Asymptotic behaviour of the length of local cohomology

Cutkosky, Steven Dale; Ha, Huy Tai; Srinivasan, Hema; Theodorescu, Emanoil

Asymptotic behaviour of the length of local cohomology

Authors

Cutkosky, Steven Dale

Ha, Huy Tai

Srinivasan, Hema

Theodorescu, Emanoil

Description

Let k be a field of characteristic 0, R = k[x_1, ..., x_d] 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 H^0_m(R/I^n) and n^d. 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 H^0_m(R/I^n) is not asymptotically a polynomial in n.
To appear in Canadian Journal of Mathematics. 13 pages