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On the Degree of Hilbert Polynomials of Derived Functors

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Abstract

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 ≤ i ≤ d, 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 R i (·) and μ i R (·) are the ith Betti number and the ith Bass number, respectively.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Sanandaj Branch, Islamic Azad University, Sanandaj, P.O. Box 618, Iran

    H. Saremi

  2. Department of Mathematics, University of Kurdistan, Sanandaj, P.O. Box 416, Iran

    A. Mafi

Authors
  1. H. Saremi
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  2. A. Mafi
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Correspondence to H. Saremi or A. Mafi.

Additional information

Russian Text © The Author(s), 2019, published in Matematicheskie Zametki, 2019, Vol. 106, No. 3, pp. 450–456.

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Saremi, H., Mafi, A. On the Degree of Hilbert Polynomials of Derived Functors. Math Notes 106, 423–428 (2019). https://doi.org/10.1134/S0001434619090116

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  • DOI: https://doi.org/10.1134/S0001434619090116

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