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.
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  and T. Oaku and N. Takayama  provide a utility for such computation and are both
implemented in the package D-modules  for Macaulay 2 .
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