Lemma 96.12.4. In the situation of Lemma 96.12.1. Assume that $G$, $H$ are representable by algebraic spaces and étale. Then $\mathcal{H}_ d(\mathcal{X}'/\mathcal{Y}') \to \mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ is representable by algebraic spaces and étale. If also $H$ is surjective and the induced functor $\mathcal{X}' \to \mathcal{Y}' \times _\mathcal {Y} \mathcal{X}$ is surjective, then $\mathcal{H}_ d(\mathcal{X}'/\mathcal{Y}') \to \mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ is surjective.

Proof. Set $\mathcal{X}'' = \mathcal{Y}' \times _\mathcal {Y} \mathcal{X}$. By Lemma 96.4.1 the $1$-morphism $\mathcal{X}' \to \mathcal{X}''$ is representable by algebraic spaces and étale (in particular the condition in the second statement of the lemma that $\mathcal{X}' \to \mathcal{X}''$ be surjective makes sense). We obtain a $2$-commutative diagram

$\xymatrix{ \mathcal{X}' \ar[r] \ar[d] & \mathcal{X}'' \ar[r] \ar[d] & \mathcal{X} \ar[d] \\ \mathcal{Y}' \ar[r] & \mathcal{Y}' \ar[r] & \mathcal{Y} }$

It follows from Lemma 96.12.2 that $\mathcal{H}_ d(\mathcal{X}''/\mathcal{Y}')$ is the base change of $\mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ by $\mathcal{Y}' \to \mathcal{Y}$. In particular we see that $\mathcal{H}_ d(\mathcal{X}''/\mathcal{Y}') \to \mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ is representable by algebraic spaces and étale, see Algebraic Stacks, Lemma 93.10.6. Moreover, it is also surjective if $H$ is. Hence if we can show that the result holds for the left square in the diagram, then we're done. In this way we reduce to the case where $\mathcal{Y}' = \mathcal{Y}$ which is the content of Lemma 96.12.3. $\square$

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Your email address will not be published.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 05XS. Beware of the difference between the letter 'O' and the digit '0'.