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arXiv · 1412.5300

Rigid cohomology over Laurent series fields II: Finiteness and Poincar\'e duality for smooth curves

Christopher Lazda·Ambrus Pál
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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 p-adic local monodromy theory over $\mathcal{E}^\dagger_K$, and then using an \'{e}tale 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 Poincar\'{e} duality for smooth curves, but with restrictions on the coefficients.

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Christopher Lazda, Ambrus Pál. 2014-12-17. Rigid cohomology over Laurent series fields II: Finiteness and Poincar\'e duality for smooth curves. https://arxiv.org/abs/1412.5300

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