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

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.