We define the twisted de Rham cohomology and show how to use it to define the
notion of 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 homological definition of integral. We show how to use the twisted de
Rham cohomology in order to define the Frobenius map on the p-adic cohomology.
Finally, we consider two-dimensional topological quantum field theories with
general coefficients.Comment: 15 page
