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

Flach and Morin constructed in (Doc. Math. 23 (2018), 1425–1560) Weil-'etale cohomology $H_{\text{W,c}}^i(X,\mathbb{Z}(n))$ for a proper, regular arithmetic scheme $X$ (i.e. separated and of finite type over $\mathrm{Spec}\mathbb{Z}$) 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 $H^i(X_{\acute{\text{e}}\text{t}},{\mathbb{Z}}^c(n))$, 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 $H_{\text{W,c}}^i(X,\mathbb{Z}(n))$ is defined unconditionally.