Asymptotic behaviour of the length of local cohomology

Details Asymptotic behaviour of the length of local cohomology Asymptotic behaviour of the length of local cohomology

2004-03-22

arxiv.org/abs/math/0403370v1

Canadian J. Math. 57 (2005), no. 6, 1178-1192.

Mathematics

Commutative Algebra

To appear in Canadian Journal of Mathematics. 13 pages

Scientific paper

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.

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