### 95.14.5 Principal homogeneous spaces

Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. We define a category $G\textit{-Principal}$ as follows:

An object of $G\textit{-Principal}$ is a triple $(U, b, X)$ where

$U$ is an object of $(\mathit{Sch}/S)_{fppf}$,

$b : U \to B$ is a morphism over $S$, and

$X$ is a principal homogeneous $G_ U$-space over $U$ where $G_ U = U \times _{b, B} G$.

See Groupoids in Spaces, Definition 78.9.3.

A morphism $(U, b, X) \to (U', b', X')$ is given by a pair $(f, g)$, where $f : U \to U'$ is a morphism of schemes over $B$, and $g : X \to U \times _{f, U'} X'$ is an isomorphism of principal homogeneous $G_ U$-spaces.

Thus $G\textit{-Principal}$ is a category and

\[ p : G\textit{-Principal} \longrightarrow (\mathit{Sch}/S)_{fppf}, \quad (U, b, X) \longmapsto U \]

is a functor. Note that the fibre category of $G\textit{-Principal}$ over $U$ is the disjoint union over $b : U \to B$ of the categories of principal homogeneous $U \times _{b, B} G$-spaces, hence is a groupoid.

In the special case $S = B$ the objects are simply pairs $(U, X)$ where $U$ is a scheme over $S$, and $X$ is a principal homogeneous $G_ U$-space over $U$. Moreover, morphisms are simply cartesian diagrams

\[ \xymatrix{ X \ar[d] \ar[r]_ g & X' \ar[d] \\ U \ar[r]^ f & U' } \]

where $g$ is $G$-equivariant.

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