Cup products in sheaf cohomology, pure injectives, and a substitute for projective resolutions

Abstract

Two new definitions are given for cup products in the cohomology of sheaves on an arbitrary Grothendieck topology. These products are shown to have the usual properties. The existence of pure injectives is shown under very general conditions and it is shown how to do many constructions involving projective resolutions under more general conditions when there may not be enough projectives.