Descent morphisms and Galois theory
Bachuki Mesablishvili – 8 November 2000
Let C be a category with pullbacks and let E be a pullback-stable class of morphisms of C which is closed under composition with the isomorphisms. E defines a pseudofunctor from Cop to Cat, also denoted by E, which sends an object C to E/C, the full subcategory of the slice category C/C consisting of arrows in E with codomain C. We may then consider the category of E-descent morphisms in C as defined in G. Janelidze and W. Tholen, Facets of Descent I, Applied Categorical Structures 2, 1994:1-37.
Suppose J is a Grothendieck topology on C which is generated by the subcanonical pretopology J' for which a family (Ci->D) is in J' if and only if: the coproduct C of the Ci exists, it is universal and disjoint, and the induced morphism C-->D is both a universal regular epimorphism and an E-descent morphism.
Using the Yoneda embedding Y:Cop->Sh(C,J) we prove several results related to E-effective descent morphisms, Galois objects, and torsors in C. As an application, we get the following two theorems.
(For a commutative group G in C we denote by Gal(G,C) the abelian group of (isomorphism classes of) Galois G-objects.)
Theorem 1 Let C be a category with pullbacks and let J be the Grothendieck topology on C as above. If G is a commutative group in E (i.e. the unique morphism from G to the terminal object lies in E) then every Galois homC(-,G)-object (= homC(-,G)-torsor) in Sh(C,J) is representable by a Galois G-object. (This means that if A is a homC(-,G)-torsor in Sh(C,J), then there exists a Galois G-object a in C such that A is isomorphic to homC(-,a).
Theorem 2(a) Let B be a Grothendieck topos, and let R be a commutative ring in B. Denote by C the opposite of the category of commutative R-algebras, and let E be the class of all morphisms in C. If G is a cocommutative finite Hopf R-algebra with antipode, then there is an isomorphism between Gal(G,C) and Ext1Ab(Sh(C,J))(hom(-,Gx),hom(-,R[Z]) where J is the Grothendieck topology on C which is generated by finite families of pure morphisms, Gx is the R-dual of G, and R[Z] is the group ring.
(b) Let X be a scheme, let C be the category of schemes over X, and let J be the Grothendieck topology on C which is generated by families of pure morphisms in C, in the sense of B. Mesablishvili, (Effective) descent morphisms in the category of schemes, in Abstracts of CT2000 (Como), 154-156. Deonte by E the class of quasi-compact pure morphisms. If G is a group in E which is locally a projective module over X, then Gal(G,C) is isomorphic to Ext1Ab(Sh(C,J))(hom(-,GK),hom(-,Gm) where GK is the Cartier dual of G, and Gm is the multiplication group scheme.