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*}

Lemma 13.15.15. Assumptions and notation as in Situation 13.15.1. If there exists a subset $\mathcal{I} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ such that

  1. for all $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ there exists $s : X \to X'$ in $S$ with $X' \in \mathcal{I}$, and

  2. for every arrow $s : X \to X'$ in $S$ with $X, X' \in \mathcal{I}$ the map $F(s) : F(X) \to F(X')$ is an isomorphism,

then $RF$ is everywhere defined and every $X \in \mathcal{I}$ computes $RF$. Dually, if there exists a subset $\mathcal{P} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ such that

  1. for all $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ there exists $s : X' \to X$ in $S$ with $X' \in \mathcal{P}$, and

  2. for every arrow $s : X \to X'$ in $S$ with $X, X' \in \mathcal{P}$ the map $F(s) : F(X) \to F(X')$ is an isomorphism,

then $LF$ is everywhere defined and every $X \in \mathcal{P}$ computes $LF$.

Proof. Let $X$ be an object of $\mathcal{D}$. Assumption (1) implies that the arrows $s : X \to X'$ in $S$ with $X' \in \mathcal{I}$ are cofinal in the category $X/S$. Assumption (2) implies that $F$ is constant on this cofinal subcategory. Clearly this implies that $F : (X/S) \to \mathcal{D}'$ is essentially constant with value $F(X')$ for any $s : X \to X'$ in $S$ with $X' \in \mathcal{I}$. $\square$


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