Mathematical Notes

, Volume 106, Issue 3–4, pp 423–428 | Cite as

On the Degree of Hilbert Polynomials of Derived Functors

  • H. SaremiEmail author
  • A. MafiEmail author


Given a d-dimensional Cohen–Macaulay local ring (R,m), let I be an m-primary ideal, and let J be a minimal reduction ideal of I. If M is a maximal Cohen–Macaulay R-module, then, for n large enough and 1 ≤ id, the lengths of the modules ExtRi(R/J,M/InM) and ToriR(R/J,M/InM) are polynomials of degree d − 1. It is also shown that
$$\deg \beta _i^R(M/{I^n}M) = \deg \mu _R^i(M/{I^n}M) = d - 1,$$
where β i R (·) and μ R i (·) are the ith Betti number and the ith Bass number, respectively.


Hilbert–Samuel polynomial derived functors 


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Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Sanandaj BranchIslamic Azad UniversitySanandajIran
  2. 2.Department of MathematicsUniversity of KurdistanSanandajIran

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