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\begin{document}

\title{Hilbert Schemes of Quotient Stacks}
\author[Olsson]{Martin Olsson}
\address{Department of Mathematics \\ Massachusetts Institute of
Technology \\ Cambridge MA 02139}
\email{olsson@math.mit.edu}
\author[Starr]{Jason Starr} 
\address{Department of Mathematics \\ Massachusetts Institute of
Technology \\ Cambridge MA 02139} 
\email{jstarr@math.mit.edu}
\date{\today}

\maketitle

\tableofcontents

\begin{abstract}

\end{abstract}

\section{Statement of Results}

Let $S$ be an algebraic space, $p:X\rightarrow S$ a separated, locally
finitely-presented $1$-morphism from a Deligne-Mumford stack $X$ to an
algebraic space $S$.  Let $F$ be a quasi-coherent $\OO_X$-module (in
the \'etale site of $X$) such that $F$ is locally finitely-presented.
We define a contravariant functor
\begin{equation}
Q=Q(F/X/S): S-\text{schemes} \rightarrow \text{Sets}
\end{equation}
as follows:  For each $S$-scheme $f:Z\rightarrow S$, we define $X_Z$
to be $X\times_S Z$, and we define $F_Z$ to be the pullback of $F$ to
$X_Z$.  We define $Q(Z)$ to be the set of $\OO_{X_Z}$-module quotients
$F_Z\rightarrow G$ which satisfy
\begin{enumerate}
\item $G$ is a quasi-coherent $\OO_{X_Z}$-module which is locally
finitely-presented, 
\item $G$ is flat over $Z$,
\item the support of $G$ is proper over $Z$.
\end{enumerate}
All of these properties are preserved by base-change on $Z$, therefore
pullback maps $Q$ into a contravariant functor.  

\begin{thm}~\label{thm-thm1}  $Q$ is represented by an algebraic space
which is separated and locally finitely-presented over $S$.
\end{thm}

Suppose now that $S$ is an affine scheme such that $\QQ\subset \OO_S$,
suppose that $X=[Y/G]$ is a quotient of a separated,
finitely-presented $S$-scheme $Y$ by an affine, finite-type $S$-group
scheme $G$, and suppose that the coarse moduli space $X=Y//G$ is a
quasi-projective $S$-scheme (respectively projective $S$-scheme).  

\begin{thm}~\label{thm-thm2}  Under the above conditions, the connected
components of $Q$ are quasi-projective $S$-schemes (respectively
projective $S$-schemes).
\end{thm}

\section{Hilbert Polynomials}~\label{sec-HP}
Suppose $k$ is a field and ${\mc X}$ is a separated, locally
finitely-presented Deligne-Mumford stack over $\SP{k}$.  Suppose that
${\mc F}$ is a coherent $\OO_{{\mc X}}$-module with proper support.
Then the sum
\begin{equation}
\chi({\mc X},{\mc F}) = \sum_{i=0}^\infty (-1)^i \text{dim}_k
H^i_{\'et}({\mc X},{\mc F}) 
\end{equation}
is finite.  For each locally free sheaf $E$ of finite rank on ${\mc X}$, we
also have that $E\otimes_{\OO_{\mc X}} {\mc F}$ is again coherent with proper
support.  By standard arguments (reference), the function
\begin{equation}
P_{\mc F}:K^0({\mc X})\rightarrow \ZZ, [E]\mapsto \chi({\mc
  X},E\otimes_{\OO_{\mc X}} {\mc F}), 
\end{equation}
is defined and gives a group homomorphism.  

\begin{defn}~\label{defn-HP} Suppose $A$ is an Abelian group and
  $i:A\rightarrow K^0({\mc X})$ is a group homomorphism.  Let ${\mc
  X},{\mc F}$ be as above.  Define the $A$-Hilbert polynomial of ${\mc
  F}$, $P_{A,{\mc F}}$, to be the function $P_{\mc F}\circ i$.
\end{defn}

\textbf{Remarks:}
(1) If ${\mc L}$ is an invertible sheaf on ${\mc X}$, if
    $A=\ZZ[t,t^{-1}]$ 
    and if $i:A\rightarrow K^0({\mc X})$ is given by $i(t^d) =
    \lt[{\mc L}^d\rt]$, then the $A$-Hilbert polynomial of ${\mc F}$ is
    the usual Hilbert 
    polynomial of ${\mc F}$ with respect to ${\mc L}$.  However we will
    need to consider cases which cannot be reduced to this case.

