Asymptotic behaviour of the length of local cohomology
Abstract
Let k be a field of characteristic 0, R = k[x1, ..., xd] be a polynomial ring, and m its maximal homogeneous ideal. Let I be a homogeneous ideal in R. In this paper we investigate asymptotic behaviour of the quotient between the length of local cohomology group H0m(R/In) and nd. We show that this quantity always has a limit as n goes to infinity. We also give an example for which the limit is irrational; in particular, this proves that the length of H0m(R/In) is not asymptotically a polynomial in n.
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