Weil-étale cohomology and duality for arithmetic schemes in negative weights

Paper

Abstract

Flach and Morin constructed in (Doc. Math. 23 (2018), 1425--1560) Weil-étale cohomology

for a proper, regular arithmetic scheme

(i.e. separated and of finite type over

) and

. In the case when

, we generalize their construction to an arbitrary arithmetic scheme

, thus removing the proper and regular assumption. The construction uses étale motivic cohomology groups

, as studied by Geisser (Ann. of Math. (2) 172 (2010), 1095--1126), and assumes their finite generation for

. We give a class of X for which finite generation is known, and hence

is defined unconditionally.