Hilbert-Kunz density functions and F-thresholds

JOURNAL OF ALGEBRA

卷:567

Article

Trivedi, Vijaylaxmi¹  Watanabe, Kei-Ichi² 

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

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

关键词: F-thresholds;    Reduction mod p;   

DOI  :  10.1016/j.jalgebra.2020.09.025

来源: Elsevier

【 摘 要 】

The first author had shown earlier that for a standard graded ring R and a graded ideal I in characteristic p > 0, with l(R/I) < infinity, there exists a compactly supported continuous function f(R,I) whose Riemann integral is the HK multiplicity e(HK) (R, I). We explore further some other invariants, namely the shape of the graph of f(R,m) (where m is the graded maximal ideal of R) and the maximum support (denoted as alpha(R, I)) of f(R,I). In case R is a domain of dimension d >= 2, we prove that (R, m) is a regular ring if and only if f(R,m) has a symmetry f(R,m)(x) = f(R,m)(d - x), for all x. If R is strongly F-regular on the punctured spectrum then we prove that the F-threshold c(I)(m) coincides with alpha(R, I). As a consequence, if R is a two dimensional domain and I is generated by homogeneous elements of the same degree, then we have (1) a formula for the F-threshold c(I)(m) in terms of the minimum strong Harder-Narasimhan slope of the syzygy bundle and (2) a well defined notion of the F-threshold c(I)(m) in characteristic 0. This characterisation readily computes c(I(n))(m), for the set of all irreducible plane trinomials k[x, y, z]/(h), where m = (x, y, z) and I(n) = (x(n), y(n), z(n)). (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free