Hilbert-Kunz density function for graded domains

Abstract

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 ⊂ 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 R G , m G , 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 ⊂ S L 2 ( k ) is a finite group acting linearly on the ring k [ X , Y ] .