The colorful Helly theorem and colorful resolutions of ideals

TL;DR

This paper explores the deep connection between topological Helly theorems and algebraic properties of ideals, providing algebraic generalizations and a syzygetic version of Helly's theorem.

Contribution

It establishes a novel correspondence between topological and algebraic Helly theorems and introduces algebraic generalizations, including a syzygetic version of Helly's theorem.

Findings

Unified topological and algebraic Helly theorems

Algebraic generalizations of colorful Helly theorem

A syzygetic version of Helly's theorem

Abstract

We demonstrate that the topological Helly theorem and the algebraic Auslander-Buchsbaum may be viewed as different versions of the same phenomenon. Using this correspondence we show how the colorful Helly theorem of I.Barany and its generalizations by G.Kalai and R.Meshulam translates to the algebraic side. Our main results are algebraic generalizations of these translations, which in particular gives a syzygetic version of Hellys theorem.