\

(2) An instructive example is to consider the case where $G$ is a
    finite group and ${\mc
    X}=BG\times_{\SP\ZZ} \SP k$ the
    classifying stack for principal $G$-bundles on $k$-schemes.  The
    locally finitely-presented $\OO_{\mc X}$-modules 
    correspond to finite-dimensional representations of $G$ over $k$.
    The group $K^0$ is isomorphic to the representation ring of $G$,
    which is semi-simple if $\# G$ is relatively prime to char$(k)$.
    In this case, ${\mc F}$ is determined up to isomorphism by $P_{\mc
    F}$.  

\

Recall that a Deligne-Mumford stack ${\mc X}$ is \emph{tame} if for
each algebraically-closed field $k$ and each 1-morphism $\zeta:\SP
k\rightarrow {\mc X}$, the stabilizer group 
\begin{equation}
G_{\zeta} := \lt(\SP k \times \SP k\rt)_{(\zeta,\zeta),\Delta} {\mc
  X}
\end{equation}
has order $\# G_{\zeta}$ relatively prime to char$(k)$.  The fiber
product above is over ${\mc X}\times {\mc X}$ and $\Delta:{\mc
  X}\rightarrow {\mc X}\times {\mc X}$ is the diagonal morphism.  We
remind the reader of some facts about tame Deligne-Mumford stacks.

\begin{lem}~\label{lem-tame}
Let ${\mc X}$ be a separated, tame Deligne-Mumford stack, $\pi:{\mc
  X}\rightarrow X$ its coarse moduli space.  
\begin{enumerate}
\item The additive functor $\pi_*$ from the category of Abelian
  sheaves in the \'etale site of ${\mc X}$ to the category of Abelian
  sheaves in the \'etale site of $X$ maps quasi-coherent sheaves to
  quasi-coherent sheaves and maps locally finitely-presented sheaves
  to locally finitely-presented sheaves.
\item For any quasi-coherent sheaf ${\mc F}$ on ${\mc X}$, the higher
  direct images $R^i\pi_*{\mc F} = 0$ for $i>0$.
\item Suppose $g:X\rightarrow S$ is a morphism of algebraic spaces and
  suppose ${\mc F}$ is a quasi-coherent sheaf on ${\mc X}$ which is
  flat over $S$.  Then also $\pi_* {\mc F}$ is flat over $S$.
\end{enumerate}

\begin{proof}
  (1) and (2) form ~\cite[lemma 2.3.4]{AV}.  And (3) follows by the
  same local analysis in the proof of ~\cite[lemma 2.3.4]{AV} since
  the invariant submodule $M^\Gamma$ (in the notation of ~\cite{AV})
  of an $S$-flat module $M$ is a direct summand, and so it is also
  $S$-flat.
\end{proof}

Now suppose that $f:{\mc X}\rightarrow S$ is a 1-morphism from a
tame Deligne-Mumford stack to a connected algebraic space such that $f$ is
separated and locally finitely-presented,  Define $A=K^0({\mc X})$ and
for each field $k$ and each morphism $g:\SP k\rightarrow S$, define
$i_g:A\rightarrow K^0(\SP k\times_S {\mc X})$ to be the pullback map
$K^0(\text{pr}_2)$.  

