\sektion{32}{Properties of algebraic stacks}

%  $\mu$-gerbes to cohomology classes. We got confused before. There are two points.
% 
%  (1) if $\mu$ is an injective abelian sheaf and $\G$ is a $\mu$-gerbe, then $\G$ is
%  trivial. Choose an \'etale surjection $X'\to X$ and $s'\in \G(X')$. Then consider the
%  functor $F:(\sch/X)^{op}\to \set$ given by $(V\to X)\mapsto \{(s,\iota)|s\in
%  \G(V),\iota:p^*s\xrightarrow\sim s'\text{ in } \G(X'_V)\}/\sim$, where
%  $(s_1,\iota_1)\sim (s_2,\iota_2)$ if there is a $\delta:s_1\to s_2$ in $\G(V)$ so that
%  $\iota_1=\iota_2\circ p^*\delta$.
%  \begin{remark}
%    $\delta$ is unique if it exists because $\mu(V)\to \mu(X'_V)$ is injective (because
%    $X_V'\to V$ is a covering and $\mu$ is a sheaf. This implies that $F$ is a sheaf, and
%    in fact a $p_*\mu/\mu$-torsor (we saw this last time). This implies that $F$ is
%    trivial, so $\G$ is trivial
%  \end{remark}
% 
%  (2) if $\mu$ is arbitrary, then choose $\mu\hookrightarrow I$ with $I$ injective. Then
%  $\G\times^\mu I$ is trivial and we have a map $j:\G\to \G\times^\mu I$. Fix $s_0\in
%  \G(\G\times^\mu I)(X)$ and define $P:(\sch/X)^{op}\to \set$ by $(V\to X)\mapsto \{(s\in
%  \G(V),\iota:j(x)\xrightarrow\sim s_0\text{ in } \G\times^\mu I(V))\}$. Then $P$ is an
%  $I/\mu$-torsor. Then you get a map $H^1(X,I/\mu)\to H^2(X,\mu)$. You have the map
%  $\G\mapsto \partial(P)$.
% 
%  \begin{proposition}
%    Let $\mu\to X$ be a smooth (commutative) group scheme. Then any $\mu$-gerbe is
%    algebraic.
%  \end{proposition}
%  We need the following lemma.
%  \begin{lemma}
%    Any $\mu$-torsor is an algebraic space with finite type diagonal.
%  \end{lemma}
%  \begin{proof}
%    We have to check representability of the diagonal and the \'etale cover. Say $P\to X$
%    is a $\mu$-torsor. Consider the diagonal, and let $T$ be a scheme.
%    \[\xymatrix{
%     Z\ar[r]\ar[d] & T\ar[d]^{(p_1,p_2)}\\
%     P\ar[r]^<>(.5)\Delta & P\times_X P
%    }\]
%    The $p_1$ defines a trivialization $\mu_T\xrightarrow P_T$. This identifies $p_2$
%    with an element $g\in \mu(T)$. Let $Z$ be the fiber product
%    \[\xymatrix{
%     Z\ar[r]\ar[d] & T\ar[d]^g\\
%     T\ar[r]^e & G_T
%    }\qquad\qquad
%    \xymatrix{
%     Z\ar[r]\ar[d] & T\ar[d]\\
%     G_T\ar[r]^<>(.5)\Delta & G_T\times_T G_T
%    }\]
%    Since $G_T$ is locally of finite presentation, something is. $X\to S$ locally of
%    finite presentation implies that $X\to X\times_S X$ is quasi-compact and of finite
%    type. It requires an argument, but we won't give it here.
% 
%    The \'etale cover: choose a trivialization of $P$ over some \'etale cover $X'\to X$
%    and take $\mu\times_X X'\cong P\times_X X'\to P$.
%  \end{proof}
%  \begin{proof}[Proof of Proposition]
%    Let $T$ be a scheme
%    \[\xymatrix{
%     \isom(s_1,s_2)\ar[r]\ar[d]& T\ar[d]^{(s_1,s_2)}\\
%     \G\ar[r]^\Delta & \G\times \G
%    }\]
%     $\isom$ is a $\mu$-torsor, so by the lemma it is an algebraic space, which implies
%    representability of the diagonal
% 
%    Smooth surjection. Find $X'\to X$ \'etale so that $\G(X')$ is non-empty, and let $s'$
%    be an object.
%    \[\xymatrix{
%     X'\ar[d]\\
%     [X'/\mu]\ar@{}[r]|{=}& B_{X'}\mu\ar@{}[r]|{=} &\G\times_X X' \ar[r]\ar[d] & X'\ar[d]^{et} \ar@/_/[l]_{s'}\\
%     & & \G\ar[r] & X
%    }\]
%    To check that the map $X'\to [X/\mu]$ is smooth, check that for any $T$ and let $P$ be
%    a $\mu$-torsor.
%    \[
% %   \xymatrix{
% %    X'\ar[d] & Z\ar[l]\ar[d] & \mu_{X'} & Z'\ar[d]\ar@/_/[ll]\\
% %    [X'/\mu] & T\ar[l]_P & X\ar@/^/[ll]^{P_0} & T'\ar@/^/[ll]\ar[l]
% %   }
%    \xymatrix @!0 @C+1pc{
%     & Z\ar[rr]\ar[dd]|\hole& & X'\ar[dd]\\
%     Z'\ar[rr]\ar[dd]\ar[ur] & & \mu_{X'}\ar[dd]\ar[ur]\\
%     & T\ar[rr]|\hole^(.3){P} & & [X'/\mu]\\
%     T'\ar[rr]\ar[ur] & & X \ar[ur]_{P_0}
%    }\]
%  \end{proof}

