Cohomology ring of non-compact abelian arrangements

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