The author continues his recent investigation of some aspects of descent theory for schemes. Let $\CH$ be a category of schemes. We show that quasi-compact pure morphisms of schemes are effective descent morphisms with respect to $\CH$-indexed categories given by (i) quasi-coherent modules of finite type, (ii) flat quasi-coherent modules, (iii) flat quasi-coherent modules of finite type, (iv) locally projective quasi-coherent modules of finite type. Moreover, we prove that a quasi-compact morphism of schemes is pure precisely when it is a stable regular epimorphism in $\CH$. Finally, we present an alternative characterization of pure morphisms of schemes.
Scheme, pure morphism, descent theory.
MSC 2000: 14A15, 18A20, 18A32