TY - BOOK

T1 - The overconvergent de Rham-Witt complex

AU - Davis, Christopher James

N1 - Advised by Kiran Kedlaya. The link is given below, but notice that the results of the first four chapters are improved and expanded in my joint work with Andreas Langer and Thomas Zink, and the last two chapters are improved and expanded in my joint work with Kiran Kedlaya. http://dspace.mit.edu/handle/1721.1/50593

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

M3 - Ph.D. thesis

BT - The overconvergent de Rham-Witt complex

PB - Department of Mathematical Sciences, Faculty of Science, University of Copenhagen

ER -