Cohomología local con soporte un ideal monomial (D-módulos y combinatoria)

Abstract

We study, by using the theory of algebraic D-modules, the local cohomology modules supported on a monomial ideal I of the polynomial ring R = k[x1, . . . , xn], where k is a field of characteristic zero. We compute the characteristic cycle of Hr I (R) and H p P(Hr I (R)), where P is an homogeneous prime ideal of R. By using these results we can describe the support of these modules, in particular we can decide when the local cohomology module Hr I (R) vanishes in terms of the minimal primary decomposition of the monomial ideal I, compute the Bass numbers of Hr I (R) and describe its associated primes. The characteristic cycles also give some invariants of the ring R/I. We use these invariants to compute the Hilbert function of R/I, the minimal free resolutions of squarefree monomial ideals and the cohomology groups of the complement of an arrangement of linear varieties given by the monomial ideal I. Finally, we determine the local cohomology modules by using the category introduced by Galligo-GrangerMaisonobe [3] and compute its Hilbert function.