Topological Representation of Sheaf Cohomology of Sites

Abstract

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