Vanishing Of Local Cohomology Modules Research Articles

Toward an efficient algorithm for deciding the vanishing of local cohomology modules in prime characteristic

Let $$R=k[x_1,\dots ,x_n]$$ be a ring of polynomials over a field k of characteristic $$p>0$$ . There is an algorithm due to Lyubeznik for deciding the vanishing of local cohomology modules $$H^i_I(R)$$ where $$I\subset R$$ is an ideal. This algorithm has not been implemented because its complexity grows very rapidly with the growth of p which makes it impractical. In this paper we produce a modification of this algorithm that consumes a modest amount of memory.

Local cohomology modules supported at determinantal ideals

We provide new results on the vanishing of local cohomology modules supported at ideals of minors of matrices over arbitrary commutative Noetherian rings. In the process, we compute the local cohomology of rings of polynomials with integer coefficients---supported at generic determinantal ideals---and also obtain results on F-modules and D-modules that are likely to be of independent interest.

Computing the support of local cohomology modules

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 ∈ R that avoids a direct computation of R f as a D -module. Based on this approach, we develop an algorithm for computing the characteristic cycle of the local cohomology modules H I r ( R ) for any ideal I ⊆ R using the Čech complex. The algorithm, in particular, is useful for answering questions regarding vanishing of local cohomology modules and computing Lyubeznik numbers. These applications are illustrated by examples of computations using our implementation of the algorithm in Macaulay 2.

On the Associated Primes of Matlis Duals of Top Local Cohomology Modules

After motivating the question, we prove various results about the set of associated primes of Matlis duals of top local cohomology modules. In some cases, we can calculate this set. An easy application of this theory is the well-known fact that Krull dimension can be expressed by the vanishing of local cohomology modules.

Cofiniteness and vanishing of local cohomology modules

Let R be a noetherian local ring with maximal ideal m and residue field k. If M is a finitely generated R-module then the local cohomology modules are known to be Artinian. Grothendieck [3], exposé 13, 1·2 made the following conjecture:If I is an ideal of R and M is a finitely generated R-module, then HomR (R/I, ) is finitely generated.

Some Results on the Vanishing of Local Cohomology Modules

Some Results on the Vanishing of Local Cohomology Modules