Purity in categories of sheaves

Abstract

We consider categorical and geometric purity for sheaves of modules over a scheme satisfying some mild conditions, both for the category of all sheaves and for the category of quasicoherent sheaves. We investigate the relations between these four purities; for example, we give a characterisation of geometric pure-injectives in both the quasicoherent and non-quasicoherent case. We also compute a number of examples, in particular describing both the geometric and categorical Ziegler spectra for the category of quasicoherent sheaves over the projective line over a field.