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 18.28.8. Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings. Let

\[ 0 \to \mathcal{F}'' \to \mathcal{F}' \to \mathcal{F} \to 0 \]

be a short exact sequence of presheaves of $\mathcal{O}$-modules. Let $\mathcal{G}$ be a presheaf of $\mathcal{O}$-modules.

  1. If $\mathcal{F}$ is a flat presheaf of modules, then the sequence

    \[ 0 \to \mathcal{F}'' \otimes _{p, \mathcal{O}} \mathcal{G} \to \mathcal{F}' \otimes _{p, \mathcal{O}} \mathcal{G} \to \mathcal{F} \otimes _{p, \mathcal{O}} \mathcal{G} \to 0 \]

    is exact.

  2. If $\mathcal{C}$ is a site, $\mathcal{O}$, $\mathcal{F}$, $\mathcal{F}'$, $\mathcal{F}''$, and $\mathcal{G}$ are sheaves, and $\mathcal{F}$ is flat as a sheaf of modules, then the sequence

    \[ 0 \to \mathcal{F}'' \otimes _\mathcal {O} \mathcal{G} \to \mathcal{F}' \otimes _\mathcal {O} \mathcal{G} \to \mathcal{F} \otimes _\mathcal {O} \mathcal{G} \to 0 \]

    is exact.

Proof. Choose a flat presheaf of $\mathcal{O}$-modules $\mathcal{G}'$ which surjects onto $\mathcal{G}$. This is possible by Lemma 18.28.7. Let $\mathcal{G}'' = \mathop{\mathrm{Ker}}(\mathcal{G}' \to \mathcal{G})$. The lemma follows by applying the snake lemma to the following diagram

\[ \begin{matrix} & & 0 & & 0 & & 0 & & \\ & & \uparrow & & \uparrow & & \uparrow & & \\ & & \mathcal{F}'' \otimes _{p, \mathcal{O}} \mathcal{G} & \to & \mathcal{F}' \otimes _{p, \mathcal{O}} \mathcal{G} & \to & \mathcal{F} \otimes _{p, \mathcal{O}} \mathcal{G} & \to & 0 \\ & & \uparrow & & \uparrow & & \uparrow & & \\ 0 & \to & \mathcal{F}'' \otimes _{p, \mathcal{O}} \mathcal{G}' & \to & \mathcal{F}' \otimes _{p, \mathcal{O}} \mathcal{G}' & \to & \mathcal{F} \otimes _{p, \mathcal{O}} \mathcal{G}' & \to & 0 \\ & & \uparrow & & \uparrow & & \uparrow & & \\ & & \mathcal{F}'' \otimes _{p, \mathcal{O}} \mathcal{G}'' & \to & \mathcal{F}' \otimes _{p, \mathcal{O}} \mathcal{G}'' & \to & \mathcal{F} \otimes _{p, \mathcal{O}} \mathcal{G}'' & \to & 0 \\ & & & & & & \uparrow & & \\ & & & & & & 0 & & \end{matrix} \]

with exact rows and columns. The middle row is exact because tensoring with the flat module $\mathcal{G}'$ is exact. The proof in the case of sheaves is exactly the same. $\square$


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