  We study a graded algebra D=D(L,G) defined by a finite lattice L and a subset
G in L, a so-called building set. This algebra is a generalization of the
cohomology algebras of hyperplane arrangement compactifications found in work
of De Concini and Procesi. Our main result is a representation of D, for an
arbitrary atomic lattice L, as the Chow ring of a smooth toric variety that we
construct from L and G. We describe this variety both by its fan and
geometrically by a series of blowups and orbit removal. Also we find a Groebner
basis of the relation ideal of D and a monomial basis of D over Z.
