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 21.18.11. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\alpha : \mathcal{P}^\bullet \to \mathcal{Q}^\bullet $ be a quasi-isomorphism of K-flat complexes of $\mathcal{O}$-modules. For every complex $\mathcal{F}^\bullet $ of $\mathcal{O}$-modules the induced map

\[ \text{Tot}(\text{id}_{\mathcal{F}^\bullet } \otimes \alpha ) : \text{Tot}(\mathcal{F}^\bullet \otimes _\mathcal {O} \mathcal{P}^\bullet ) \longrightarrow \text{Tot}(\mathcal{F}^\bullet \otimes _\mathcal {O} \mathcal{Q}^\bullet ) \]

is a quasi-isomorphism.

Proof. Choose a quasi-isomorphism $\mathcal{K}^\bullet \to \mathcal{F}^\bullet $ with $\mathcal{K}^\bullet $ a K-flat complex, see Lemma 21.18.10. Consider the commutative diagram

\[ \xymatrix{ \text{Tot}(\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{P}^\bullet ) \ar[r] \ar[d] & \text{Tot}(\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{Q}^\bullet ) \ar[d] \\ \text{Tot}(\mathcal{F}^\bullet \otimes _\mathcal {O} \mathcal{P}^\bullet ) \ar[r] & \text{Tot}(\mathcal{F}^\bullet \otimes _\mathcal {O} \mathcal{Q}^\bullet ) } \]

The result follows as by Lemma 21.18.3 the vertical arrows and the top horizontal arrow are quasi-isomorphisms. $\square$


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