Frobenius and homological dimensions of complexes

A Correction to this article was published on 08 November 2019

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Abstract

It is proved that a module M over a Noetherian local ring R of prime characteristic and positive dimension has finite flat dimension if \({\text {Tor}}_i^R({}^{e}\!R, M)=0\) for \({\text {dim}}\,R\) consecutive positive values of i and infinitely many e. Here \({}^{e}\!R\) denotes the ring R viewed as an R-module via the eth iteration of the Frobenius endomorphism. In the case R is Cohen–Macualay, it suffices that the Tor vanishing above holds for a single \(e\geqslant \log _p e(R)\), where e(R) is the multiplicity of the ring. This improves a result of Dailey et al. (J Commut Algebra), as well as generalizing a theorem due to Miller (Contemp Math 331:207–234, 2003) from finitely generated modules to arbitrary modules. We also show that if R is a complete intersection ring then the vanishing of \({\text {Tor}}_i^R({}^{e}\!R, M)\) for single positive values of i and e is sufficient to imply M has finite flat dimension. This extends a result of Avramov and Miller (Math Res Lett 8(1–2):225–232, 2001).

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  • 08 November 2019

    The proof of Theorem 3.2 in the paper contains an error

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Correspondence to Taran Funk.

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Funk, T., Marley, T. Frobenius and homological dimensions of complexes. Collect. Math. 71, 287–297 (2020). https://doi.org/10.1007/s13348-019-00260-7

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Keywords

  • Frobenius endomorphism
  • Flat dimension
  • Injective dimension
  • Complete intersection

Mathematics Subject Classification

  • 13D05
  • 13D07
  • 13A35