LENGTH OF LOCAL COHOMOLOGY OF POWERS OF IDEALS

摘要

Let R be a polynomial ring over a field k with irrelevant ideal m and dimension d. Let I be a homogeneous ideal in R. We study the asymptotic behavior of the length of the modules H-m(i)(R/I-n) for n 0. We show that for a fixed number alpha is an element of Z, lim sup(n -infinity) lambda H-m(i) (R/I-n)=-alpha n)/n(d) infinity. It follows that lim sup(n -infinity) lambda H-m(i) (R/I-n)=))/n(d) infinity when X = Proj R/ I is locally a complete intersection (lci) and i = dimX. We also establish that the actual limit exists and is rational for certain classes of monomial ideals I such that the lengths of local cohomology of I-n are eventually finite. Our proofs use Grobner deformation and Presburger arithmetic. Finally, we utilize more traditional commutative algebra techniques to show that lim inf(n -infinity) lambda H-m(i) (R/I-n)=)/n(d) 0 under certain conditions when R/I is either F-pure or lci.