Frobenius Betti numbers and modules of finite projective dimension

الأصل · EN

Let (R,m,K) be a local ring, and let M be an R-module of finite length. We study asymptotic invariants, βᶠᵢ(M,R), defined by twisting with Frobenius the free resolution of M. This family of invariants includes the Hilbert-Kunz multiplicity (eHK(m,R)=βᶠ₀(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 βᶠᵢ(M,R) implies that M has finite projective dimension. In particular, we give a complete characterization of the vanishing of βᶠᵢ(M,R) for one-dimensional rings. As a consequence of our methods, we give conditions for the non-existence of syzygies of finite length.