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arXiv · 1412.4266

Frobenius Betti numbers and modules of finite projective dimension

Alessandro De Stefani·Craig Huneke·Luis Núñez-Betancourt
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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.

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Alessandro De Stefani, Craig Huneke, Luis Núñez-Betancourt. 2014-12-13. Frobenius Betti numbers and modules of finite projective dimension. https://arxiv.org/abs/1412.4266

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