Rigid cohomology over Laurent series field I: First definitions and basic properties
Abstract
This is the first in a series of papers in which we construct and study a new $p$adic cohomology theory for varieties over Laurent series fields $k(\!(t)\!)$ in characteristic $p$. This will be a version of rigid cohomology, taking values in the bounded Robba ring $\mathcal{E}_K^\dagger$, and in this paper, we give the basic definitions and constructions. The cohomology theory we define can be viewed as a relative version of Berthelot's rigid cohomology, and is constructed by compactifying $k(\!(t)\!)$varieties as schemes over $k[\![ t]\!]$ rather than over $k(\!(t)\!)$. We reprove the foundational results necessary in our new context to show that the theory is well defined and functorial, and we also introduce a category of `twisted' coefficients. In latter papers we will show some basic structural properties of this theory, as well as discussing some arithmetic applications including the weight monodromy conjecture and independence of $\ell$ results for equicharacteristic local fields.
 Publication:

arXiv eprints
 Pub Date:
 November 2014
 arXiv:
 arXiv:1411.7000
 Bibcode:
 2014arXiv1411.7000L
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Algebraic Geometry;
 14F30;
 11G20
 EPrint:
 45 pages, comments very welcome!