The singular Riemann-Roch theorem and Hilbert-Kunz functions (Q853934)

In this paper, via the singular Riemann-Roch theorem, the author proves that the class of the \(e\)-th Frobenius power can be described using the class of the canonical module \(\omega_A\) for a normal local ring \(A\) of positive characteristic. As a corollary, the author proves that the coefficient \(\beta(I,M)\) of the second term of the Hilbert-Kunz function \(\ell_A(M/I^{[p^e]}M)\) of \(e\) vanishes if \(A\) is a \(\mathbb{Q}\)-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 \(e\)-th Frobenius power \(F^e_*{\mathcal O}_X\) can be described using the canonical divisor \(K_X\). In addition, the author gives also two interesting examples for the results of this paper.