The singular Riemann-Roch theorem and Hilbert-Kunz functions

Details The singular Riemann-Roch theorem and Hilbert-Kunz functions The singular Riemann-Roch theorem and Hilbert-Kunz functions

2005-06-24

arxiv.org/abs/math/0506492v2

Mathematics

Commutative Algebra

12 pages. to appear in J. Algebra

Scientific paper

In the paper, by the singular Riemann-Roch theorem, it is proved that the class of the e-th Frobenius power can be described using the class of the canonical module for a normal local ring of positive characteristic. As a corollary, we prove that the coefficient of the second term of the Hilbert-Kunz function of a finitely generated A-module M vanishes if A is a Q-Gorenstein ring and M is 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 direct image of the structure sheaf via e-th Frobenius power can be described using the canonical divisor of X.

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