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| 概要 |
Lichtenbaum has conjectured (Ann of Math. (2) 170(2) (2009), 657–683) the existence of a Grothendieck topology for an arithmetic scheme X such that the Euler characteristic of the cohomology groups of... the constant sheaf Z with compact support at infinity gives, up to sign, the leading term of the zeta function ζ_X(s) at s = 0. In this paper we consider the category of sheaves X^-_L on this conjectural site for X = Spec(O_F) the spectrum of a number ring. We show that X^^-_L has, under natural topological assumptions, a welldefined fundamental group whose abelianization is isomorphic, as a topological group, to the Arakelov–Picard group of F. This leads us to give a list of topological properties that should be satisfied by X^^-_L. These properties can be seen as a global version of the axioms for the Weil group. Finally, we show that any topos satisfying these properties gives rise to complexes of étale sheaves computing the expected Lichtenbaum cohomology.続きを見る
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