## The overconvergent de Rham-Witt complex

##### Author(s)

Davis, Christopher (Christopher James)
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##### Alternative title

Over convergent de Rham-Witt complex

##### Other Contributors

Massachusetts Institute of Technology. Dept. of Mathematics.

##### Advisor

Kiran S. Kedlaya.

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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 dim C < 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 ... (See Section 4.1 for the notation.) We cover our smooth affine C with affines B each of which is finite, tale over a localization of a polynomial algebra. For these particular affines, we decompose ... 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(...) is not surjective for p > 2.

##### Description

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2009. Includes bibliographical references (p. 83-84).

##### Date issued

2009##### Department

Massachusetts Institute of Technology. Dept. of Mathematics.##### Publisher

Massachusetts Institute of Technology

##### Keywords

Mathematics.