The Stacks project

Lemma 93.10.2. Let $S$ be an object of $\mathit{Sch}_{fppf}$. Let $\mathcal{P}$ be as in Definition 93.10.1. Consider a $2$-commutative diagram

\[ \xymatrix{ \mathcal{X}' \ar[r] \ar[d]_{f'} & \mathcal{X} \ar[d]^ f \\ \mathcal{Y}' \ar[r] & \mathcal{Y} } \]

of $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume the horizontal arrows are equivalences and $f$ (or equivalently $f'$) is representable by algebraic spaces. Then $f$ has $\mathcal{P}$ if and only if $f'$ has $\mathcal{P}$.

Proof. Note that this makes sense by Lemma 93.9.3. Proof omitted. $\square$

Comments (3)

Comment #26 by David Zureick-Brown on

Typo: representably.

Comment #5939 by Dario Weißmann on

@#26: seconded

Comment #6128 by on

Thanks for finding this ancient spelling error. The fix is here.

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Your email address will not be published.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 0459. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 93.10.2 as pdf