Mora's holy grail: Algorithms for computing in localizations at prime ideals

TL;DR

This paper develops algorithms for computing localizations at prime ideals and their associated graded rings using standard basis techniques, with implementations in Singular, advancing Mora's foundational work.

Contribution

It provides alternative proofs and expands applications of Mora's algorithms for localizations, including resolutions, parameters, Hilbert polynomials, and dimension analysis.

Findings

Algorithms implemented in Singular's graal.lib

Enhanced methods for ideal resolutions and Hilbert polynomial computation

Expanded applications to dimension and regularity analysis

Abstract

This article discusses a computational treatment of the localization A_L of an affine coordinate ring A at a prime ideal L and its associated graded ring Gr_a(A_L) with the means of standard basis techniques. Building on Mora's work, we present alternative proofs on two of the central statements and expand on the applications mentioned by Mora: resolutions of ideals, systems of parameters and Hilbert polynomials, as well as dimension and regularity of A_L. All algorithms are implemented in the library graal.lib for the computer algebra system Singular.