Principal homogeneous space

A principal -object in the category of algebraic varieties or schemes. If is a scheme and is a group scheme over , then a principal -object in the category of schemes over is said to be a principal homogeneous space. If is the spectrum of a field (cf. Spectrum of a ring) and is an algebraic -group (cf. Algebraic group), then a principal homogeneous space over is an algebraic -variety acted upon (from the left) by such that if is replaced by its separable algebraic closure , then each point defines an isomorphic mapping of the varieties and . A principal homogeneous space is trivial if and only if is non-empty. The set of classes of isomorphic principal homogeneous spaces over a smooth algebraic group can be identified with the set of Galois cohomology . In the general case the set of classes of principal homogeneous spaces over an -group scheme coincides with the set of one-dimensional non-Abelian cohomology . Here is some Grothendieck topology on the scheme [2].

Principal homogeneous spaces have been computed in a number of cases. If is a finite field, then each principal homogeneous space over a connected algebraic -group is trivial (Lang's theorem). This theorem also holds if is a -adic number field and is a simply-connected semi-simple group (Kneser's theorem). If is a multiplicative -group scheme, then the set of classes of principal homogeneous spaces over becomes identical with the Picard group of . In particular, if is the spectrum of a field, this group is trivial. If is an additive -group scheme, then the set of classes of principal homogeneous spaces over becomes identical with the one-dimensional cohomology group of the structure sheaf of . In particular, this set is trivial if is an affine scheme. If is a global field (i.e. an algebraic number field or a field of algebraic functions in one variable), then the study of the set of classes of principal homogeneous spaces over an algebraic -group is based on the study of the Tate–Shafarevich set , which consists of the principal homogeneous spaces over with rational points in all completions with respect to the valuations of . If is an Abelian group over the field , then the set of classes of principal homogeneous spaces over forms a group (cf. Weil–Châtelet group).

References

[1]	J.-P. Serre, "Cohomologie Galoisienne" , Springer (1973) MR0404227 Zbl 0259.12011
[2]	M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , Masson (1970) MR0302656 MR0284446 Zbl 0223.14009 Zbl 0203.23401
[3]	S. Lang, J. Tate, "Principal homogeneous spaces over abelian varieties" Amer. J. Math. , 80 (1958) pp. 659–684 MR0106226 Zbl 0097.36203

Comments

The notion of a principal homogeneous space is not restricted to algebraic geometry. For instance, it is defined in the category of -sets, where is a group. Let be a finite (profinite, etc.) group. Let be a -set, i.e. a set with an action of on it. Let be a -group, i.e. a group object in the category of -sets, which means that is a group and that the action of on is by group automorphisms of : for , . One says that operates compatibly with the -action from the left on if there is a -action on such that for , , . A principal homogeneous space over in this setting is a -set on which acts compatibly with the -action and such that for all there is a such that . (This is the property to which the word "principal" refers; one also says that is an affine space over .) In this case there is a natural bijective correspondence between and isomorphism classes of principal homogeneous spaces over and, in fact, (for non-Abelian ) is sometimes defined this way.