Coefficient ideals in and blowups of a commutative noetherian domain

Abstract

The Ratliff-Rush ideal associated to a nonzero ideal I in a commutative Noetherian domain R with unity is I = ⋃n=1∞ (In+1:RIn = ⋂ {IS⋂R:SϵB(I)}, where B(I) = {R[I/a]P:aϵI−0, PϵSpec(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 “S2-ification.” This allows us to generalise the notion of coefficient ideas. We investigate these ideas in the specific context of a two-dimentional regular local ring, observing the interaction of these notions with the Zariski theory of complete ideals.

Department(s)

Mathematics

Document Type

Article

DOI

https://doi.org/10.1006/jabr.1993.1261

Publication Date

12-15-1993

Journal Title

Journal of Algebra

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