Multiplicities of graded components of local cohomology modules

Abstract

Let M be a finitely generated graded module over a Noetherian homogeneous ring R with local base ring ( R 0 , m 0 ) . Then, the nth graded component H R + i ( M ) n of the ith local cohomology module of M with respect to the irrelevant ideal R + of R is a finitely generated R 0 -module which vanishes for all n ≫ 0 . In various situations we show that, for an m 0 -primary ideal q 0 ⊆ R 0 , the multiplicity e q 0 ( H R + i ( M ) n ) of H R + i ( M ) n is antipolynomial in n of degree less than i. In particular we consider the following three cases: (a) i < g ( M ) , where g ( M ) is the so-called cohomological finite length dimension of M; (b) i = g ( M ) ; (c) dim ( R 0 ) = 2 . In cases (a) and (b) we express the degree and the leading coefficient of the representing polynomial in terms of local cohomological data of M (e.g. the sheaf induced by M) on Proj ( R ) . We also show that the lengths of the graded components of various graded submodules of H R + i ( M ) are antipolynomial of degree less than i and prove invariance results on these degrees.