On the Degree of Hilbert Polynomials of Derived Functors

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 Ext_Ri(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_Rⁱ (·) are the ith Betti number and the ith Bass number, respectively.