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COMPUTING THE SUPPORT OF LOCAL COHOMOLOGY MODULES JOSEP `ALVAREZ MONTANER AND ANTON LEYKIN
 

Summary: COMPUTING THE SUPPORT OF LOCAL COHOMOLOGY MODULES
JOSEP `ALVAREZ MONTANER AND ANTON LEYKIN
Abstract. For a polynomial ring R = k[x1, ..., xn], we present a method to compute the characteristic
cycle of the localization Rf for any nonzero polynomial f R that avoids a direct computation of Rf
as a D-module. Based on this approach, we develop an algorithm for computing the characteristic cycle
of the local cohomology modules Hr
I (R) for any ideal I R using the Cech 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.
1. Introduction
Let k be a field of characteristic zero and R = k[x1, . . . , xn] the ring of polynomials in n variables. For
any ideal I R, the local cohomology modules Hr
I (R) have a natural finitely generated module structure
over the Weyl algebra An. Recently, there has been an effort made towards effective computation of these
modules by using the theory of Gr¨obner bases over rings of differential operators. Algorithms given by
U. Walther [23] and T. Oaku and N. Takayama [20] provide a utility for such computation and are both
implemented in the package D-modules [16] for Macaulay 2 [10].
Walther's algorithm is based on the construction of the Cech complex in the category of An-modules.
So it is necessary to give a description of the localization Rf at a polynomial f R. An algorithm

  

Source: Alvarez Montaner, Josep - Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya

 

Collections: Mathematics