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.


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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