Topological representation of sheaf cohomology of sites

Abstract

For a site S (with enough points), we construct a topological space X(S) and a full embedding ' of the category of sheaves on S into those on X(S) (i.e., a morphism of toposes ': Sh(X(S)) ! Sh(S)). The embedding will be shown to induce a full embedding of derived categories, hence isomorphisms H(S; A) = H(X(S); 'A) for any abelian sheaf A on S. 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(Yet; A) and the ordinary sheaf cohomology H(X(Y ); 'A), for any sheaf A for the etale topology on Y .