Frobenius-Poincaré function and Hilbert-Kunz multiplicity

Abstract

We generalize the notion of Hilbert-Kunz multiplicity of a graded triple ( M , R , I ) (M,R,I) in characteristic p > 0 p>0 by proving that for any complex number y y , the limit lim n → ∞ ( 1 p n ) dim ⁡ ( M ) ∑ j = − ∞ ∞ λ ( ( M I [ p n ] M ) j ) e − i y j / p n \begin{equation*} \underset {n \to \infty }{\lim }(\frac {1}{p^n})^{\operatorname {dim}(M)}\sum \limits _{j= -\infty }^{\infty }\lambda \left ( (\frac {M}{I^{[p^n]}M})_j\right )e^{-iyj/p^n} \end{equation*} exists. We prove that the limiting function in the complex variable y y is entire and name this function the Frobenius-Poincaré function. We establish various properties of Frobenius-Poincaré functions including its relation with the tight closure of the defining ideal I I ; and relate the study Frobenius-Poincaré functions to the behaviour of graded Betti numbers of R I [ p n ] \frac {R}{I^{[p^n]}} as n n varies. Our description of Frobenius-Poincaré functions in dimension one and two and other examples raises questions on the structure of Frobenius-Poincaré functions in general.