# The Stacks Project

## Tag: 06NR

This tag has label spaces-descent-lemma-curiosity and it points to

The corresponding content:

Lemma 52.7.1. Let $S$ be a scheme. Let $X \to Y \to Z$ be morphism of algebraic spaces. Let $P$ be one of the following properties of morphisms of algebraic spaces over $S$: flat, locally finite type, locally finite presentation. Assume that $X \to Z$ has $P$ and that $X \to Y$ is a surjection of sheaves on $(\textit{Sch}/S)_{fppf}$. Then $Y \to Z$ is $P$.

Proof. Choose a scheme $W$ and a surjective étale morphism $W \to Z$. Choose a scheme $V$ and a surjective étale morphism $V \to W \times_Z Y$. Choose a scheme $U$ and a surjective étale morphism $U \to V \times_Y X$. By assumption we can find an fppf covering $\{V_i \to V\}$ and lifts $V_i \to X$ of the morphism $V_i \to Y$. Since $U \to X$ is surjective étale we see that over the members of the fppf covering $\{V_i \times_X U \to V\}$ we have lifts into $U$. Hence $U \to V$ induces a surjection of sheaves on $(\textit{Sch}/S)_{fppf}$. By our definition of what it means to have property $P$ for a morphism of algebraic spaces (see Morphisms of Spaces, Definition 46.27.1, Definition 46.22.1, and Definition 46.26.1) we see that $U \to W$ has $P$ and we have to show $V \to W$ has $P$. Thus we reduce the question to the case of morphisms of schemes which is treated in Descent, Lemma 31.10.8. $\square$

\begin{lemma}
\label{lemma-curiosity}
Let $S$ be a scheme. Let $X \to Y \to Z$ be morphism of algebraic spaces.
Let $P$ be one of the following properties of morphisms of algebraic spaces
over $S$:
flat, locally finite type, locally finite presentation.
Assume that $X \to Z$ has $P$ and that
$X \to Y$ is a surjection of sheaves on $(\Sch/S)_{fppf}$.
Then $Y \to Z$ is $P$.
\end{lemma}

\begin{proof}
Choose a scheme $W$ and a surjective \'etale morphism $W \to Z$.
Choose a scheme $V$ and a surjective \'etale morphism $V \to W \times_Z Y$.
Choose a scheme $U$ and a surjective \'etale morphism $U \to V \times_Y X$.
By assumption we can find an fppf covering $\{V_i \to V\}$ and
lifts $V_i \to X$ of the morphism $V_i \to Y$. Since $U \to X$ is surjective
\'etale we see that over the members of the fppf covering
$\{V_i \times_X U \to V\}$ we have lifts into $U$. Hence $U \to V$ induces
a surjection of sheaves on $(\Sch/S)_{fppf}$.
By our definition of what it means to have property $P$ for a
morphism of algebraic spaces (see
Morphisms of Spaces,
Definition \ref{spaces-morphisms-definition-flat},
Definition \ref{spaces-morphisms-definition-locally-finite-type}, and
Definition \ref{spaces-morphisms-definition-locally-finite-presentation})
we see that $U \to W$ has $P$ and we have to show $V \to W$ has $P$.
Thus we reduce the question to the case of morphisms of schemes
which is treated in
Descent, Lemma \ref{descent-lemma-curiosity}.
\end{proof}


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