Advertisement

Frobenius and homological dimensions of complexes

  • Taran FunkEmail author
  • Thomas Marley
Article
  • 3 Downloads

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).

Keywords

Frobenius endomorphism Flat dimension Injective dimension Complete intersection 

Mathematics Subject Classification

13D05 13D07 13A35 

Notes

References

  1. 1.
    Auslander, M., Buchsbaum, D.A.: Homological dimension in Noetherian rings. II. Trans. Am. Math. Soc. 88(1), 94–206 (1958)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Avramov, L.L., Miller, C.: Frobenius powers of complete intersections. Math. Res. Lett. 8(1–2), 225–232 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Avramov, L.L., Foxby, H.-B.: Homological dimensions of unbounded complexes. J. Pure Appl. Algebra 71, 129–155 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bass, H.: Injective dimension in Noetherian rings. Trans. Am. Math. Soc. 102, 19–29 (1962)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Christensen, L.W., Iyengar, S.B., Marley, T.: Rigidity of ext and tor with coefficients in residue fields of a commutative Noetherian ring. Proceedings of the Edinburgh Mathematical Society (to appear)Google Scholar
  6. 6.
    Dailey, D.J., Iyengar, S.B., Marley, T.: Detecting finite flat dimension of modules via iterates of the Frobenius endomorphism. J. Commut. Algebra (to appear) Google Scholar
  7. 7.
    Dutta, S.P.: On modules of finite projective dimension over complete intersections. Proc. Am. Math. Soc. 131(1), 113–116 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Herzog, J.: Ringe der Charakteristik \(p\) und Frobeniusfunktoren. Math. Z. 140, 67–78 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jensen, C.: On the vanishing of \(\displaystyle \lim _{\longleftarrow }{}^{(i)}\). J. Algebra 15, 151–166 (1970)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Koh, J., Lee, K.: Some restrictions on the maps in minimal resolutions. J. Algebra 202, 671–689 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kunz, E.: Characterizations of regular local rings for characteristic \(p\). Am. J. Math. 91, 772–784 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lech, C.: Inequalities related to certain couples of local rings. Acta Math. 112, 69–89 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Marley, T.: The Frobenius functor and injective modules. Proc. Am. Math. Soc. 142(6), 1911–1923 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Miller, C.: The Frobenius endomorphism and homological dimensions, commutative algebra (Grenoble/Lyon, 2001). Contemp. Math. 331, 207–234 (2003) CrossRefGoogle Scholar
  15. 15.
    Marley, T., Webb, M.: The acyclicity of the Frobenius functor for modules of finite flat dimension. J. Pure Appl. Algebra 220(8), 2886–2896 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Peskine, C., Szpiro, L.: Dimension projective finie et cohomologie locale. Applications à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck, Inst. Hautes Études Sci. Publ. Math. No. 42, pp. 47–119 (1973)Google Scholar
  17. 17.
    Raynaud, M., Gruson, L.: Critères de platitude et de projectivé. Techniques de "platification" d’un module. Invent. Math. 13, 1–89 (1971)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Universitat de Barcelona 2019

Authors and Affiliations

  1. 1.University of Nebraska-LincolnLincolnUSA

Personalised recommendations