Related papers: Groebner deformations, connectedness and cohomolog…

We study properties of graded maximal Cohen-Macaulay modules over an N-graded locally finite, Auslander Gorenstein, and Cohen-Macaulay algebra of dimension two. As a consequence, we extend a part of McKay correspondence in dimension two to…

By extending some basic results of Grothendieck and Foxby about local cohomology to commutative DG-rings, we prove new amplitude inequalities about finite DG-modules of finite injective dimension over commutative local DG-rings,…

In this note we extend connectedness results to formal properties of inverse images under proper maps of Schubert varieties and of the diagonal in products of projective rational homogeneous spaces

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…

The notion of linkage with respect to a semidualizing module is introduced. It is shown that over a Cohen-Macaulay local ring with canonical module, every Cohen-Macaulay module of finite Gorenstein injective dimension is linked with respect…

Let R be a commutative Noetherian (not necessarily local) ring with identity and a be a proper ideal of R. We introduce a notion of a-relative system of parameters and characterize them by using the notion of cohomological dimension. Also,…

We establish a "second vanishing theorem" for local cohomology modules over regular rings of unramified mixed characteristic, which relates the connectedness of the spectrum of a ring with the vanishing of local cohomology. Applying this,…

Let $I\subseteq S=K[x_1,\ldots,x_n]$ be a homogeneous ideal equipped with a monomial order $<$. We show that if $\operatorname{in}_<(I)$ is a square-free monomial ideal, then $S/I$ and $S/\operatorname{in}_<(I)$ have the same connectedness…

This thesis is comprised of three chapters. The first chapter deals with bounded complexes of Gorenstein projective and Gorenstein injective modules. Deploying methods of relative homological algebra, we approximate such complexes with…

Let R be a Cohen-Macaulay local ring. It is shown that under some mild conditions, the Cohen-Macaulayness property is preserved under linkage. We also study the connection of (S_n) locus of a horizontally linked module and the attached…

Let R be a commutative noetherian local ring. As an analogue of the notion of the dimension of a triangulated category defined by Rouquier, the notion of the dimension of a subcategory of finitely generated R-modules is introduced in this…

Let $(R,\mathfrak{m})$ be a complete local ring, and $G={\rm gr}_{\mathfrak{m}}(R)$ be its associated graded ring. We introduce a homogenization technique which allows to relate $G$ to the special fiber and $R$ to the generic fiber of a…

We provide a Hochster type formula for the local cohomology modules of binomial edge ideals. As a consequence we obtain a simple criterion for the Cohen-Macaulayness of these ideals and we describe their Castelnuovo-Mumford regularity and…

Let $\Delta$ be a simplicial complex. We study the expansions of $\Delta$ mainly to see how the algebraic and combinatorial properties of $\Delta$ and its expansions are related to each other. It is shown that $\Delta$ is Cohen-Macaulay,…

This work builds on earlier work of the first three authors where a notion of congruence modules in higher codimension is introduced. The main new results are a criterion for detecting regularity of local rings in terms of congruence…

Motivated by the notion of geometrically linked ideals, we show that over a Gorenstein local ring $R$, if a Cohen-Macaulay $R$-module $M$ of grade $g$ is linked to an $R$-module $N$ by a Gorenstein ideal $c$, such that $Ass_R(M)\cap…

The overall goal is to approach the Cohen--Macaulay property of the special fiber $\mathcal{F}(I)$ of an equigenerated homogeneous ideal $I$ in a standard graded ring over an infinite field. When the ground ring is assumed to be local, the…

This book contains notes of a seminar on Ofer Gabber's work on the etale cohomology and uniformization of quasi-excellent schemes. His main results include (cf. introduction) constructibility theorems (for abelian or non-abelian…

The aim of this paper is to extend the main result of C. Huneke and G. Lyubeznik in [Adv. Math. 210 (2007), 498--504] to the class of rings that are images of Cohen-Macaulay local rings. Namely, let $R$ be a local Noetherian domain of…

Some descriptions of linked ideals in a commutative Notherian ring $R$ are provided in terms of the Associated prime ideals of $R$. Then, among other things, we make some characterization of Cohen-Macaulay, Gorenstein and regular local…