper ideal of $R$. The aim of this paper is to study the structure…