Topological representation of sheaf cohomology of sites

Abstract

For a site & (with enough points), we construct a topological space X(&) and a full embedding ϕ* of the category of sheaves on & into those on X (&) (i.e., a morphism of toposes ϕ:Sh (X(&)) →Sh(&)). The embedding will be shown to induce a full embedding of derived categories, hence isomorphisms H*(&,A) = H*(X(&), ϕ*A) for any Abelian sheaf A on &. As a particular case, this will give for any scheme Y a topological space X (Y) and a functorial isomorphism between the etale cohomology H*(Y et,A) and the ordinary sheaf cohomology H*(X((Y),),ϕ*A), for any sheaf A for the etale topology on Y.