Question about closed immersion of algebraic spaces This is Prop 3.11 of Knutson's Algebraic Spaces:
Let $X\rightarrow Y$ be a map of algebraic spaces. Let $g:Y\rightarrow X$ be a section of f, i.e, a map satisfying $fg=id$. Show that if $f$ is separated, then $g$ is a closed immersion.
Can one give me a proof of this? The proof should be fairly simple (unfortunately, I have no experience with alg. spaces).
 A: In order to apply Knutson's Lemma I 1.21 we need to know that the identity $id_Y$ is a closed immersion (of algebraic spaces), the composition of two closed immersions is a closed immersion and that the base change of a closed immersion is a closed immersion.Let's show that the base change of a closed immersion is a closed immersion.
Let $X \to Y$ be a closed immersion of algebraic spaces and $Z \to Y$ any map.
Then there are representable etale coverings $V \to X$ and $U \to Y$ and a closed immersion of schemes $V \to U$. Let $W \to Z$ be a representable etale covering.
First assume that $W \to Z \to Y$ factors through $U$. Now $W \times_U V \to Z\times_Y X$ is a representable etale covering and $W \times_U V \to W$ is a closed immersion of schemes since the base change of a closed immersion of schemes is a closed immersion. So $Z\times_Y X \to Z$ is a closed immersion of algebraic spaces.
For the general case, set $W'=W\times_Y U$. Note that $W'$ is a scheme, of course $W' \to Y$ factors through $U$ and $W' \to W$ is an etale covering. So  $W' \times_U V \to W'$ is a closed immersion of schemes and $W' \times_U V=W'\times_Y V \to W\times_Y V$ is an etale covering. Also $W\times_Y V$ is a scheme and $W\times_Y V=W\times_Z(Z\times_Y V) \to Z \times_Y X $ is a representable etale covering.
So the maps $W'\times_U V \to W\times_Y V \to W$ and $W'\times_U V \to W' \to W$
form a cartesian diagram of schemes (haven't figured out to how draw diagrams here yet) s.t. both horizontal arrows ( $W'\times_U V \to W\times_Y V$ and $W' \to W$) are etale covering maps and the left vertical arrow ($W'\times_U V \to W'$) a closed immersion. Since closed immersions satisfy effective descent in the etale topology the right vertical arrow ($W\times_Y V \to W$) is a closed immersion. Hence $Z\times_Y X \to Z$ is a closed immersion of algebraic spaces.
