Asymptotic Hilbert-Kunz multiplicity

[摘要] For a pair (M, I), where M is finitely generated graded module over a standard graded ring R of dimension d >= 2, and I is a graded ideal with l(R/I) < infinity and generated by elements of the same degree, we prove that lim(q ->infinity) e(1)(M, I-[q])/q(d) exists, where e(1)(M, I-[q]) denotes the first coefficient of the Hilbert-Samuel polynomial of (M, I-[q]). We use this to get an expression for lim(k ->infinity) [e(HK)(M, I-k) - e(0)(M, I-k)/d!]/k(d-1), where eHK denotes the Hilbert-Kunz multiplicity. In particular, if dim M = d then we deduce that the difference e(HK)-(M, I-k) - e(M, I-k)/d! grows at least as a fixed positive multiple of k(d-1) as k -> infinity. This is proved using `renormalizedt HK density functions. (C) 2017 Elsevier Inc. All rights reserved.

[发布日期] 2017-12-15

[关键词] Hilbert-Kunz density;Hilbert-Kunz multiplicity;Hilbert-Samuel polynomial