Alapan Mukhopadhyay : Frobenius-Poincare Function and Hilbert-Kunz Multiplicity

Posted by Kevin Tucker , part of the Commutative Algebra Seminar.

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April 13, 2022, 3 p.m.

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Abstract

We shall discuss a natural generalization of the classical Hilbert-Kunz multiplicity theory when the underlying objects are graded. More precisely, given a graded ring $R$ and a finite co-length homogeneous ideal $I$ in a positive characteristic $p$ and for any complex number $y$, we shall show that the limit $$\underset{n \to \infty}{\lim}(\frac{1}{p^n})^{\text{dim}(R)}\sum \limits_{j= -\infty}^{\infty}\lambda \left( (\frac{R}{I^{[p^n]}R})_j\right)e^{-iyj/p^n}$$ exists. This limit as a function in the complex variable $y$ is a natural refinement of the Hilbert-Kunz multiplicity of the pair $(R,I)$: the value of the limiting function at the origin is the Hilbert-Kunz multiplicity of the pair $(R,I)$. We name this limiting function the <i>Frobenius-Poincare function</i> of $(R,I)$. We shall establish that Frobenius-Poincare functions are holomorphic everywhere in the complex plane. We shall discuss properties of Frobenius-Poincare functions, give examples and describe these functions in terms of the sequence of graded Betti numbers of $\frac{R}{I^{[p^n]}R}$. On the way, we shall mention some questions on the structure and properties of Frobenius-Poincare functions.