The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

100.12 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 91.16.12.

Lemma 100.12.1. 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 91.16.5. 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 91.16.1 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 91.4.3). 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 100.12.2. 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 100.12.1. To prove that $\mathcal{C}\! \mathit{omplexes}_{X/B} \to B$ is locally of finite presentation, we have to show that $\mathcal{C}\! \mathit{omplexes}_{X/B} \to B$ is limit preserving, see Limits of Stacks, Proposition 94.3.8. This follows from Quot, Lemma 91.16.8 (small detail omitted). $\square$

Comments (3)

Comment #3013 by arp on

should be

Comment #3014 by arp on

whoops, previous comment was intended for tag 0DPX

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