The Stacks project

Lemma 94.10.2. Let $S$ be an object of $\mathit{Sch}_{fppf}$. Let $\mathcal{P}$ be as in Definition 94.10.1. 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 94.9.3. 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.

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