On Associated Graded Rings Having Almost Maximal Depth

. We generalize a recent result of Rossi and Valla, and independently Wang, about the depth of G(m) where m is the maximal ideal of a d-dimensional Cohen-Macaulay local ring R having embedding dimesion e 0 (m) + d \Gamma 2. The generalization removes the restriction on the embedding dimension and replaces it with the condition that (m 3 =Jm 2 ) 1 where J is a d-generated minimal reduction of m. The main theorem also applies to m-primary ideals I satisfying J " I 2 = JI and (I 3 =JI 2 ) 1, where J is a d-generated reduction of I. An example of such an I in a 5-dimensional regular local ring is included as a nontrivial illustration of the theorem. 1. Introduction The purpose of this article is to generalize a recent result proved independently by Rossi and Valla in [RV], and Wang in [W]. Their result answered a question posed by J.D. Sally in [S3] about the nature of depth(G(m)) where (R; m) is a Cohen-Macaulay local ring of embedding dimension e 0 (m) + d \Gamma 2, e 0 (..

A d-Dimensional Extension Of A Lemma Of Huneke's And Formulas For The Hilbert Coefficients

. A d-dimensional version is given of a 2-dimensional result due to C. Huneke. His result produced a formula relating the length (I n+1 =JI n ) to the difference P I (n + 1) \Gamma H I (n+ 1), where I is primary for the maximal ideal of a 2-dimensional Cohen-Macaulay local ring R, J is a minimal reduction of I, H I (n) = (R=I n ), and P I (n) is the Hilbert-Samuel polynomial of 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. We also include a characterization, in terms of the Hilbert coefficients of I, of the condition depth(G(I)) d \Gamma 1. 1. Introduction If (R; m) is a d-dimensional (Noetherian) local ring and I is an m-primary ideal of R then let (R=I n ) denote the length of the R-module R=I n . The Hilbert-Samuel function of I is the function H I (n) = (R=I n ). It follows from the theory of Hilbert functions applied to the associated graded ring G(I) = \Phi n0 I n =..

Length One Ideal Extensions And Their Associated Graded Rings

Let (R; m) be a d-dimensional Cohen-Macaulay local ring. Given m-primary ideals J I of R such that I is contained in the integral closure of J and (I=J) = 1, we compare depth G(J) and depth G(I). For example, if J has reduction number one, JI = I and (J) d + 1, we prove that depth G(I) d 1. If, in addition, (I) = d + 1, we show that I has reduction number one, and hence G(I) is Cohen-Macaulay. These results, besides leading to statements comparing depths of associated graded rings along a composition series, make visible the possibility of studying powers of an ideal by using reductions that are not minimal reductions

REDUCTION NUMBERS AND IDEALS OF ANALYTIC SPREAD ONE

Let (R,M) be a commutative Noetherian local ring having an infinite residue field and suppose I is an ideal in R. The reduction number of I (denoted r(I)) is defined, and conditions on R which force r(I) (LESSTHEQ) 1 for every regular analytic spread one ideal I of R are obtained. Several examples are given illustrating r(I) and the role it sometimes plays in connection with other concepts. The situation for ideals having analytic spread larger than one is also discussed, and an interpretation of r(I) involving the relations on the generators of I is investigated

Hilbert Coefficients And The Depths Of Associated Graded Rings

this article is to pursue a better understanding of the interplay between between the Hilbert coefficients and the depth of G(I), where G(I) = R=I \Phi I=