Cohomology ring of non‐compact abelian arrangements

Abstract We give an Orlik–Solomon‐type presentation for the cohomology ring of arrangements in a non‐compact abelian Lie group. The new insight consists of comparing arrangements in different abelian groups. Our work is based on the Varchenko–Gel'fand ring for real hyperplane arrangements and from that we deduce the cohomology rings of all other abelian arrangements. As a by‐product, we obtain a new proof of the Orlik–Solomon relations and those of De Concini–Procesi.