# The Stacks Project

## Tag: 06Q8

This tag has label stacks-morphisms-lemma-descent-finite-presentation and it points to

The corresponding content:

Lemma 70.18.7. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $\mathcal{Z} \to \mathcal{Y}$ be a surjective, flat, locally finitely presented morphism of algebraic stacks. If the base change $\mathcal{Z} \times_\mathcal{Y} \mathcal{X} \to \mathcal{Z}$ is locally of finite presentation, then $f$ is locally of finite presentation.

Proof. Choose an algebraic space $W$ and a surjective smooth morphism $W \to \mathcal{Z}$. Then $W \to \mathcal{Z}$ is surjective, flat, and locally of finite presentation (Morphisms of Spaces, Lemmas 46.33.7 and 46.33.5) hence $W \to \mathcal{Y}$ is surjective, flat, and locally of finite presentation (by Properties of Stacks, Lemma 69.5.2 and Lemmas 70.17.2 and 70.18.2). Since the base change of $\mathcal{Z} \times_\mathcal{Y} \mathcal{X} \to \mathcal{Z}$ by $W \to \mathcal{Z}$ is locally of finite presentation (Lemma 70.17.3) we may replace $\mathcal{Z}$ by $W$.

Choose an algebraic space $V$ and a surjective smooth morphism $V \to \mathcal{Y}$. Choose an algebraic space $U$ and a surjective smooth morphism $U \to V \times_\mathcal{Y} \mathcal{X}$. We have to show that $U \to V$ is locally of finite presentation. Now we base change everything by $W \to \mathcal{Y}$: Set $U' = W \times_\mathcal{Y} U$, $V' = W \times_\mathcal{Y} V$, $\mathcal{X}' = W \times_\mathcal{Y} \mathcal{X}$, and $\mathcal{Y}' = W \times_\mathcal{Y} \mathcal{Y} = W$. Then it is still true that $U' \to V' \times_{\mathcal{Y}'} \mathcal{X}'$ is smooth by base change. Hence by our definition of locally finitely presented morphisms of algebraic stacks and the assumption that $\mathcal{X}' \to \mathcal{Y}'$ is locally of finite presentation, we see that $U' \to V'$ is locally of finite presentation. Then, since $V' \to V$ is surjective, flat, and locally of finite presentation as a base change of $W \to \mathcal{Y}$ we see that $U \to V$ is locally of finite presentation by Descent on Spaces, Lemma 52.10.8 and we win. $\square$

\begin{lemma}
\label{lemma-descent-finite-presentation}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
Let $\mathcal{Z} \to \mathcal{Y}$ be a surjective, flat, locally finitely
presented morphism of algebraic stacks. If the base change
$\mathcal{Z} \times_\mathcal{Y} \mathcal{X} \to \mathcal{Z}$
is locally of finite presentation, then $f$ is locally of finite
presentation.
\end{lemma}

\begin{proof}
Choose an algebraic space $W$ and a surjective smooth morphism
$W \to \mathcal{Z}$. Then $W \to \mathcal{Z}$ is surjective, flat,
and locally of finite presentation
(Morphisms of Spaces, Lemmas
\ref{spaces-morphisms-lemma-smooth-flat} and
\ref{spaces-morphisms-lemma-smooth-locally-finite-presentation})
hence $W \to \mathcal{Y}$ is surjective, flat, and locally of finite
presentation (by
Properties of Stacks, Lemma
\ref{stacks-properties-lemma-composition-surjective}
and
Lemmas \ref{lemma-composition-flat} and
\ref{lemma-composition-finite-presentation}).
Since the base change of
$\mathcal{Z} \times_\mathcal{Y} \mathcal{X} \to \mathcal{Z}$
by $W \to \mathcal{Z}$ is
locally of finite presentation
(Lemma \ref{lemma-base-change-flat})
we may replace $\mathcal{Z}$ by $W$.

\medskip\noindent
Choose an algebraic space $V$ and a surjective smooth morphism
$V \to \mathcal{Y}$. Choose an algebraic space $U$ and a surjective
smooth morphism $U \to V \times_\mathcal{Y} \mathcal{X}$.
We have to show that $U \to V$ is locally of finite presentation.
Now we base change everything by $W \to \mathcal{Y}$: Set
$U' = W \times_\mathcal{Y} U$,
$V' = W \times_\mathcal{Y} V$,
$\mathcal{X}' = W \times_\mathcal{Y} \mathcal{X}$,
and $\mathcal{Y}' = W \times_\mathcal{Y} \mathcal{Y} = W$.
Then it is still true that $U' \to V' \times_{\mathcal{Y}'} \mathcal{X}'$
is smooth by base change. Hence by our definition of locally finitely
presented morphisms of algebraic stacks and the assumption that
$\mathcal{X}' \to \mathcal{Y}'$ is locally of finite presentation,
we see that $U' \to V'$ is locally of finite presentation. Then, since
$V' \to V$ is surjective, flat, and locally of finite presentation
as a base change of $W \to \mathcal{Y}$ we see that $U \to V$ is
locally of finite presentation by
Descent on Spaces, Lemma
\ref{spaces-descent-lemma-descending-property-locally-finite-presentation}
and we win.
\end{proof}


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