Coefficient Ideals in and Blowups of a Commutative Noetherian Domain Original Research Article

The Ratliff-Rush ideal associated to a nonzero ideal I in a commutative Noetherian domain R with unity is image = union operator∞n=1 (In+1:RIn = intersection operator {IS∩R:Sset membership, variantimage(I)}, where image(I) = {R[I/a]P:aset membership, variantI−0, Pset membership, variantSpec(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 image(I)(k), which is obtained from image(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.