The singular Riemann–Roch theorem and Hilbert–Kunz functions

In the paper, via the singular Riemann–Roch theorem, it is proved that the class of the eth Frobenius power can be described using the class of the canonical module ωA for a normal local ring A of positive characteristic. As a corollary, we prove that the coefficient β(I,M) of the second term of the Hilbert–Kunz function ℓA(M/I[pe]M) of e vanishes if A is a -Gorenstein ring and M is a finitely generated A-module of finite projective dimension. For a normal algebraic variety X over a perfect field of positive characteristic, it is proved that the first Chern class of the eth Frobenius power can be described using the canonical divisor KX.