The separable transfer

A separable transfer map is defined for finite surjectivemorphisms from integral schemes to a Noetherian integral normal base, for both chain complexes of sheaves and presheaves of spectra. The spectrum level transfer is a map from a direct image paired with the classifying space of a translation groupoid, and taking values in the original presheaf. More traditional transfer maps are recovered from global sections of the stack associated to this groupoid. These constructions can be made in a big site model for the motivic stable category.