 \underline{Properties of algebraic stacks over $S$}.

 By \emph{a presentation} of an algebraic stack $S$, we mean a smooth surjective morphism
 $X\to \X$, with $X$ an algebraic space.

 Let $\P$ be a property of algebraic spaces over $S$ which is local in the smooth
 topology (if $X'\to X$ is a smooth cover, then $X$ has $\P$ if and only if $X'$ has
 $\P$), then an algebraic stack $\X$ over $S$ \emph{has $\P$} if there exists a
 presentation $X\to \X$ where $X$ has $\P$.
 \begin{example}
   Locally noetherian, reduced, normal, regular, characteristic $p$, etc.
 \end{example}
 This is coarser than \'etale locally (e.g.~dimension is no good)
 \begin{remark}
   If an algebraic stack has $\P$, then for every presentation $X'\to \X$, $X'$ has $\P$.
   \[\xymatrix{
    X\times_\X X'\ar[r]\ar[d] & X\rlap{$\quad\P$}\ar[d]\\
    X'\ar[r] & \X
   }\]
   All morphisms are smooth surjections, so we can push $\P$ around like we did before.
 \end{remark}
 \begin{definition}
   $\X$ over $S$ is \emph{quasi-compact} if there exists a presentation $X\to \X$ with
   $X$ quasi-compact.
 \end{definition}
 \begin{definition}
   Say $\X$ is an algebraic stack over $S$. An open (resp.~closed, resp.~locally closed)
   substack of $\X$ is a fully faithful substack $\Y\subseteq \X$ such that $\Y\to \X$ is
   representable by an open (resp.~closed, resp.~locally closed) immersion.
 \end{definition}
 \begin{lemma}
   Let $\Y\to \X$ be a representable morphism of stacks with $\X$ algebraic. Then $\Y$ is
   algebraic.
 \end{lemma}
 Aside: $\M_{g,n}\to \M_g$ is representable because for any $T$,
 \[\xymatrix{
    C\times_T\cdots\times_T C\ar@{}[r]|<>(.5){\txt{$\supseteq$}} & U\ar[d] \ar[r] & T\ar[d]\\
    & \M_{g,n} \ar[r] & \M_g
 }\]
 Then you get that $\M_{g,n}$ is algebraic.
 \begin{proof}
   Claim: $\Delta:\Y\to \Y\times_\X \Y$ is representable.
   \[\xymatrix{
    P\ar[r]\ar[d] & T\ar[d]\\
    \Y\ar[r]  & \Y\times_\X \Y
   }\qquad\qquad
   \xymatrix{
    P\ar[r]\ar[d]& T\ar[d]\\
   \Y\times_\X T\ar[r]^<>(.5)\Delta & (\Y\times_\X T)\times_T (\Y\times_\X T)
   }\]
   so $P$ is a scheme.

