Rigid cohomology over Laurent series fields II: Finiteness and Poincaré duality for smooth curves
Abstract
In this paper we prove that the $\mathcal{E}^\dagger_K$valued cohomology, introduced in [9] is finite dimensional for smooth curves over Laurent series fields $k((t))$ in positive characteristic, and forms an $\mathcal{E}^\dagger_K$lattice inside `classical' $\mathcal{E}_K$valued rigid cohomology. We do so by proving a suitable version of the padic local monodromy theory over $\mathcal{E}^\dagger_K$, and then using an étale pushforward for smooth curves to reduce to the case of $\mathbb{A}^1$. We then introduce $\mathcal{E}^\dagger_K$valued cohomology with compact supports, and again prove that for smooth curves, this is finite dimensional and forms an $\mathcal{E}^\dagger_K$lattice in $\mathcal{E}_K$valued cohomology with compact supports. Finally, we prove Poincaré duality for smooth curves, but with restrictions on the coefficients.
 Publication:

arXiv eprints
 Pub Date:
 December 2014
 arXiv:
 arXiv:1412.5300
 Bibcode:
 2014arXiv1412.5300L
 Keywords:

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