Frobenius Betti numbers and modules of finite projective dimension

TL;DR

This paper investigates Frobenius-based asymptotic invariants of modules over local rings, exploring their properties, implications for finite projective dimension, and characterizations in one-dimensional cases.

Contribution

It introduces and analyzes the invariants eta^F_i(M,R), linking their vanishing to finite projective dimension and providing a complete characterization in one-dimensional rings.

Findings

eta^F_i(M,R) include Hilbert-Kunz multiplicity as a special case.

Vanishing of eta^F_i(M,R) characterizes finite projective dimension in certain cases.

Conditions are established for the non-existence of finite-length syzygies.

Abstract

Let $(R,\mathfrak{m},K)$ be a local ring, and let $M$ be an $R$-module of finite length. We study asymptotic invariants, $\beta^F_i(M,R),$ defined by twisting with Frobenius the free resolution of $M$. This family of invariants includes the Hilbert-Kunz multiplicity ($e_{HK}(\mathfrak{m},R)=\beta^F_0(K,R)$). We discuss several properties of these numbers that resemble the behavior of the Hilbert-Kunz multiplicity. Furthermore, we study when the vanishing of $\beta^F_i(M,R)$ implies that $M$ has finite projective dimension. In particular, we give a complete characterization of the vanishing of $\beta^F_i(M,R)$ for one-dimensional rings. As a consequence of our methods, we give conditions for the non-existence of syzygies of finite length.