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

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 $5$ .
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).