The integer cohomology algebra of toric arrangements

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.