The overconvergent de Rham-Witt complex

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.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2009

Author dc:creator

• Davis, Christopher (Christopher James)

Advisor dc:contributor.advisor

• Kiran S. Kedlaya.

• Mathematics.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/50593

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/50593