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 10.72.1. Given a flat ring map $R \to R'$, an $R$-module $M$, and an $R'$-module $N'$ the natural map

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_{R'}(M \otimes _ R R', N') \to \text{Ext}^ i_ R(M, N') \]

is an isomorphism for $i \geq 0$.

Proof. Choose a free resolution $F_\bullet $ of $M$. Since $R \to R'$ is flat we see that $F_\bullet \otimes _ R R'$ is a free resolution of $M \otimes _ R R'$ over $R'$. The statement is that the map

\[ \mathop{\mathrm{Hom}}\nolimits _{R'}(F_\bullet \otimes _ R R', N') \to \mathop{\mathrm{Hom}}\nolimits _ R(F_\bullet , N') \]

induces an isomorphism on homology groups, which is true because it is an isomorphism of complexes by Lemma 10.13.3. $\square$


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