Hilbert-Kunz density function for graded domains

JOURNAL OF PURE AND APPLIED ALGEBRA

卷:226

Article

Trivedi, Vijaylaxmi¹  Watanabe, Kei-Ichi^2,3 

[1] Tata Inst Fundamental Res, Sch Math, Homi Bhabha Rd, Mumbai 40005, Maharashtra, India

[2] Meiji Univ, Org Strateg Coordinat Res & Intellectual Properti, Tokyo, Japan

[3] Nihon Univ, Coll Humanities & Sci, Dept Math, Setagaya Ku, Tokyo 1560045, Japan

关键词: N-graded domains;    Char p methods;    Hilbert-Kunz density functions;    Reflexive sheaves;    Q-divisors;    F-thresholds;   

DOI  :  10.1016/j.jpaa.2021.106835

来源: Elsevier

【 摘 要 】

We prove the existence of HK density function for a graded pair (R, I), where R is an N-graded domain of finite type over a perfect field and I C R is a graded ideal of finite colength. This generalizes our earlier result where one proves the existence of such a function for a pair (R, I), where, in addition R is standard graded. Other properties of the HK density functions also hold for the graded pairs: for example, it is a multiplicative function for Segre products, its maximum support is the F-threshold of an m-primary ideal provided Proj R is smooth, it has a closed formula when either I is generated by a system of parameters or R is of dimension two. As one of the consequences we show that if G is a finite group scheme acting linearly on a polynomial ring R of dimension d then the HK density function f(RG,mG), of the pair (R-G, m(G)), is a piecewise polynomial function of degree d - 1. We also compute the HK density functions for (R-G, m(G)), where G subset of SL2(k) is a finite group acting linearly on the ring k[X, Y]. (C) 2021 Elsevier B.V. All rights reserved.

【 授权许可】

Free