Characteristic cycles of local cohomology modules of monomial ideals

Abstract

We study, by using the theory of algebraic D -modules, the local cohomology modules supported on a monomial ideal I of the local regular ring R=k[[x 1,…,x n]] , where k is a field of characteristic zero. We compute the characteristic cycle of H I r(R) and H m p(H I r(R)) , where m is the maximal ideal of R and I is a squarefree monomial ideal. As a consequence, we can decide when the local cohomology module H I r(R) vanishes and compute the cohomological dimension cd(R,I) in terms of the minimal primary decomposition of the monomial ideal I. We also give a Cohen–Macaulayness criterion for the local ring R/I and compute the Lyubeznik numbers λ p,i(R/I)=dim k Ext R p(k,H I n−i(R)) .