The Cohen–Macaulay and Gorenstein Properties of Rings Associated to Filtrations

Abstract

Let (R, m) be a Cohen–Macaulay local ring, and let ℱ = {F _i } _i∈ℤ be an F ₁-good filtration of ideals in R. If F ₁ is m-primary we obtain sufficient conditions in order that the associated graded ring G(ℱ) be Cohen–Macaulay. In the case where R is Gorenstein, we use the Cohen–Macaulay result to establish necessary and sufficient conditions for G(ℱ) to be Gorenstein. We apply this result to the integral closure filtration ℱ associated to a monomial parameter ideal of a polynomial ring to give necessary and sufficient conditions for G(ℱ) to be Gorenstein. Let (R, m) be a Gorenstein local ring, and let F ₁ be an ideal with ht(F ₁) = g > 0. If there exists a reduction J of ℱ with μ(J) = g and reduction number u: = r _J (ℱ), we prove that the extended Rees algebra R′(ℱ) is quasi-Gorenstein with a-invariant b if and only if J ⁿ : F _u  = F _{n+b−u+g−1} for every n ∈ ℤ. Furthermore, if G(ℱ) is Cohen–Macaulay, then the maximal degree of a homogeneous minimal generator of the canonical module ω _G(ℱ) is at most g and that of the canonical module ω _R′(ℱ) is at most g − 1; moreover, R′(ℱ) is Gorenstein if and only if J ^u : F _u  = F _u . We illustrate with various examples cases where G(ℱ) is or is not Gorenstein.

Key Words:

• Associated graded ring
• Cohen–Macaulay ring
• Filtration
• Gorenstein ring
• Monomial parameter ideal
• Reduction number

2000 Mathematics Subject Classification:

• Primary: 13A30, 13C05
• Secondary: 13E05, 13H15

ACKNOWLEDGMENT

Bernd Ulrich is partially supported by the National Science Foundation (DMS-0501011).

Notes

Communicated by S. Goto.