Weil-\'{e}tale cohomology and duality for arithmetic schemes in negative weights

TL;DR

This paper extends Weil-étale cohomology to all arithmetic schemes in negative weights by leveraging étale motivic cohomology, providing new duality results and conditions for finite generation.

Contribution

It generalizes Weil-étale cohomology construction to arbitrary arithmetic schemes for negative weights, removing previous restrictions and establishing new duality theorems.

Findings

Defines Weil-étale cohomology for all arithmetic schemes in negative weights.

Identifies classes of schemes with known finite generation of motivic cohomology.

Provides duality results for these cohomology groups.

Abstract

Flach and Morin constructed in (Doc. Math. 23 (2018), 1425--1560) Weil-\'etale cohomology $H^i_\text{W,c} (X, \mathbb{Z} (n))$ for a proper, regular arithmetic scheme $X$ (i.e. separated and of finite type over $\operatorname{Spec} \mathbb{Z}$) and $n \in \mathbb{Z}$. In the case when $n < 0$, we generalize their construction to an arbitrary arithmetic scheme $X$, thus removing the proper and regular assumption. The construction uses \'etale motivic cohomology groups $H^i(X_\text{\'et}, \mathbb{Z}^c(n))$, as studied by Geisser (Ann. of Math. (2) 172 (2010), 1095--1126), and assumes their finite generation for $n < 0$. We give a class of X for which finite generation is known, and hence $H^i_\text{W,c} (X, \mathbb{Z} (n))$ is defined unconditionally.