Related papers: Computing the support of local cohomology modules

Let $R$ be a noetherian ring, $\fa$ an ideal of $R$ such that $\dim R/\fa=1$ and $M$ a finite $R$--module. We will study cofiniteness and some other properties of the local cohomology modules $\lc^{i}_{\fa}(M)$. For an arbitrary ideal $\fa$…

For $K$ a field, consider a finite subgroup $G$ of $\operatorname{GL}_n(K)$ with its natural action on the polynomial ring $R:=K[x_1,\dots,x_n]$. Let $\mathfrak{n}$ denote the homogeneous maximal ideal of the ring of invariants $R^G$. We…

Let R be a commutative Noetherian ring. We introduce a theory of formal local cohomology for complexes of R-modules. As an application, we establish some relations between formal local cohomology, local homology, local cohomology and local…

The main focus of this paper is on determining the highest non-vanishing local cohomology modules of \Omega_(B/R), \Omega_(B/V)(\Omega_(B/k)) where R is either a complete regular local ring or a complete local normal domain with coefficient…

Let $A$ be a regular ring containing a field $K$ of characteristic zero and let $R = A[X_1,\ldots, X_m]$. Consider $R$ as standard graded with $\deg A = 0$ and $\deg X_i = 1$ for all $i$. Let $G$ be a finite subgroup of $GL_m(A)$. Let $G$…

We describe an algorithm for computing parameter-test-ideals in certain local Cohen-Macaulay rings. The algorithm is based on the study of a Frobenius map on the injective hull of the residue field of the ring and on the application of…

Let $X=\C^n$. In this paper we present an algorithm that computes the de Rham cohomology groups $H^i_{dR}(U,\C)$ where $U$ is the complement of an arbitrary Zariski-closed set $Y$ in $X$. Our algorithm is a merger of the algorithm given by…

Let $(R,m, \kappa)$ be a local ring. We give a characterization of $R$-modules $M$ whose local cohomology is finite length up to some index in terms of asymptotic vanishing of Koszul cohomology on parameter ideals up to the same index. In…

Local rings are ubiquitous in algebraic geometry. Not only are they naturally meaningful in a geometric sense, but also they are extremely useful as many problems can be attacked by first reducing to the local case and taking advantage of…

Given a $D$-module $M$ generated by a single element, and a polynomial $f$, one can construct several $D$-modules attached to $M$ and $f$ and can define the notion of the (generalized) $b$-function following M. Kashiwara. These modules are…

Let $R$ be a commutative Noetherian ring that is a smooth $\mathbb Z$-algebra. For each ideal $I$ of $R$ and integer $k$, we prove that the local cohomology module $H^k_I(R)$ has finitely many associated prime ideals. This settles a crucial…

Let $(R, \mathfrak m)$ denote an $n$-dimensional Gorenstein ring. For an ideal $I \subset R$ of height $c$ we are interested in the endomorphism ring $B = \Hom_R(H^c_I(R), H^c_I(R)).$ It turns out that $B$ is a commutative ring. In the case…

Lyubeznik's conjecture, (\cite{Ly1}, Remark 3.7) asserts the finiteness of the set ssociated primes of local cohomology modules for regular rings. But, in the case of ramified regular local ring, it is open. Recently, in Theorem 1.2 of…

The topology of periodic spaces has attracted a lot of interest in recent years in order to study and classify crystalline structures and other large homogeneous data sets, such as the distribution of galaxies in cosmology. In practice,…

In this note I give a description of Lyubeznik's local cohomology invariants for a certain natural class of local rings, namely the ones which have the same local cohomology vanishing as one expects from an isolated singularity. This…

Lyubeznik conjectured that local cohomology modules of regular rings have finitely many associated primes. We examine this conjecture for polynomial rings over the integers, and record some equational identities that arise from studying…

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 $(R,\mathfrak{m})$ be a Noetherian regular local ring containing a field of characteristic $p>0$ and $I$ a nonzero ideal of $R$. In this short note, we prove that if $\operatorname{H}^i_I(R)\neq 0$, then $…

For positive integers m >= n >= p, we compute the GL_m x GL_n-equivariant description of the local cohomology modules of the polynomial ring S of functions on the space of m x n matrices, with support in the ideal of p x p minors. Our…

We compute the sheaf cohomology with constant $\mathbb{Z}_2$ coefficients of a concrete class of locally profinite sets of independent interest. We introduce $k$-sheer partitions to aid in constructions. It is also shown that questions of…