The integer cohomology algebra of toric arrangements

TL;DR

This paper computes the integer cohomology algebra of toric arrangement complements, revealing how it depends on combinatorial data, and introduces new combinatorial tools and results in the study of arrangements.

Contribution

It provides a detailed computation of the cohomology ring for toric arrangements with integer coefficients and extends combinatorial methods to this setting.

Findings

Cohomology ring explicitly computed for toric arrangements

A combinatorial version of Brieskorn's lemma established

Uniqueness result for realizations of arithmetic matroids

Abstract

We compute the cohomology ring of the complement of a toric arrangement with integer coefficients and investigate its dependency from the arrangement's combinatorial data. To this end, we study a morphism of spectral sequences associated to certain combinatorially defined subcomplexes of the toric Salvetti category in the complexified case, and use a technical argument in order to extend the results to full generality. As a byproduct we obtain: -a "combinatorial" version of Brieskorn's lemma in terms of Salvetti complexes of complexified arrangements, -a uniqueness result for realizations of arithmetic matroids with at least one basis of multiplicity 1.