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

TL;DR

This paper establishes finiteness, lattice structure, and Poincaré duality for $ abla$-cohomology of smooth curves over Laurent series fields in positive characteristic, extending p-adic local monodromy theory.

Contribution

It proves finiteness and duality properties of $ abla$-cohomology over Laurent series fields, introducing new techniques for p-adic monodromy and cohomology with compact supports.

Findings

$ abla$-cohomology is finite dimensional for smooth curves

Cohomology forms an $ abla$-lattice inside classical rigid cohomology

Poincaré duality holds with restrictions on coefficients

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.