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.
Citation
Alessandro De Stefani. Craig Huneke. Luis Núñez-Betancourt. "Frobenius Betti numbers and modules of finite projective dimension." J. Commut. Algebra 9 (4) 455 - 490, 2017. https://doi.org/10.1216/JCA-2017-9-4-455
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