The Weil-étale topology for number rings

Abstract

There should be a Grothendieck topology for an arithmetic scheme $X$ such that the Euler characteristic of the cohomology groups of the constant sheaf $\mathbb Z$ with compact support at infinity gives, up to sign, the leading term of the zeta-function of $X$ at $s = 0$. We construct a topology (the Weil-étale topology) for the ring of integers in a number field whose cohomology groups $H^i(\mathbb Z) $ determine such an Euler characterstic if we restrict to $i\leq 3$.