  We continue the study of cluster algebras initiated in math.RT/0104151 and
math.RA/0208229. We develop a new approach based on the notion of an upper
cluster algebra, defined as an intersection of certain Laurent polynomial
rings. Strengthening the Laurent phenomenon from math.RT/0104151, we show that,
under an assumption of "acyclicity", a cluster algebra coincides with its
"upper" counterpart, and is finitely generated. In this case, we also describe
its defining ideal, and construct a standard monomial basis. We prove that the
coordinate ring of any double Bruhat cell in a semisimple complex Lie group is
naturally isomorphic to the upper cluster algebra explicitly defined in terms
of relevant combinatorial data.
