On the Weil-étale topos of regular arithmetic schemes (Q455491)

Let \(F\) be a number field. The author defines a topology \(\overline{\mathrm{Spec}(\mathcal O_F)}_W\) so that the groups \(H^i_c(\overline{\mathrm{Spec}(\mathcal O_F)}_W,\mathbb Z)\) recover the groups introduced by Lichtenbaum in his study of Weil-étale cohomology groups. Let \(\mathcal X\) be a separated scheme of finite type over \(\mathbb Z\). Artin-Verdier defines a topo \(\overline{\mathcal X}_{\mathrm{et}}\) so that there are complementary open and closed immersions NEWLINE\[NEWLINE\mathcal X_{\mathrm{et}}\to \overline{\mathcal X}_{\mathrm{et}} \leftarrow Sh(\mathcal X_\infty),NEWLINE\]NEWLINE where \(\mathcal X_\infty\) is the topological quotient space \(\mathcal X(\mathbb C)/ {\mathrm{Gal}(\mathbb C/\mathbb R)}\). For \(\mathcal Y=\mathcal X\) or \(\overline{\mathcal X}\), let NEWLINE\[NEWLINE{\mathcal Y}_W={\mathcal Y}_{\mathrm{et}}\times _{\overline{\mathrm{Spec}(\mathbb Z)}_{\mathrm{et}}} \overline{\mathrm{Spec}(\mathbb Z)}_W.NEWLINE\]NEWLINE In the case where \(\mathcal X\) is regular and proper over \(\mathbb Z\), the author proves that the cohomology groups of \(\mathcal X_W\) satisfy some (but not all) properties that one hopes for the Weil-étale cohomology groups.