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

De Stefani, Alessandro; Huneke, Craig; Núñez-Betancourt, Luis

J. Commut. Algebra 9, No. 4, 455-490 (2017).

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.)

Keywords:

Hilbert-Kunz multiplicity; projective dimension; Krull dimension of syzygies; \(F\)-contributors