Frobenius Betti numbers and modules of finite projective dimension

Abstract

Abstract. Let (R, m, K) be a local ring, and let M be an R-module of finite length. We study asymptotic invariants, β 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 (m, R) = β 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 β F i (M, R) implies that M has finite projective dimension. In particular, we give a complete characterization of the vanishing of β 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.

Frobenius Betti numbers and syzygies of finite length modules

“…As a consequence we generalize the result in [7,Corollary 4.9] to all Noetherian local rings of dimension d ≥ 1.…”

“…We also show that Question 1.2 has a positive answer for rings in which the socle dimension of H 0 m (R) is large relative to the total length of H 0 m (R), generalizing the case of Buchsbaum rings done in [7]. See Theorem 4.4 for the precise statement.…”

“…If dim(R) = d, we may consider the Frobenius Betti numbers 1 Technically, they asked the question only in the case that R is F-finite (see section 2), but, in the manner β F i is defined here, that hypothesis is not necessary. They then showed that if R has a finitely generated regular algebra of the same dimension and if d + 1 consecutive β F i (M, R) (with i positive) vanish, then pd R M < ∞ (see [7,Proposition 4.1]). Moreover, if dim R = 1 and R is Cohen-Macaulay, then a single vanishing β F i (M, R) (for i > 0) suffices ( [7,Corollary 4.8]).…”

“…They then showed that if R has a finitely generated regular algebra of the same dimension and if d + 1 consecutive β F i (M, R) (with i positive) vanish, then pd R M < ∞ (see [7,Proposition 4.1]). Moreover, if dim R = 1 and R is Cohen-Macaulay, then a single vanishing β F i (M, R) (for i > 0) suffices ( [7,Corollary 4.8]).…”

“…There are very few satisfying results in this direction, although in dimension one it is shown in [7] that if a module of finite length has an ith syzygy, Ω i , of finite length, then λ(Ω i ) may be computed as an alternating sum of lengths of certain Tor modules. In this way, they are able to show that no such Ω 3 may be finite length.…”