Abstract: Into how many pieces is R^n cut, by n hyperplanes in general position in R^n? Generalizing the answer to this question, one defines for any set of hyperplanes in a vector space, an invariant called the Orlik-Solomon algebra.

When the base field is the p-adic numbers Q_p, there is another factor in the game: the Bruhat-Tits building of the general linear group over Q_p. It is a highly symmetric simplicial complex on which the group acts. E. De Shalit attached to a collection of hyperplanes in a p-adic vecor space a certain local system of algebras, which are local versions of the Orlik-Solomon algebra. We will define the above notions, calculate the cohomology of the above system (for a finite number of hyperplanes), and show a recent combinatorial application.