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 and 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
