On the Weil-étale cohomology of the ring of $S$-integers

Abstract

In this article, we first briefly introduce the history of the Weil-étale cohomology theory of arithmetic schemes and review some important results established by Lichtenbaum, Flach and Morin. Next we generalize the Weil-etale cohomology to $S$-integers and compute the cohomology for constant sheaves $\mathbb{Z}$ or $\mathbb{R}$. We also define a Weil-étale cohomology with compact support $H_c(Y_W, -)$ for $Y=Spec \mathcal{O}_{F,S}$ where $F$ is a number field, and computed them. We verify that these cohomology groups satisfy the axioms state by Lichtenbaum. As an application, we derive a canonical representation of Tate sequence from $RGamma_c(Y_W,\mathbb{Z})$. Motivated by this result, in the final part, we define an étale complex $RGm$, such that the complexes $\mathbb{Z}$-dual of the complex $\RG(U_{et},R\Gm),\,\mathbb{Z})[2]$ is canonically quasi-isomorphic to $τ^{\leq 3}\RG_c(U_W,\mathbb{Z})$ for arbitrary étale $U$ over $Spec \mathcal{O}_{F}$. This quasi-isomorphism provides a possible approach to define the Weil-etale cohomology for higher dimensional arithmetic schemes, as the Weil groups are not involved in the definition of $R\Gm$.