A 𝑑-dimensional extension of a lemma of Huneke’s and formulas for the Hilbert coefficients

Abstract

A d d -dimensional version is given of a 2 2 -dimensional result due to C. Huneke. His result produced a formula relating the length λ ( I n + 1 / J I n ) \lambda (I^{n+1}/JI^{n}) to the difference P I ( n + 1 ) − H I ( n + 1 ) P_{I}(n+1)-H_{I}(n+1) , where I I is primary for the maximal ideal of a 2 2 -dimensional Cohen-Macaulay local ring R R , J J is a minimal reduction of I I , H I ( n ) = λ ( R / I n ) H_{I}(n)=\lambda (R/I^{n}) , and P I ( n ) P_{I}(n) is the Hilbert-Samuel polynomial of I I . We produce a formula that is valid for arbitrary dimension, and then use it to establish some formulas for the Hilbert coefficients of I I . We also include a characterization, in terms of the Hilbert coefficients of I I , of the condition d e p t h ( G ( I ) ) ≥ d − 1 depth(G(I))\geq d-1 .