Finitely Supported *-Simple Complete Ideals in a Regular Local Ring

TL;DR

This paper investigates the structure and properties of finitely supported *-simple complete ideals in regular local rings, focusing on their factorization, valuations, and monomial transforms, revealing new insights into their generators and classifications.

Contribution

It introduces new criteria for when monomial ideals are *-simple or special, and characterizes their generators and transforms in regular local rings.

Findings

Minimal number of generators determined by the order of the ideal

Conditions for inverse transforms to preserve *-product structure

Examples of finitely supported *-simple monomial ideals that are not special

Abstract

Let I be a finitely supported complete m-primary ideal of a regular local ring (R, m). A theorem of Lipman implies that I has a unique factorization as a *-product of special *-simple complete ideals with possibly negative exponents for some of the factors. The existence of negative exponents occurs if the dimension of R is at least 3 because of the existence of finitely supported *-simple ideals that are not special. We consider properties of special *-simple complete ideals such as their Rees valuations and point basis. Let (R, m) be a d-dimensional equicharacterstic regular local ring with m = (x_1, ..., x_d)R. We define monomial quadratic transforms of R and consider transforms and inverse transforms of monomial ideals. For a large class of monomial ideals I that includes complete inverse transforms, we prove that the minimal number of generators of I is completely determined by the order of I. We give necessary and sufficient conditions for the complete inverse transform of a *-product of monomial ideals to be the *-product of the complete inverse transforms of the factors. This yields examples of finitely supported *-simple monomial ideals that are not special. We prove that a finitely supported *-simple monomial ideal with linearly ordered base points is special *-simple.