The Weil-étale topology for number rings


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 $i\leq 3$.


Stephen Lichtenbaum

Department of Mathematics
Brown University
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151 Thayer Street
Providence, RI 02912
United States