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 10.117.6. Let $R \to S$ be a ring map. Let $M$ be an $S$-module. Let $U \subset \mathop{\mathrm{Spec}}(R)$ be a dense open. Assume there is a covering $U = \bigcup _{i \in I} D(f_ i)$ of opens such that $U(R_{f_ i} \to S_{f_ i}, M_{f_ i})$ is dense in $D(f_ i)$ for each $i \in I$. Then $U(R \to S, M)$ is dense in $\mathop{\mathrm{Spec}}(R)$.

Proof. In view of Lemma 10.117.5 this is a purely topological statement. Namely, by that lemma we see that $U(R \to S, M) \cap D(f_ i)$ is dense in $D(f_ i)$ for each $i \in I$. By Topology, Lemma 5.21.4 we see that $U(R \to S, M) \cap U$ is dense in $U$. Since $U$ is dense in $\mathop{\mathrm{Spec}}(R)$ we conclude that $U(R \to S, M)$ is dense in $\mathop{\mathrm{Spec}}(R)$. $\square$


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