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The singular Riemann-Roch theorem and Hilbert-Kunz functions

Kazuhiko Kurano
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Abstract

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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Kazuhiko Kurano. 2005-11-22. The singular Riemann-Roch theorem and Hilbert-Kunz functions. https://arxiv.org/abs/math/0506492

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