Related papers: Some duality and equivalence results

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…

This paper shows the existence of ideals whose localizations and completions at prime ideals are parameter test ideals of the localized and completed rings. We do this for Cohen-Macaulay localizations (resp., completions) of non-local…

Let $A$ be a Cohen-Macaulay local ring with $\operatorname{dim} A = d\ge 3$, possessing the canonical module ${\mathrm K}_A$. Let $a_1, a_2, \ldots, a_r$ $(3 \le r \le d)$ be a subsystem of parameters of $A$ and set $Q= (a_1, a_2, \ldots,…

Given two determinantal rings over a field k. We consider the Rees algebra of the diagonal ideal, the kernel of the multiplication map. The special fiber ring of the diagonal ideal is the homogeneous coordinate ring of the join variety.…

For a coherent filtered D-module we show that the dual of each graded piece over the structure sheaf is isomorphic to a certain graded piece of the ring-theoretic local cohomology complex of the graded quotient of the dual of the filtered…

The relationships between the invariants and the homological properties of $I$, ${\rm Gin}(I)$ and $I^{\rm lex}$ have been studied extensively over the past decades. A result of A. Conca, J. Herzog and T. Hibi points out some rigid…

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d>0$ with infinite residue field. Let $M$ be a finitely generated proper $R$-submodule of a free $R$-module $F$ with $\ell (F/M) < \infty$ and having rank $r$. In this article,…

This is the first in a series of papers highlighting the applications of reduced and coreduced modules. Let $R$ be a commutative unital ring and $I$ an ideal of $R$. We show that $I$-reduced $R$-modules and $I$-coreduced $R$-modules provide…

Let R be a local ring and C a semidualizing module of R. We investigate the behavior of certain classes of generalized Cohen-Macaulay R-modules under the Foxby equivalence between the Auslander and Bass classes with respect to C. In…

Let $(R,\fm)$ be a local ring and $\fa$ be an ideal of $R$. The inequalities $$\begin{array}{ll} \ \Ht(\fa) \leq \cd(\fa,R) \leq \ara(\fa) \leq l(\fa) \leq \mu(\fa) \end{array}$$ are known. It is an interesting and long-standing problem to…

This article investigates the relationship between Betti numbers of finitely generated modules over a Noetherian local ring $(R, \mathfrak{m})$ and the structure of formal local cohomology modules. We establish a connection between the…

In this paper we show that a large class of one-dimensional Cohen-Macaulay local rings $(A,\mathfrak{m})$ has the property that if $M$ is a maximal Cohen-Macaulay $A$-module then the Hilbert function of $M$ ( with respect to $\mathfrak{m}$)…

Let $R = \bigoplus_{n \in \mathbb{N}_{0}} R_{n}$ be a standard graded ring, $M$ be a finite graded $R$-module and $J$ be a homogenous ideal of $R$. In this paper we study the graded structure of the $i$-th local cohomology module of $M$…

We introduce a concept of formal local homology modules which is in some sense dual to P. Schenzel's concept of formal local cohomology modules. The dual theorem and the non-vanishing theorem of formal local homology modules will be shown.…

Let $\mathfrak a$ denote an ideal of a local ring $(R, \mathfrak m).$ Let $M$ be a finitely generated $R$-module. There is a systematic study of the formal cohomology modules $\varprojlim \HH^i(M/\mathfrak a^nM), i \in \mathbb Z.$ We…

For a Cohen-Macaulay ring $R$, we exhibit the equivalence of the bounded derived categories of certain resolving subcategories, which, amongst other results, yields an equivalence of the bounded derived category of finite length and finite…

Let \fa be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. We explore the behavior of the two notions f_{\fa}(M), the finiteness dimension of M with respect to \fa, and, its dual notion q_{\fa}(M), the…

For a polynomial ring $R=k[x_1,...,x_n]$, we present a method to compute the characteristic cycle of the localization $R_f$ for any nonzero polynomial $f\in R$ that avoids a direct computation of $R_f$ as a $D$-module. Based on this…

Over a noetherian ring, it is a classic result of Matlis that injective modules admit direct sum decompositions into injective hulls of quotients by prime ideals. We show that over a Cohen-Macaulay ring admitting a dualizing module,…

The objective of this paper is to introduce and study completions and local homology of comodules over Hopf algebroids, extending previous work of Greenlees and May in the discrete case. In particular, we relate module-theoretic to…