# Principal homogeneous space

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A principal $G$- object in the category of algebraic varieties or schemes. If $S$ is a scheme and $\Gamma$ is a group scheme over $S$, then a principal $G$- object in the category of schemes over $\Gamma$ is said to be a principal homogeneous space. If $S$ is the spectrum of a field $k$( cf. Spectrum of a ring) and $\Gamma$ is an algebraic $k$- group (cf. Algebraic group), then a principal homogeneous space over $\Gamma$ is an algebraic $k$- variety $V$ acted upon (from the left) by $\Gamma$ such that if $k$ is replaced by its separable algebraic closure $\overline{k}$, then each point $v \in V ( \overline{k} )$ defines an isomorphic mapping $g \rightarrow gv$ of the varieties $V _{\overline{ {k}} }$ and $\Gamma _{\overline{ {k}} }$. 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 $\Gamma$ 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 $S$- group scheme $\Gamma$ 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 $S$[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 $p$- adic number field and $\Gamma$ is a simply-connected semi-simple group (Kneser's theorem). If $\Gamma = \Gamma _{m,S}$ is a multiplicative $S$- group scheme, then the set of classes of principal homogeneous spaces over $\Gamma$ becomes identical with the Picard group $\mathop{\rm Pic}\nolimits (S)$ of $S$. In particular, if $S$ is the spectrum of a field, this group is trivial. If $\Gamma = \Gamma _{a,S}$ is an additive $S$- group scheme, then the set of classes of principal homogeneous spaces over $\Gamma$ becomes identical with the one-dimensional cohomology group $H ^{1} (S,\ {\mathcal O} _{S} )$ of the structure sheaf ${\mathcal O} _{S}$ of $S$. In particular, this set is trivial if $S$ 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 $\Gamma$ $\def\Sha{ {\mathop{\amalg\kern-0.30em\amalg}}}$ is based on the study of the Tate–Shafarevich set $\Sha ( \Gamma )$, which consists of the principal homogeneous spaces over $\Gamma$ with rational points in all completions $k _{V}$ with respect to the valuations of $k$. If $\Gamma$ is an Abelian group over the field $k$, then the set of classes of principal homogeneous spaces over $\Gamma$ 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

The notion of a principal homogeneous space is not restricted to algebraic geometry. For instance, it is defined in the category of $G$- sets, where $G$ is a group. Let $G$ be a finite (profinite, etc.) group. Let $E$ be a $G$- set, i.e. a set $E$ with an action $G \times E \rightarrow E$ of $G$ on it. Let $\Gamma$ be a $G$- group, i.e. a group object in the category of $G$- sets, which means that $\Gamma$ is a group and that the action of $G$ on $\Gamma$ is by group automorphisms of $\Gamma$: $(xy) ^ \gamma = x ^ \gamma y ^ \gamma$ for $\gamma \in G$, $x,\ y \in \Gamma$. One says that $\Gamma$ operates compatibly with the $G$- action from the left on $E$ if there is a $\Gamma$- action $\Gamma \times E \rightarrow E$ on $E$ such that $( \gamma x ) ^{g} = ( \gamma ^{g} )(x ^{g} )$ for $g \in G$, $\gamma \in \Gamma$, $x \in E$. A principal homogeneous space over $\Gamma$ in this setting is a $G$- set $P$ on which $\Gamma$ acts compatibly with the $G$- 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 $\Gamma$.) In this case there is a natural bijective correspondence between $H ^{1} (G,\ \Gamma )$ and isomorphism classes of principal homogeneous spaces over $\Gamma$ and, in fact, $H ^{1} (G,\ \Gamma )$( for non-Abelian $\Gamma$) is sometimes defined this way.