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 Ĩ = ⋃^∞_n=1 (Iⁿ⁺¹:_RIⁿ = ⋂ {IS∩R:S∈$\text{B}$(I)}, where $\text{B}$(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 $\text{B}$(I)^(k), which is obtained from $\text{B}$(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.