User:Maximilian Janisch/latexlist/Algebraic Groups/Principal homogeneous space

This page is a copy of the article Principal homogeneous space in order to test automatic LaTeXification. This article is not my work.

A principal $k$-object in the category of algebraic varieties or schemes. If $5$ is a scheme and $I$ is a group scheme over $5$, then a principal $k$-object in the category of schemes over $I$ is said to be a principal homogeneous space. If $5$ is the spectrum of a field $k$ (cf. Spectrum of a ring) and $I$ is an algebraic $k$-group (cf. Algebraic group), then a principal homogeneous space over $I$ is an algebraic $k$-variety $V$ acted upon (from the left) by $I$ such that if $k$ is replaced by its separable algebraic closure $k$, then each point $v \in V ( \vec { k } )$ defines an isomorphic mapping $g \rightarrow g v$ of the varieties $V _ { k }$ and $\Gamma _ { F }$. A principal homogeneous space $V$ is trivial if and only if $V ( k )$ is non-empty. The set of classes of isomorphic principal homogeneous spaces over a smooth algebraic group $I$ can be identified with the set of Galois cohomology $H ^ { 1 } ( k , \Gamma )$. In the general case the set of classes of principal homogeneous spaces over an $5$-group scheme $I$ coincides with the set of one-dimensional non-Abelian cohomology $H ^ { 1 } ( S _ { T } , \Gamma )$. Here $S _ { T }$ is some Grothendieck topology on the scheme $5$ [2].

Principal homogeneous spaces have been computed in a number of cases. If $k$ is a finite field, then each principal homogeneous space over a connected algebraic $k$-group is trivial (Lang's theorem). This theorem also holds if $k$ is a $D$-adic number field and $I$ is a simply-connected semi-simple group (Kneser's theorem). If $\Gamma = \Gamma _ { m , S }$ is a multiplicative $5$-group scheme, then the set of classes of principal homogeneous spaces over $I$ becomes identical with the Picard group $\operatorname { Pic } ( S )$ of $5$. In particular, if $5$ is the spectrum of a field, this group is trivial. If $\Gamma = \Gamma _ { \alpha , S }$ is an additive $5$-group scheme, then the set of classes of principal homogeneous spaces over $I$ becomes identical with the one-dimensional cohomology group $H ^ { 1 } ( S , O _ { S } )$ of the structure sheaf $O _ { S }$ of $5$. In particular, this set is trivial if $5$ is an affine scheme. If $k$ 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 $k$-group $I$ is based on the study of the Tate–Shafarevich set $\square ( \Gamma )$, which consists of the principal homogeneous spaces over $I$ with rational points in all completions $k _ { V }$ with respect to the valuations of $k$. If $I$ is an Abelian group over the field $k$, then the set of classes of principal homogeneous spaces over $I$ 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 $k$-sets, where $k$ is a group. Let $k$ be a finite (profinite, etc.) group. Let $k$ be a $k$-set, i.e. a set $k$ with an action $G \times E \rightarrow E$ of $k$ on it. Let $I$ be a $k$-group, i.e. a group object in the category of $k$-sets, which means that $I$ is a group and that the action of $k$ on $I$ is by group automorphisms of $I$: $( x y ) ^ { \gamma } = x ^ { \gamma } y ^ { \gamma }$ for $\gamma \in G$, $x , y \in \Gamma$. One says that $I$ operates compatibly with the $k$-action from the left on $k$ if there is a $I$-action $\Gamma \times E \rightarrow E$ on $k$ such that $( \gamma x ) ^ { g } = ( \gamma ^ { g } ) ( x ^ { g } )$ for $g \in G$, $\gamma \in \Gamma$, $x \in E$. A principal homogeneous space over $I$ in this setting is a $k$-set $P$ on which $I$ acts compatibly with the $k$-action and such that for all $x , y \in P$ there is a $\gamma \in \Gamma$ such that $y = \gamma x$. (This is the property to which the word "principal" refers; one also says that $P$ is an affine space over $I$.) In this case there is a natural bijective correspondence between $H ^ { 1 } ( G , \Gamma )$ and isomorphism classes of principal homogeneous spaces over $I$ and, in fact, $H ^ { 1 } ( G , \Gamma )$ (for non-Abelian $I$) is sometimes defined this way.