## Frobenius Betti numbers and modules of finite projective dimension.(English)Zbl 1471.13014

Summary: Let $$(R,\mathfrak{m} ,K)$$ be a local ring, and let $$M$$ be an $$R$$-module of finite length. We study asymptotic invariants, $$\beta^Fi(M,R)$$, 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.

### MSC:

 13A35 Characteristic $$p$$ methods (Frobenius endomorphism) and reduction to characteristic $$p$$; tight closure 13D02 Syzygies, resolutions, complexes and commutative rings 13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.) 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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