Pure morphisms of schemes
by
Bachuki Mesablishvili
Department des Mathématiques, Université Catholique Louvain-La-Neuve, Belgium

We show that the class M of closed immersions of schemes is the second part of a factorization system (E, M) on the category SCH of schemes. We call a morphism in SCH pure if it lies in the stabilization E' of E [1]. We prove that, for any morphism f : R --> S of commutative rings, the induced morphism f* : Spec(S) --> Spec(R) of affine schemes is pure iff f is pure as a morphism of R-modules. Moreover, f* is a M-universal E-morphism [2] iff every ideal v of R is the intersection with R of vS.

Using these facts and the results of [3, 4], we obtain our main result.

Theorem. A quasi-compact morphism of schemes is an effective descent morphism [5] with respect to the (SCH) ^o -indexed category given by quasi-coherent modules if and only if it lies in E'.

The sufficient part of the theorem generalizes Grothendieck's result that faithfully flat quasi-compact morphisms of schemes are effective descent morphisms [6].

References

[1] A. Carboni, G. Janelidze, M. Kelly, R. Paré, On Localization and Stabilization for Factorization Systems, Appl. Categorical Structures, 5 (1997), 1-58.

[2] G. Janelidze, W. Tholen, Facets of Descent, 1, Appl. Categorical Structures, 2 (1994), 1-37.

[3] B. Mesablishvili, Pure Morphisms of Commutative Rings are Effective descent Morphisms for Modules - a New Proof, Theory and Applications of Categories, 7 (2000), 38-42.

[4] B. Mesablishvili, On Some Propeties of Pure Morphisms of Commutative Rings, Theory and Applications of Categories, 10 (2002), 180-186.

[5] G. Janelidze, W. Tholen, Facets of Descent, 2, Appl. Categorical Structures, 5 (1997), 229-248.

[6] A. Grothendieck, Technique de Descente et Theorems d'existence en Geometrie Algebrique, 1, Seminaire Bourbaki, 190, 1959.

Date received: May 31, 2002