Computations over Local Rings in Macaulay2

TL;DR

This paper introduces Macaulay2 packages for computations over local rings, enabling algebraic geometry researchers to perform homological calculations such as resolutions, syzygies, and invariants more efficiently.

Contribution

It provides new software tools for computations over local rings in Macaulay2, including methods for resolutions, syzygies, and local ring properties, enhancing algebraic geometry research capabilities.

Findings

Implemented packages for local ring computations in Macaulay2

Enabled efficient calculation of resolutions, syzygies, and invariants

Facilitated homological methods in algebraic geometry

Abstract

Local rings are ubiquitous in algebraic geometry. Not only are they naturally meaningful in a geometric sense, but also they are extremely useful as many problems can be attacked by first reducing to the local case and taking advantage of their nice properties. Any localization of a ring $R$, for instance, is flat over $R$. Similarly, when studying finitely generated modules over local rings, projectivity, flatness, and freeness are all equivalent. We introduce the packages PruneComplex, Localization and LocalRings for Macaulay2. The first package consists of methods for pruning chain complexes over polynomial rings and their localization at prime ideals. The second package contains the implementation of such local rings. Lastly, the third package implements various computations for local rings, including syzygies, minimal free resolutions, length, minimal generators and presentation, and the Hilbert--Samuel function. The main tools and procedures in this paper involve homological methods. In particular, many results depend on computing the minimal free resolution of modules over local rings.