Nagoya Mathematical Journal

Hilbert-Samuel polynomials for the contravariant extension functor

Andrew Crabbe, Daniel Katz, Janet Striuli, and Emanoil Theodorescu

Full-text: Open access


Let (R,m) be a local ring, and let M and N be finite R-modules. In this paper we give a formula for the degree of the polynomial giving the lengths of the modules ExtRi(M,N/mnN). A number of corollaries are given, and more general filtrations are also considered.

Article information

Nagoya Math. J., Volume 198 (2010), 1-22.

First available in Project Euclid: 10 May 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13D07: Homological functors on modules (Tor, Ext, etc.) 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 13H15: Multiplicity theory and related topics [See also 14C17] 13C11: Injective and flat modules and ideals


Crabbe, Andrew; Katz, Daniel; Striuli, Janet; Theodorescu, Emanoil. Hilbert-Samuel polynomials for the contravariant extension functor. Nagoya Math. J. 198 (2010), 1--22. doi:10.1215/00277630-2009-005.

Export citation


  • [1] A. Corso, C. Huneke, D. Katz, and W. Vasconcelos, Integral closure of ideals and annihilators of homology, Lect. Notes Pure Appl. Math. 227 (2005), 33–48.
  • [2] A. Crabbe and J. Striuli, Constructing big indecomposable modules, Proc. Amer. Math. Soc. 137 (2009), 2181–2189.
  • [3] S. Iyengar and T. Puthenpurakal, Hilbert-Samuel functions of modules over Cohen-Macaulay local rings, Proc. Amer. Math. Soc. 135 (2007), 637–648.
  • [4] S. Goto, F. Hayasaka, and R. Takahashi, On vanishing of certain Ext modules, J. Math. Soc. Japan 60 (2008), 1045–1064.
  • [5] D. Katz and E. Theodorescu, On the degree of Hilbert polynomials associated to the torsion functor, Proc. Amer. Math. Soc. 135 (2007), 3073–3082.
  • [6] V. Kodiyalam, Homological invariants of powers of an ideal, Proc. Amer. Math. Soc. 118 (1993), 757–764.
  • [7] E. Theodorescu, Derived functors and Hilbert polynomials, Math. Proc. Cambridge Philos. Soc. 132 (2002), 75–88.