Twisted de Rham cohomology, homological definition of the integral and “Physics over a ring”

We use the notion of the twisted de Rham cohomology to give meaning to an integral of the form ∫g(x)ef(x)dx over an arbitrary ring. We discuss also a definition of a family of integrals and some properties of the above homological definition of the integral. We show how to use the twisted de Rham cohomology to define the Frobenius map on the p-adic cohomology. Finally, we consider two-dimensional topological quantum field theories with general coefficients.