Lichtenbaum conjectured the existence of a Weil-\'etale cohomology in order to describe the vanishing order and the special value of the Zeta function of an arithmetic scheme $\mathcal{ X}$ at $s=0$ in terms of Euler-Poincar\'e characteristics. Assuming the (conjectured) finite generation of some \'etale motivic cohomology groups we construct such a cohomology theory for regular schemes proper over $\mathrm{ Spec}(\mathbb{ Z})$. In particular, we obtain (unconditionally) the right Weil-\'etale cohomology for geometrically cellular schemes over number rings. We state a conjecture expressing the vanishing order and the special value up to sign of the Zeta function $\zeta(\mathcal{ X},s)$ at $s=0$ in terms of a perfect complex of abelian groups $R\Gamma_{ W,c}(\mathcal{ X},\mathbb{ Z})$. Then we relate this conjecture to Soul\'e's conjecture and to the Tamagawa number conjecture of Bloch-Kato, and deduce its validity in simple cases.
Categories: math.NT math.AG