The Ratliff-Rush ideal associated to a nonzero ideal I in a commutative Noetherian domain R with unity is I = ∞ n=1 (I n+1 : R I n) = {IS ∩ R : S ∈ B(I)}, where B(I) = {R[I/a] P : a ∈ I − 0, P ∈ Spec(R[I/a])} is the blowup of I. We observe that certain ideals are minimal or even unique in the class of ideals having the same associated Ratliff-Rush ideal. If (R, M) is local, quasi-unmixed, and analytically unramified, and if I is M-primary, then we show that the coefficient ideal I {k} of I, i.e., the largest ideal containing I whose Hilbert polynomial agrees with that of I in the highest k terms, is also contracted from a blowup B(I) (k) , which is obtained from B(I) by a process similar to "S 2-ification". This allows us to generalize the notion of coefficient ideals. We investigate these ideals in the specific context of a two-dimensional regular local ring, observing the interaction of these notions with the Zariski theory of complete ideals.
Abstract. We compare the Castelnuovo regularity defined with respect to different homogeneous ideals in a graded ring and use the result we obtain to prove a generalized Goto-Shimoda theorem for ideals of positive height in a Cohen-Macaulay local ring.
Let (R, m) be a 2-dimensional regular local ring and I an m-primary ideal. The aim of this paper is to find conditions on I so that the associated graded ring of I,and the Rees ring of I,where t is an indeterminate, are Cohen–Macaulay (resp. Gorenstein). To this end, we use the results and techniques from Zariski's theory of complete ideals ([14], appendix 5) and its later generalizations and refinements due to Huneke [7] and Lipman[8]. The main result is an application of three deep theorems: (i) a generalization of Macaulay's classical theorem on Hilbert series of Gorenstein graded rings [13], (ii) a generalization of the Briançon–Skoda theorem due to Lipman and Sathaye [9], and (iii) a formula for the length of R/I, where I is a complete m-primary ideal, due to Hoskin[4] and Deligne[1].
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