Related papers: Cousin complexes and applications

We extend the Auslander-Buchweitz axioms and prove Cohen-Macaulay approximation results for fibred categories. Then we show that these axioms apply for the fibred category of pairs consisting of a finite type flat family of Cohen-Macaulay…

In this paper we study the local cohomology of all finitely generated bigraded modules over a standard bigraded polynomial ring which have only one nonvanishing local cohomology with respect to one of the irrelevant bigraded ideals.

We study syzygies of (maximal) Cohen-Macaulay modules over one dimensional Cohen-Macaulay local rings. We compare these modules to Cohen-Macaulay modules over the endomorphism ring of the maximal ideal. After this comparison, we give…

Let $K$ be a field and $S=K[x_1,\ldots,x_m, y_1,\ldots,y_n]$ be the standard bigraded polynomial ring over $K$. In this paper, we explicitly describe the structure of finitely generated bigraded "sequentially Cohen--Macaulay" $S$-modules…

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…

A local Cohen--Macaulay ring is called Ulrich-split if any short exact sequence of Ulrich modules split. In this paper we initiate the study of Ulrich split rings. We prove several necessary or sufficient criteria for this property, linking…

It is shown that a module is sequentially Cohen-Macaulay if and only if the index of reducibility for distinguished parameter ideals are eventually constant with special value. As corollaries to the main theorem we given to characterize the…

In this paper, we study the two different topics related to sequentially Cohen-Macaulay modules. The questions are when the sequentially Cohen-Macaulay property preserve the localization and the module-finite extension of rings.

Let $(R, \frak m)$ be a local ring and $M$ a finitely generated $R$-module. It is shown that if $M$ is relative Cohen-Macaulay with respect to an ideal $\frak a$ of $R$, then $\text{Ann}_R(H_{\mathfrak{a}}^{\text{cd}(\mathfrak{a},…

Using the theory of cohomology annihilators, we define a family of topologies on the set of isomorphism classes of maximal Cohen-Macaulay modules over a Gorenstein ring. We study compactness of these topologies.

Let $\mathfrak{q}$ denote an ideal of a local ring $(A,\mathfrak{m})$. For a system of elements $\underline{a} = a_1,\ldots,a_t$ such that $a_i \in \mathfrak{q}^{c_i}, i = 1, \ldots,t,$ and $n \in \mathbb{Z}$ we investigate a subcomplex…

The notion of weakly Laskerian modules was introduced recently by the authors. Let $R$ be a commutative Noetherian ring with identity, $\fa$ an ideal of $R$, and $M$ a weakly Laskerian module. It is shown that if $\fa$ is principal, then…

The main result of the paper states that for a graded ideal I in a polynomial ring R over a field of characteristic 0, the Hilbert functions of the local cohomology modules of R/I and of R/Gin(I) coincide if and only if R/I is sequentially…

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 develop a formalism of unit $F$-modules in the style of Lyubeznik and Emerton-Kisin for rings which have finite $F$-representation type after localization and completion at every prime ideal. As applications, we show that if $R$ is such…

The Cohen-Macaulay type of idealizations of maximal Cohen-Macaulay modules over Cohen-Macaulay local rings is explored. There are two extremal cases, one of which is closely related to the theory of Ulrich modules \cite{BHU, GOTWY1, GOTWY2,…

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…

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 $\mathfrak{a}$ 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.

Inspired by a question raised by Eisenbud-Musta\c{t}\u{a}-Stillman regarding the injectivity of maps from ${\rm Ext}$ modules to local cohomology modules and the work by the third author with Pham, we introduce a class of rings which we…