**B. Mesablishvili**

## More on Descent Theory for Schemes

**Abstract:**

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.

**Keywords:**

Scheme, pure morphism, descent theory.

**MSC 2000:** 14A15, 18A20, 18A32