# The Stacks Project

## Tag 0DPV

### 98.12. Properties of moduli of complexes on a proper morphism

Let $f : X \to B$ be a morphism of algebraic spaces which is proper, flat, and of finite presentation. Then the stack $\mathcal{C}\!\mathit{omplexes}_{X/B}$ parametrizing relatively perfect complexes with vanishing negative self-exts is algebraic. See Quot, Theorem 89.16.12.

Lemma 98.12.1. The diagonal of $\mathcal{C}\!\mathit{omplexes}_{X/B}$ over $B$ is affine and of finite presentation.

Proof. The representability of the diagonal (by algebraic spaces) was shown in Quot, Lemma 89.16.5. From the proof we find that we have to show: given a scheme $T$ over $B$ and objects $E, E' \in D(\mathcal{O}_{X_T})$ such that $(T, E)$ and $(T, E')$ are objects of the fibre category of $\mathcal{C}\!\mathit{omplexes}_{X/B}$ over $T$, then $\mathit{Isom}(E, E') \to T$ is affine and of finite presentation. Here $\mathit{Isom}(E, E')$ is the functor $$(\mathit{Sch}/T)^{opp} \to \textit{Sets},\quad T' \mapsto \{\varphi : E_{T'} \to E'_{T'} \text{ isomorphism in }D(\mathcal{O}_{X_{T'}})\}$$ where $E_{T'}$ and $E'_{T'}$ are the derived pullbacks of $E$ and $E'$ to $X_{T'}$. Consider the functor $H = \mathop{\mathcal{H}\!\mathit{om}}\nolimits(E, E')$ defined by the rule $$(\mathit{Sch}/T)^{opp} \to \textit{Sets},\quad T' \mapsto \mathop{\mathrm{Hom}}\nolimits_{\mathcal{O}_{X_{T'}}}(E_T, E'_T)$$ By Quot, Lemma 89.16.1 this is an algebraic space affine and of finite presentation over $T$. The same is true for $H' = \mathop{\mathcal{H}\!\mathit{om}}\nolimits(E', E)$, $I = \mathop{\mathcal{H}\!\mathit{om}}\nolimits(E, E)$, and $I' = \mathop{\mathcal{H}\!\mathit{om}}\nolimits(E', E')$. Therefore we see that $$\mathit{Isom}(E, E') = (H' \times_T H) \times_{c, I \times_T I', \sigma} T$$ where $c(\varphi', \varphi) = (\varphi \circ \varphi', \varphi' \circ \varphi)$ and $\sigma = (\text{id}, \text{id})$ (compare with the proof of Quot, Proposition 89.4.3). Thus $\mathit{Isom}(E, E')$ is affine over $T$ as a fibre product of schemes affine over $T$. Similarly, $\mathit{Isom}(E, E')$ is of finite presentation over $T$. $\square$

Lemma 98.12.2. The morphism $\mathcal{C}\!\mathit{omplexes}_{X/B} \to B$ is quasi-separated and locally of finite presentation.

Proof. To check $\mathcal{C}\!\mathit{omplexes}_{X/B} \to B$ is quasi-separated we have to show that its diagonal is quasi-compact and quasi-separated. This is immediate from Lemma 98.12.1. To prove that $\mathcal{C}\!\mathit{omplexes}_{X/B} \to B$ is locally of finite presentation, we have to show that $\mathcal{C}\!\mathit{oh}_{X/B} \to B$ is limit preserving, see Limits of Stacks, Proposition 92.3.8. This follows from Quot, Lemma 89.16.8 (small detail omitted). $\square$

The code snippet corresponding to this tag is a part of the file moduli.tex and is located in lines 1992–2068 (see updates for more information).

\section{Properties of moduli of complexes on a proper morphism}
\label{section-complexes}

\noindent
Let $f : X \to B$ be a morphism of algebraic spaces which is proper,
flat, and of finite presentation. Then the stack
$\Complexesstack_{X/B}$ parametrizing relatively perfect complexes
with vanishing negative self-exts is algebraic. See
Quot, Theorem \ref{quot-theorem-complexes-algebraic}.

\begin{lemma}
\label{lemma-complexes-diagonal-affine-fp}
The diagonal of $\Complexesstack_{X/B}$ over $B$ is affine
and of finite presentation.
\end{lemma}

\begin{proof}
The representability of the diagonal (by algebraic spaces)
was shown in Quot, Lemma \ref{quot-lemma-complexes-diagonal}.
From the proof we find that we have to show:
given a scheme $T$ over $B$ and objects
$E, E' \in D(\mathcal{O}_{X_T})$ such that
$(T, E)$ and $(T, E')$ are objects of the fibre category
of $\Complexesstack_{X/B}$ over $T$, then
$\mathit{Isom}(E, E') \to T$
is affine and of finite presentation.
Here $\mathit{Isom}(E, E')$ is the functor
$$(\Sch/T)^{opp} \to \textit{Sets},\quad T' \mapsto \{\varphi : E_{T'} \to E'_{T'} \text{ isomorphism in }D(\mathcal{O}_{X_{T'}})\}$$
where $E_{T'}$ and $E'_{T'}$ are the derived pullbacks of $E$ and $E'$
to $X_{T'}$. Consider the functor $H = \SheafHom(E, E')$ defined
by the rule
$$(\Sch/T)^{opp} \to \textit{Sets},\quad T' \mapsto \Hom_{\mathcal{O}_{X_{T'}}}(E_T, E'_T)$$
By Quot, Lemma \ref{quot-lemma-complexes-open-neg-exts-vanishing}
this is an algebraic space affine and of finite presentation over $T$.
The same is true for $H' = \SheafHom(E', E)$, $I = \SheafHom(E, E)$, and
$I' = \SheafHom(E', E')$. Therefore we see that
$$\mathit{Isom}(E, E') = (H' \times_T H) \times_{c, I \times_T I', \sigma} T$$
where $c(\varphi', \varphi) = (\varphi \circ \varphi', \varphi' \circ \varphi)$
and $\sigma = (\text{id}, \text{id})$ (compare with the proof of
Quot, Proposition \ref{quot-proposition-isom}). Thus
$\mathit{Isom}(E, E')$ is affine over $T$ as a fibre product of
schemes affine over $T$. Similarly, $\mathit{Isom}(E, E')$ is
of finite presentation over $T$.
\end{proof}

\begin{lemma}
\label{lemma-complexes-qs-lfp}
The morphism $\Complexesstack_{X/B} \to B$ is quasi-separated and
locally of finite presentation.
\end{lemma}

\begin{proof}
To check $\Complexesstack_{X/B} \to B$ is quasi-separated we have to
show that its diagonal is quasi-compact and quasi-separated.
This is immediate from Lemma \ref{lemma-complexes-diagonal-affine-fp}.
To prove that $\Complexesstack_{X/B} \to B$ is locally of finite
presentation, we have to show that $\Cohstack_{X/B} \to B$
is limit preserving, see
Limits of Stacks, Proposition
\ref{stacks-limits-proposition-characterize-locally-finite-presentation}.
This follows from Quot, Lemma \ref{quot-lemma-complexes-limits}
(small detail omitted).
\end{proof}

\input{chapters}

Comment #3013 by arp on December 5, 2017 a 2:17 am UTC

$\mathcal{C}\!\mathit{oh}_{X/B} \to B$ should be $\mathcal{C}\!\mathit{omplexes}_{X/B} \to B$

Comment #3014 by arp on December 5, 2017 a 2:19 am UTC

whoops, previous comment was intended for tag 0DPX

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