Related papers: A local homology theory for linearly compact modul…

For the Cousin complex of certain modules, we investigate finiteness of cohomology modules, local duality property and injectivity of its terms. The existence of canonical modules of Noetherian non-local rings and the Cousin complexes of…

We introduce and investigate using Hilbert modules the properties of the {\em Fourier algebra} $A(G)$ for a locally compact groupoid $G$. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This…

Given a Serre class $\mathcal{S}$ of modules, we compare the containment of the Koszul homology, Ext modules, Tor modules, local homology, and local cohomology in $\mathcal{S}$ up to a given bound $s \geq 0$. As some applications, we give a…

We give an elementary proof of Grothendieck's non-vanishing Theorem: For a finitely generated non-zero module $M$ over a Noetherian local ring $A$ with maximal ideal $\m$, the local cohomology module $H^{\dim M}_{\m}(M)$ is non-zero.

For a finite module $M$ over a local, equicharacteristic ring $(R,m)$, we show that the well-known formula $\cohdim(m,M)=\dim M$ becomes trivial if ones uses Matlis duals of local cohomology modules together with spectral sequences. We also…

A duality theorem for the category of locally compact Hausdorff spaces and continuous maps which generalizes the well-known Duality Theorem of de Vries is proved.

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,…

We propose a homology theory for locally compact spaces with ends in which the ends play a special role. The approach is motivated by results for graphs with ends, where it has been highly successful. But it was unclear how the original…

This paper studies Moore's measurable cohomology theory for locally compact groups and Polish modules. An elementary dimension-shifting argument is used to show that all classes in that theory have representatives with considerable extra…

In this article, we pursue two main objectives. The first is to show that the fundamental results of Green-Lazarsfeld (1987, 1991) on generic vanishing theorems, and works of Budur-Wang (2015, 2020) on cohomology jumping loci, can be…

Let $(R,\fm)$ be a relative Cohen-Macaulay local ring with respect to an ideal $\fa$ of $R$ and set $c:=\h\fa$. In this paper, we investigate some properties of the Matlis dual $\H_{\fa}^c(R)^{\vee}$ of the $R$-module $\H_{\fa}^c(R)$ and we…

We use a Mayer-Vietoris-like spectral sequence to establish vanishing results for the cohomology of complements of linear and elliptic hyperplane arrangements, as part of a more general framework involving duality and abelian duality…

Let $\textbf{H} = ((H, F^{\bullet}), L)$ be a polarized variation of Hodge structure on a smooth quasi-projective variety $U.$ By M. Saito's theory of mixed Hodge modules, the variation of Hodge structure $\textbf{H}$ can be viewed as a…

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.

We give several related versions of global Grothendieck Duality for unbounded complexes on noetherian formal schemes. The proofs, based on a non-trivial adaptation of Deligne's method for the special case of ordinary schemes, are reasonably…

Let $\fa$ be an ideal of a Noetherian local ring $(R,\fm)$ and $M$ a finitely generated $R$-module. In this paper we introduce some criterions on Artinianness of formal local cohomology, in particular vanishing and finiteness of local…

An associative ring with 1 is said to be semilocal provided it is semisimple artinian modulo its Jacobson radical, that is, modulo its Jacobson radical it is isomorphic to a finite product of matrices over division rings. Modules with a…

Let $\mathfrak{a}$ denote an ideal of a commutative Noetherian ring $R$. Let $M$ and $N$ be two $R$-modules. In this paper, we give partial answers on the extension of Hartshorne's conjecture about the cofiniteness of torsion and extension…

In this paper, we introduce the theory of local cohomology and local duality to Notherian connected cochain DG algebras. We show that the notion of local cohomology functor can be used to detect the Gorensteinness of a homologically smooth…

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…