Computing Direct Sum Decompositions

TL;DR

This paper introduces two practical algorithms for decomposing finitely generated modules over certain algebras, enabling applications in algebraic geometry and singularity theory, with new insights into module summands.

Contribution

It presents two novel algorithms for indecomposable summand computation in graded and local algebra settings, expanding computational tools in algebra and geometry.

Findings

Algorithms successfully compute indecomposable summands in various algebraic contexts.

Examples reveal new phenomena in module decomposition, including Frobenius pushforwards.

Applications include decompositions of coherent sheaves and syzygies over Artinian rings.

Abstract

We describe and prove correctness of two practical algorithms for finding indecomposable summands of finitely generated modules over a finitely generated k-algebra R. The first algorithm applies in the (multi)graded case, which enables the computation of indecomposable summands of coherent sheaves on subvarieties of toric varieties (in particular, for varieties embedded in projective space); the second algorithm applies when R is local and k is a finite field, opening the door to computing decompositions in singularity theory. We also present multiple examples, including some which present previously unknown phenomena regarding the behavior of summands of Frobenius pushforwards (including in the non-graded case) and syzygies over Artinian rings.