Stein manifolds are the complex-analytic analogues of affine algebraic varieties. A classical result of Forster shows that one can recover a Stein manifold S from its algebra A of global holomorphic functions. In this talk, I will present a purely algebraic description of the cohomology of S with finite coefficients in terms of A. This provides a bridge between the topology of S and the algebraic geometry of A. I will discuss applications concerning analytic cycles and sums of squares in analytic geometry.