The Stacks project

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:

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

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

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

    3. $X$ 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.

  2. 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.

Remark 95.14.6. We conjecture that up to a replacement as in Stacks, Remark 8.4.9 the functor

\[ p : G\textit{-Principal} \longrightarrow (\mathit{Sch}/S)_{fppf} \]

defines a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$. This would follow if one could show that given

  1. a covering $\{ U_ i \to U\} _{i \in I}$ of $(\mathit{Sch}/S)_{fppf}$,

  2. an group algebraic space $H$ over $U$,

  3. for every $i$ a principal homogeneous $H_{U_ i}$-space $X_ i$ over $U_ i$, and

  4. $H$-equivariant isomorphisms $\varphi _{ij} : X_{i, U_ i \times _ U U_ j} \to X_{j, U_ i \times _ U U_ j}$ satisfying the cocycle condition,

there exists a principal homogeneous $H$-space $X$ over $U$ which recovers $(X_ i, \varphi _{ij})$. The technique of the proof of Bootstrap, Lemma 80.11.8 reduces this to a set theoretical question, so the reader who ignores set theoretical questions will “know” that the result is true. In https://math.columbia.edu/~dejong/wordpress/?p=591 there is a suggestion as to how to approach this problem.

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