Propagating weights of tori along free resolutions

TL;DR

This paper develops methods to determine how torus actions propagate through free resolutions of graded modules, enabling explicit calculations and algorithmic implementations in algebraic geometry software.

Contribution

It introduces a systematic approach to track torus weights along free resolutions, enhancing computational tools in equivariant algebraic geometry.

Findings

Algorithms for propagating torus weights are developed.

Implementation in Macaulay2 demonstrates practical utility.

Provides explicit methods for analyzing torus actions on free resolutions.

Abstract

The action of a torus on a graded module over a polynomial ring extends to the entire minimal free resolution of the module. We explain how to determine the action of the torus on the free modules in the resolution, when the resolution can be calculated explicitly. The problem is reduced to analyzing how the weights of a torus propagate along an equivariant map of free modules. The results obtained are used to design algorithms which have been implemented in the software system Macaulay2.