The Stacks project

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

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.