   For the smooth surjection, choose a presentation $X\to \X$, then let $Y=\Y\times_\X
   X$, which is an algebraic space which smoothly surjects onto $\Y$.
 \end{proof}
 \begin{lemma}
   Let $\Z\xrightarrow g \Y \xleftarrow f \X$ be a diagram of algebraic stacks. Then the
   stack $\X\times_\Y \Z$ is algebraic.
 \end{lemma}
 \begin{proof}
   Representability of the diagonal:
   \[\xymatrix{
    I=\isom\bigl((x_1,z_1,\iota_1),(x_2,z_2,\iota_2)\bigr)\ar[r]\ar[d]& T\ar[d]^{(x_1,z_1,\iota_1)\times (x_2,z_2,\iota_2)}\\
    \X\times_\Y\Z\ar[r]& (\X\times_\Y \Z)\times (\X\times_\Y \Z)
   }\]
   Where $x_i\in\X(T)$, $z_i\in \Z(T)$ and $\iota_i:f(x_i)\xrightarrow\sim g(z_i)$. An
   isomorphism between two such collections of data is a pair
   $(\rho_x,\rho_z):(x_1,z_1,\iota_1)\to (x_2,z_2,\iota_2)$, where
   $\rho_x:x_1\xrightarrow\sim x_2$ in $\X(T)$ and $\rho_z:z_1\xrightarrow\sim z_2$ in
   $\Z(T)$ such that
   \[\xymatrix{
    f(x_1)\ar[r]^{\rho_x}\ar[d]_{\iota_1} & f(x_2)\ar[d]^{\iota_2}\\
    g(z_1)\ar[r]^{\rho_z} & g(z_2)
   }\]
    So we see that $I$ is the equalizer of $\isom_\X(x_1,x_2)\times
   \isom_\Z(z_1,z_2)\rightrightarrows \isom(f(x_1),g(z_2))$

   Smooth surjection. Choose a presentation $Y\to \Y$ and choose $Z\to Y_\Z$ be a smooth
   surjection\anton{I didn't quite catch how you get this}.
   \[
%    \xymatrix@!0{
%     & & & & X\ar[dd]\\  \\
%     & & & & Y_\X\ar[dd]\ar[dr]\\
%     & & & & & \X\ar[dd]\\
%     Z\ar[rr]& & Y_\Z\ar[dr] \ar[rr] & & Y\ar[dr]\\
%     & Z'\ar@{.>}[rr]& & \Z\ar[rr] & & \Y
%    }
   \xymatrix@!0{
    & & & & X\ar[dd]\\
    & & & & & \X\ar[dd]\\
    & & & & Y_\X\ar[dd]\ar[ur]\\
    & Z'\ar[rr]^{sm} & & \Z \ar[rr]|\hole & & \Y\\
    \llap{$Z'\times_{\Z}Y_\Z=\;$}Z\ar[ur]\ar[rr] & & Y_\Z \ar[rr]\ar[ur] & & Y\ar[ur]
   }\]
    Then $Z\times_Y X\to \Z\times_\Y \X$ is a smooth surjection.
 \end{proof}
 Let $\P$ be a property of morphisms of algebraic spaces which is local on source and
 target in the smooth topology.
 \[\xymatrix{
    X''\ar[r]^{f'}\ar[dr]_{f''}& X'\ar[r]\ar[d]\ar@{}[dr]|(.25)\pb & X\ar[d]^f\\
    & Y'\ar[r]^q & Y
 }\]
 If $q$ and $p$ ar smooth surjections then $f$ has $\P$ if and only if $f''$ has $\P$.
 \begin{definition}
   A morphisms of algebraic stacks $f:\X\to \Y$ has $\P$ if there exists a diagram
   \[\xymatrix{
    X'\ar[r]^{P'}\ar[dr]_{f'}&\X'\ar[r]\ar[d] & \X\ar[d]^f\\
    & Y\ar[r]^Q & \Y
   }\]
   $P',Q$ presentations, then $f'$ has $\P$.
 \end{definition}
 So we can talk about surjective, universally open, locally of finite presentation
 (resp.~type), flat, smooth.
