Subsection 95.14.5 (04UN): 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:

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

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

a covering $\{ U_ i \to U\} _{i \in I}$ of $(\mathit{Sch}/S)_{fppf}$ ,
an group algebraic space $H$ over $U$ ,
for every $i$ a principal homogeneous $H_{U_ i}$ -space $X_ i$ over $U_ i$ , and
$H$ -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.

The Stacks project

95.14.5 Principal homogeneous spaces

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