This work was done with Mark de Longueville.
The ring structure of the integral cohomology of complements of real linear subspace arrangements is considered. While the additive structure of the cohomology is given in terms of the intersection poset and dimension function by a theorem of Goresky and MacPherson, we describe the multiplicative structure in terms of the intersection poset, the dimension function and orientations of the participating subspaces for the class of arrangements without pairs of intersections of codimension one. In particular, this yields a description of the integral cohomology ring of complex arrangements conjectured by Yuzvinsky. For general real arrangements a weaker result is obtained. The approach is geometric and the methods are elementary.
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