\

\begin{lem}~\label{lem-HPconst}
Suppose that ${\mc F}$ is a locally finitely-presented quasi-coherent
sheaf on ${\mc X}$ which is flat over $S$ and which has proper support
over $S$.  Then there exists a function $P:A\rightarrow \ZZ$ such that
for all $g:\SP k\rightarrow S$, $P_{A,g^*{\mc F}} = P$.
\end{lem}

\begin{proof}
We need to show that for each locally free sheaf ${\mc E}$ on ${\mc
  X}$, the function 
\begin{equation}
(g:\SP k\rightarrow S) \mapsto \chi\lt(g^*\lt({\mc E}\otimes_{\OO_{\mc
    X}}{\mc F}\rt)\rt)
\end{equation}
is constant.  Since $R^i\pi_*$ vanishes on all quasi-coherent modules
for $i>0$, we have
\begin{equation}
\chi\lt(g^*\lt({\mc E}\otimes_{\OO_{\mc X}}{\mc F}\rt)\rt) =
\chi\lt(g^*\pi_*\lt( {\mc E}\otimes_{\OO_{\mc X}}{\mc F}\rt)\rt).
\end{equation}
And by lemma~\ref{lem-tame} (3), we know that $\pi_*({\mc
  E}\otimes_{\OO_{\mc X}}{\mc F})$ is an $S$-flat, locally
  finitely-presented sheaf with proper support over $S$.  So by
  ~\cite[corollary, p. 50]{Mum}, we conclude that there is some
  $P([{\mc E}])\in \ZZ$ such that for all $g:\SP k\rightarrow S$, we
  have
\begin{equation}
\chi\lt(g^*\pi_*\lt( {\mc E}\otimes_{\OO_{\mc X}}{\mc F}\rt)\rt) =
  P([{\mc E}]).
\end{equation}
\end{proof}

Fix an additive homomorphism $P:K^0({\mc X})\rightarrow \ZZ$ and
define $Q^P = Q^P({\mc F}/{\mc X}/S)$ to be the subfunctor of $Q({\mc
  F}/{\mc X}/S)$ such that for each $S$-scheme, $g:Z\rightarrow S$ we
have $Q^P(g:Z\rightarrow S)$ is the set of quotients $[g^*{\mc
  F}\rightarrow {\mc G}]\in Q(g:Z\rightarrow S]$ such that for each
field $k$ and each morphism $h:\SP k \rightarrow Z$, we have
$P_{A,h^*{\mc G}}=P$.  By lemma~\ref{lem-HPcont}, we see that $Q^P$ is
an open and closed subfunctor (possibly empty) of $Q$ and that $Q$ is
the disjoint union of $Q^P$ as $P$ ranges over all $P$.

\

We now reformulate theorem~\ref{thm-thm2} as follows.

\begin{thm}~\label{thm-thm3} If $S$ is an affine scheme, $\QQ\subset
  \OO_S$, if ${\mc X}=[Y/G]$ is a quotient of a separated,
  finitely-presented $S$-scheme $Y$ by an affine, finite-type
  $S$-group scheme $G$, and if the coarse moduli space $X=Y//G$ is a
  quasi-projective $S$-scheme, then for each locally
  finitely-presented quasi-coherent sheaf ${\mc F}$ on ${\mc X}$ and
  for each homomorphism $P:K^0({\mc X})\rightarrow \ZZ$, the functor
  $Q^P({\mc F}/{\mc X}/S)$ is represented by a quasi-projective
  $S$-scheme.  If ${\mc F}$ has proper support over $S$, then
  $Q^P({\mc F}/{\mc X}/S)$ is represented by a projective $S$-scheme.
\end{thm}

We will prove theorem~\label{thm-thm3} in the remainder of the paper.

\section{Generating sheaves}
Let ${\mc X}$ be a separated, tame Deligne-Mumford stack with coarse
moduli space 
$\pi:{\mc X}\rightarrow X$.  For each locally free sheaf ${\mc E}$ on
${\mc X}$, define the functors





















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