Descent Theory for Schemes

In this paper we continue the investigation of some aspects of descent theory for schemes that was begun in [11]. Let SCH be a category of schemes. We show that quasi-compact pure morphisms of schemes are effective descent morphisms with respect to SCH-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 quasicoherent 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 SCH. Finally, we present an alternative characterization of pure morphisms of schemes. 2000 Mathematics Subject Classification: 14A15, 18A20, 18A32. Key words and phrases: Scheme, pure morphism, descent theory.

• Type: Journal
• Year: 2004
• Where: ACS
• Authors: Bachuki Mesablishvili
• Keywords: ACS 2004, ACS 2007, Pure Morphism, Quasi-coherent Modules, Quasi-compact Pure Morphisms