Algebraicity Picard stack

examples-stacks.tex

@@ -1554,6 +1554,12 @@ \section{The Picard stack}
 \label{section-picard-stack}
 
 \noindent
+In this section we introduce the Picard stack in complete generality.
+In the chapter on Quot and Hilb we will show that it is an algebraic
+stack under suitable hypotheses, see
+Quot, Section \ref{quot-section-picard-stack}.
+
+\medskip\noindent
 Let $S$ be a scheme.
 Let $\pi : X \to B$ be a morphism of algebraic spaces over $S$.
 We define a category $\textit{Pic}_{X/B}$ as follows:

quot.tex

@@ -1026,9 +1026,9 @@ \section{The stack of coherent sheaves}
 flat over $T'$. Set $\mathcal{F} = i^*\mathcal{F}'$. The following
 are equivalent
 \begin{enumerate}
-\item $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_{X'}$-module
+\item $\mathcal{F}'$ is a quasi-coherent $\mathcal{O}_{X'}$-module
 of finite presentation,
-\item $\mathcal{F}$ is an $\mathcal{O}_{X'}$-module of finite presentation,
+\item $\mathcal{F}'$ is an $\mathcal{O}_{X'}$-module of finite presentation,
 \item $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_X$-module
 of finite presentation,
 \item $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation,
@@ -1278,7 +1278,7 @@ \section{The stack of coherent sheaves}
 More on Algebra, Lemma \ref{more-algebra-lemma-pseudo-coherent-tensor}.
 \end{proof}
 
-\begin{theorem}[Algebraicity of stack coherent sheaves]
+\begin{theorem}[Algebraicity of stack coherent sheaves; flat case]
 \label{theorem-coherent-algebraic}
 Let $S$ be a scheme. Let $f : X \to B$ be morphism of algebraic spaces
 over $S$. Assume that $f$ is of finite presentation, separated, and
@@ -2250,6 +2250,98 @@ \section{The Hilbert functor}
 
 
 
+\section{The Picard stack}
+\label{section-picard-stack}
+
+\noindent
+The Picard stack for a morphism of algebraic spaces was introduced
+in Examples of Stacks, Section \ref{examples-stacks-section-picard-stack}.
+We will deduce it is an open substack of the stack of coherent sheaves
+(in good cases) from the following lemma.
+
+\begin{lemma}
+\label{lemma-picard-stack-open-in-coh}
+Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces
+over $S$ which is flat, of finite presentation, and proper.
+Then natural map
+$$
+\textit{Pic}_{X/B} \longrightarrow \textit{Coh}_{X/B}
+$$
+is representable by open immersions.
+\end{lemma}
+
+\begin{proof}
+Observe that the map simply sends a triple $(T, g, \mathcal{L})$
+as in Examples of Stacks, Section \ref{examples-stacks-section-picard-stack}
+to the same triple $(T, g, \mathcal{L})$ but where now we view
+this as a triple of the kind described in
+Situation \ref{situation-coherent}.
+This works because the invertible $\mathcal{O}_{X_T}$-module
+$\mathcal{L}$ is certainly a finitely presented $\mathcal{O}_{X_T}$-module,
+it is flat over $T$ because $X_T \to T$ is flat, and the support is
+proper over $T$ as $X_T \to T$ is proper
+(Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-base-change-flat}
+and \ref{spaces-morphisms-lemma-base-change-proper}).
+Thus the statement makes sense.
+
+\medskip\noindent
+Having said this, it is clear that the content of the lemma is the
+following: given an object $(T, g, \mathcal{F})$ of
+$\textit{Coh}_{X/B}$ there is an open subscheme $U \subset T$
+such that for a morphism of schemes $T' \to T$ the following
+are equivalent
+\begin{enumerate}
+\item[(a)] $T' \to T$ factors through $U$,
+\item[(b)] the pullback $\mathcal{F}_{T'}$ of
+$\mathcal{F}$ by $X_{T'} \to X_T$ is invertible.
+\end{enumerate}
+Let $W \subset |X_T|$ be the set of points $x \in |X_T|$
+such that $\mathcal{F}$ is locally free in a neighbourhood of $x$. By
+More on Morphisms of Spaces, Lemma
+\ref{spaces-more-morphisms-lemma-finite-free-open}.
+$W$ is open and formation
+of $W$ commutes with arbitrary base change.
+Clearly, if $T' \to T$ satisfies (b), then $|X_{T'}| \to |X_T|$
+maps into $W$. Hence we may replace $T$ by the open
+$T \setminus f_T(|X_T| \setminus W)$ in order
+to construct $U$. After doing so we reach the situation
+where $\mathcal{F}$ is finite locally free.
+In this case we get a disjoint union decomposition
+$X_T = X_0 \amalg X_1 \amalg X_2 \amalg \ldots$
+into open and closed subspaces such that the restriction of
+$\mathcal{F}$ is locally free of rank $i$ on $X_i$. Then clearly
+$$
+U = T \setminus f_T(|X_0| \cup |X_2| \cup |X_3| \cup \ldots )
+$$
+works. (Note that if we assume that $T$ is quasi-compact, then
+$X_T$ is quasi-compact hence only a finite number of $X_i$
+are nonempty and so $U$ is indeed open.)
+\end{proof}
+
+\begin{proposition}
+\label{proposition-pic}
+Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic
+spaces over $S$. If $f$ is flat, of finite presentation, and proper, then
+$\textit{Pic}_{X/B}$ is an algebraic stack.
+\end{proposition}
+
+\begin{proof}
+Immediate consequence of
+Lemma \ref{lemma-picard-stack-open-in-coh},
+Algebraic Stacks, Lemma
+\ref{algebraic-lemma-representable-morphism-to-algebraic}
+and either
+Theorem \ref{theorem-coherent-algebraic}
+or
+Theorem \ref{theorem-coherent-algebraic-general}
+\end{proof}
+
+
+
+
+
+
+