2026-09-19T19:46:01Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/505932022-01-13T07:54:35Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Kiran S. Kedlaya.
Davis, Christopher (Christopher James)
Massachusetts Institute of Technology. Dept. of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2010-01-07T20:58:11Z
2010-01-07T20:58:11Z
2009
2009
http://hdl.handle.net/1721.1/50593
465219096
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2009.
Includes bibliographical references (p. 83-84).
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.
by Christopher Davis.
Ph.D.
84 p.
eng
Massachusetts Institute of Technology
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.
http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
The overconvergent de Rham-Witt complex
Over convergent de Rham-Witt complex
Thesis
application/pdf
false
Publication
unknown
unknown
<Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="d3c5755e-8014-493e-8262-d503ad3e7b12">
<Type xmlns="https://www.openaire.eu/cerif-profile/vocab/COAR_Publication_Types">http://purl.org/coar/resource_type/c_1843</Type>
<Language>eng</Language>
<Title>The overconvergent de Rham-Witt complex</Title>
<Subtitle>Over convergent de Rham-Witt complex</Subtitle>
<PublishedIn>
<Publication>
</Publication>
</PublishedIn>
<PublicationDate>2009</PublicationDate>
<Authors>
<Author>
<DisplayName>Davis, Christopher (Christopher James)</DisplayName>
<Affiliation>
<OrgUnit>
</OrgUnit>
</Affiliation>
</Author>
</Authors>
<Editors>
</Editors>
<Publishers>
<Publisher>
<DisplayName>Massachusetts Institute of Technology</DisplayName>
<OrgUnit />
</Publisher>
</Publishers>
<License>http://dspace.mit.edu/handle/1721.1/7582</License>
<Keyword>Mathematics.</Keyword>
<Abstract>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.</Abstract>
<Access xmlns="http://purl.org/coar/access_right"
>
</Access>
</Publication>