# The Stacks Project

## Tag: 03MF

This tag has label spaces-morphisms-lemma-surjective-local and it points to

The corresponding content:

Lemma 45.6.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:
1. $f$ is surjective,
2. for every scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times_Y X \to Z$ is surjective,
3. for every affine scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times_Y X \to Z$ is surjective,
4. there exists a scheme $V$ and a surjective étale morphism $V \to Y$ such that $V \times_Y X \to V$ is a surjective morphism,
5. there exists a scheme $U$ and a surjective étale morphism $\varphi : U \to X$ such that the composition $f \circ \varphi$ is surjective,
6. there exists a commutative diagram $$\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }$$ where $U$, $V$ are schemes and the vertical arrows are surjective étale such that the top horizontal arrow is surjective, and
7. there exists a Zariski covering $Y = \bigcup Y_i$ such that each of the morphisms $f^{-1}(Y_i) \to Y_i$ is surjective.

Proof. Omitted. $\square$

\begin{lemma}
\label{lemma-surjective-local}
Let $S$ be a scheme.
Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.
The following are equivalent:
\begin{enumerate}
\item $f$ is surjective,
\item for every scheme $Z$ and any morphism $Z \to Y$ the morphism
$Z \times_Y X \to Z$ is surjective,
\item for every affine scheme $Z$ and any morphism
$Z \to Y$ the morphism $Z \times_Y X \to Z$ is surjective,
\item there exists a scheme $V$ and a surjective \'etale morphism
$V \to Y$ such that $V \times_Y X \to V$ is a surjective morphism,
\item there exists a scheme $U$ and a surjective \'etale morphism
$\varphi : U \to X$ such that the composition $f \circ \varphi$
is surjective,
\item there exists a commutative diagram
$$\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }$$
where $U$, $V$ are schemes and the vertical arrows are surjective \'etale
such that the top horizontal arrow is surjective, and
\item there exists a Zariski covering $Y = \bigcup Y_i$ such that
each of the morphisms $f^{-1}(Y_i) \to Y_i$ is surjective.
\end{enumerate}
\end{lemma}

\begin{proof}
Omitted.
\end{proof}


To cite this tag (see How to reference tags), use:

\cite[\href{http://stacks.math.columbia.edu/tag/03MF}{Tag 03MF}]{stacks-project}


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).