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Summary: arXiv:math.AC/0212130
v1
9
Dec
2002
THE DEPTH OF THE ASSOCIATED GRADED RING OF IDEALS
WITH ANY REDUCTION NUMBER
IAN ABERBACH, LAURA GHEZZI AND HUY T
AI H
A
Abstract. Let R be a local Cohen-Macaulay ring, let I be an R-ideal, and let G be
the associated graded ring of I. We give an estimate for the depth of G when G is not
necessarily Cohen-Macaulay. We assume that I is either equimultiple, or has analytic
deviation one, but we do not have any restriction on the reduction number. We also give
a general estimate for the depth of G involving the rst r + ` powers of I, where r denotes
the Castelnuovo regularity of G and ` denotes the analytic spread of I.
Key words. depth, associated graded ring, Rees algebra, reduction number, Castelnuovo
regularity.
0. Introduction
Let R be a Noetherian local ring with in nite residue eld k, and let I be an R-ideal. The
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