The Stacks Project

Tag 06MQ

Lemma 91.11.4. Let $\mathcal{Z}' \to \mathcal{Z}$ be a monomorphism of algebraic stacks. Assume there exists a field $k$ and a locally finitely presented, surjective, flat morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{Z}$. Then either $\mathcal{Z}'$ is empty or $\mathcal{Z}' \to \mathcal{Z}$ is an equivalence.

Proof. We may assume that $\mathcal{Z}'$ is nonempty. In this case the fibre product $T = \mathcal{Z}' \times_\mathcal{Z} \mathop{\mathrm{Spec}}(k)$ is nonempty, see Lemma 91.4.3. Now $T$ is an algebraic space and the projection $T \to \mathop{\mathrm{Spec}}(k)$ is a monomorphism. Hence $T = \mathop{\mathrm{Spec}}(k)$, see Morphisms of Spaces, Lemma 58.10.8. We conclude that $\mathop{\mathrm{Spec}}(k) \to \mathcal{Z}$ factors through $\mathcal{Z}'$. Suppose the morphism $z : \mathop{\mathrm{Spec}}(k) \to \mathcal{Z}$ is given by the object $\xi$ over $\mathop{\mathrm{Spec}}(k)$. We have just seen that $\xi$ is isomorphic to an object $\xi'$ of $\mathcal{Z}'$ over $\mathop{\mathrm{Spec}}(k)$. Since $z$ is is surjective, flat, and locally of finite presentation we see that every object of $\mathcal{Z}$ over any scheme is fppf locally isomorphic to a pullback of $\xi$, hence also to a pullback of $\xi'$. By descent of objects for stacks in groupoids this implies that $\mathcal{Z}' \to \mathcal{Z}$ is essentially surjective (as well as fully faithful, see Lemma 91.8.4). Hence we win. $\square$

The code snippet corresponding to this tag is a part of the file stacks-properties.tex and is located in lines 2519–2525 (see updates for more information).

\begin{lemma}
\label{lemma-monomorphism-into-point}
Let $\mathcal{Z}' \to \mathcal{Z}$ be a monomorphism of algebraic stacks.
Assume there exists a field $k$ and a locally finitely presented, surjective,
flat morphism $\Spec(k) \to \mathcal{Z}$. Then either $\mathcal{Z}'$
is empty or $\mathcal{Z}' \to \mathcal{Z}$ is an equivalence.
\end{lemma}

\begin{proof}
We may assume that $\mathcal{Z}'$ is nonempty. In this case the
fibre product $T = \mathcal{Z}' \times_\mathcal{Z} \Spec(k)$
is nonempty, see
Lemma \ref{lemma-points-cartesian}.
Now $T$ is an algebraic space and the projection $T \to \Spec(k)$
is a monomorphism. Hence $T = \Spec(k)$, see
Morphisms of Spaces, Lemma
\ref{spaces-morphisms-lemma-monomorphism-toward-field}.
We conclude that $\Spec(k) \to \mathcal{Z}$ factors through
$\mathcal{Z}'$. Suppose the morphism $z : \Spec(k) \to \mathcal{Z}$
is given by the object $\xi$ over $\Spec(k)$. We have just seen that
$\xi$ is isomorphic to an object $\xi'$ of $\mathcal{Z}'$ over
$\Spec(k)$. Since $z$ is
is surjective, flat, and locally of finite presentation we see that
every object of $\mathcal{Z}$ over any scheme is fppf locally isomorphic
to a pullback of $\xi$, hence also to a pullback of $\xi'$. By descent of
objects for stacks in groupoids this implies that
$\mathcal{Z}' \to \mathcal{Z}$ is essentially surjective (as well as
fully faithful, see
Lemma \ref{lemma-monomorphism}).
Hence we win.
\end{proof}

There are no comments yet for this tag.

There are also 2 comments on Section 91.11: Properties of Algebraic Stacks.

Add a comment on tag 06MQ

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 lower-right corner).