Rigid cohomology over Laurent series fields II: Finiteness and Poincaré duality for smooth curves

Abstract

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

40 pages, comments very welcome!

References in corpus (1)

• Rigid cohomology over Laurent series fields III: Absolute coefficients and arithmetic applications