Atlas: Pure morphisms of schemes by Bachuki Mesablishvili

Atlas home || Conferences | Abstracts | about Atlas

Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories
September 23-28, 2002
Fields Institute
Toronto, ON, Canada

Organizers
George Janelidze, Georgian Academy of Sciences, Bodo Pareigis, University of Munich, Walter Tholen, York University

View Abstracts
Conference Homepage

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