We define the overconvergent de Rham-Witt complex for a smooth affine variety over a perfect field in characteristic p. We show that, after tensoring with Q, its cohomology agrees with Monsky-Washnitzer cohomology. If dimC < p, we have an isomorphism integrally. One advantage of our construction is that it does not involve a choice of lift to characteristic zero.

To prove that the cohomology groups are the same, we first define a comparison map.
We cover our smooth affine with affines each of which is finite, etale over a localization of a polynomial algebra. For these particular affines, we decompose the overconvergent de Rham-Witt complex into an integral part and a fractional part and then show that the integral part is isomorphic to the Monsky-Washnitzer complex and that the fractional part is acyclic. We deduce our result from a homotopy argument and the fact that our complex is a Zariski sheaf with sheaf cohomology equal to zero in positive degrees. (For the latter, we lift the proof from [4] and include it as an appendix.)

We end with two chapters featuring independent results. In the first, we reinterpret several rings from p-adic Hodge theory in such a way that they admit analogues which use big Witt vectors instead of p-typical Witt vectors. In this generalization we check that several familiar properties continue to be valid. In the second, we offer a proof that the Frobenius map on W(O_Cp) is not surjective for p > 2.