Frobenius Betti numbers and modules of finite projective dimension

Let $(R,\mathfrak{m},K)$ be a local ring, and let $M$ be an $R$-module of finite length. We study asymptotic invariants, ${\beta }_i^F(M,R),$ defined by twisting with Frobenius the free resolution of $M$. This family of invariants includes the Hilbert-Kunz multiplicity ($e_{HK}(\mathfrak{m},R)={\beta }_0^F(K,R)$). We discuss several properties of these numbers that resemble the behavior of the Hilbert-Kunz multiplicity. Furthermore, we study when the vanishing of ${\beta }_i^F(M,R)$ implies that $M$ has finite projective dimension. In particular, we give a complete characterization of the vanishing of ${\beta }_i^F(M,R)$ for one-dimensional rings. As a consequence of our methods, we give conditions for the non-existence of syzygies of finite length.