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 18.28.6. Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings. Let $U$ be an object of $\mathcal{C}$. Consider the functor $j_ U : \mathcal{C}/U \to \mathcal{C}$.

  1. The presheaf of $\mathcal{O}$-modules $j_{U!}\mathcal{O}_ U$ (see Remark 18.19.7) is flat.

  2. If $\mathcal{C}$ is a site, $\mathcal{O}$ is a sheaf of rings, $j_{U!}\mathcal{O}_ U$ is a flat sheaf of $\mathcal{O}$-modules.

Proof. Proof of (1). By the discussion in Remark 18.19.7 we see that

\[ j_{U!}\mathcal{O}_ U(V) = \bigoplus \nolimits _{\varphi \in \mathop{Mor}\nolimits _\mathcal {C}(V, U)} \mathcal{O}(V) \]

which is a flat $\mathcal{O}(V)$-module. Hence (1) follows from Lemma 18.28.2. Then (2) follows as $j_{U!}\mathcal{O}_ U = (j_{U!}\mathcal{O}_ U)^\# $ (the first $j_{U!}$ on sheaves, the second on presheaves) and Lemma 18.28.3. $\square$


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