 \subsektion{Artin's Theorem}

%David: I can make this example clear. An fppf space is the same as an
%etale space by artin's theorem.
\begin{example}
 Say $X/S$ is an algebraic space and $G/S$ is a smooth group scheme
acting freely on $X$. Then the quotient $[X/G]$ is an algebraic space
\anton{explain}. This is not clear because it only comes with a
smooth covering.
\end{example}
\begin{corollary}
 Let $\X$ be an algebraic stack over $S$. Then the following are
equivalent.
 \begin{enumerate}
   \item $\X$ is an algebraic space,
   \item for every $x\in \X$, $\aut_x=\{\id\}$.
 \end{enumerate}
\end{corollary}
\begin{proof}
 $(1\Rightarrow 2)$ is clear. $(2\Rightarrow 1)$ Let's check that
$\Delta$ is formally unramified. This is clear because doing
something, we get a non-trivial automorphism of $x_1$.
 \[\xymatrix{
  T_0\ar@{^(->}[r]\ar[d] & T\ar[d]\ar@{-->}[dl]\\
  \isom(x_1,x_2)\ar[d]\ar[r] & Z\ar[d]^{(x_1,x_2)}\\
  \X\ar[r] & \X\times_S \X
 }\]
 So there exists an \'etale cover $X\to \X$. Also, 2 implies that
$\Delta:\X\to \X\times \X$ is a monomorphism, which implies that
$\X\cong |\X|$ (the sheaf of isomorphism classes of things in the
fibers). Thus, we can assume $\X=F$ is a sheaf, and it has an \'etale
cover $X\to F$. The remaining point is that $F\to F\times F$ is
representable (by algebraic spaces); we have to check
representability by schemes. Let $T$ be a scheme.
 \[\xymatrix{
  \Delta^{-1}(T)\ar[d]\ar@{^(->}[r] & T\ar[d]\\
  F\ar[r] & F\times F
 }\]
 where $\Delta^{-1}(T)\hookrightarrow T$ is separated and
quasi-finite (the fibers are empty or one point), so scheme.
\end{proof}
This should make you happy. Usually when we have moduli, they are
presented as something with a group action.
\begin{example}
 $B\mu_p$ over the field $\FF_p$. Recall that $\mu_p=\ker(\GG_m\to
\GG_m;u\mapsto u^p)$. We see that $B\mu_p=[\GG_m/\GG_m]$, where the
action is $u\ast v=u^pv$. So $B\mu_p$ is an Artin (algebraic) stack,
but it is not Deligne-Mumford in general.
 \[\xymatrix{
  \mu_p\ar[d]\ar[r]^{\text{not}}_{\text{unram.}} & \spec
\FF_p\ar[d]^{(p_1,p_2)} \\
  B\mu_p\ar[r] & B\mu_p\times B\mu_p
 }\]
 you can't make an \'etale cover.
\end{example}

