The Stacks project

Remark 97.21.4 (Automorphisms). Assumptions and notation as in Lemma 97.21.2. Let $x', x''$ be lifts of $x$ to $A'$. Then we have a composition map

\[ \text{Inf}(x'/x) \times \mathop{\mathrm{Mor}}\nolimits _{\textit{Lift}(x, A')}(x', x'') \times \text{Inf}(x''/x) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\textit{Lift}(x, A')}(x', x''). \]

Since $\textit{Lift}(x, A')$ is a groupoid, if $\mathop{\mathrm{Mor}}\nolimits _{\textit{Lift}(x, A')}(x', x'')$ is nonempty, then this defines a simply transitive left action of $\text{Inf}(x'/x)$ on $\mathop{\mathrm{Mor}}\nolimits _{\textit{Lift}(x, A')}(x', x'')$ and a simply transitive right action by $\text{Inf}(x''/x)$. Now the lemma says that $\text{Inf}(x'/x) = \text{Inf}_ x(I) = \text{Inf}(x''/x)$. We claim that the two actions described above agree via these identifications. Namely, either $x' \not\cong x''$ in which the claim is clear, or $x' \cong x''$ and in that case we may assume that $x'' = x'$ in which case the result follows from the fact that $\text{Inf}(x'/x)$ is commutative. In particular, we obtain a well defined action

\[ \text{Inf}_ x(I) \times \mathop{\mathrm{Mor}}\nolimits _{\textit{Lift}(x, A')}(x', x'') \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\textit{Lift}(x, A')}(x', x'') \]

which is simply transitive as soon as $\mathop{\mathrm{Mor}}\nolimits _{\textit{Lift}(x, A')}(x', x'')$ is nonempty.


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07YB. Beware of the difference between the letter 'O' and the digit '0'.