Lecture Announcement
Date: Thursday, 3 November 2022.
Time: 6:30 PM.
Venue: Zoom talk.
Frobenius-Poincare Function and Hilbert-Kunz Multiplicity
Alapan Mukhopadhyay
University of Michigan, Ann Arbour.
03-11-22

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Abstract

Hilbert-Kunz multiplicity theory is a widely studied multiplicity theory in prime characteristic. We shall discuss a natural generalization of the classical Hilbert-Kunz multiplicity theory when the underlying objects are graded. More precisely, given a prime characteristic $p$ standard graded domain $R$ and a finite co-length homogeneous ideal $I$ and for any complex number $y$, we shall show that the following limit exists.

$$\underset{n \to \infty}{\lim}(\frac{1}{p^n})^{\text{dim}(R)}\sum \limits_{j= -\infty}^{\infty}\lambda \left( (\frac{R^{1/p^n}}{IR^{1/p^n}})_{j/p^n}\right)e^{-iyj/p^n}$$

This limiting function in the complex variable $y$ is holomorphic everywhere on the complex plane, we name the limiting function the \textit{Frobenius-Poincare function}. We shall discuss properties of Frobenius-Poincar\'e functions, describe these functions in terms of the sequence of graded Betti numbers of $\frac{R^{1/p^n}}{IR^{1/p^n}}$. On the way, we shall mention some questions on the structure and properties of Frobenius-Poincare functions; and discuss examples to indicate the invariants of $(R,I)$ encoded in the Frobenius-Poincar\'e functions.

See PDF here: https://tinyurl.com/3mwvur7k