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 25.6.6. Let $Y$ be an affine scheme. Let $f \in \Gamma (Y, \mathcal{O}_ Y)$. The open subspace $D(f)$ is an affine scheme.

Proof. We may assume that $Y = \mathop{\mathrm{Spec}}(R)$ and $f \in R$. Consider the morphism of affine schemes $\phi : U = \mathop{\mathrm{Spec}}(R_ f) \to \mathop{\mathrm{Spec}}(R) = Y$ induced by the ring map $R \to R_ f$. By Algebra, Lemma 10.16.6 we know that it is a homeomorphism onto $D(f)$. On the other hand, the map $\phi ^{-1}\mathcal{O}_ Y \to \mathcal{O}_ U$ is an isomorphism on stalks, hence an isomorphism. Thus we see that $\phi $ is an open immersion. We conclude that $D(f)$ is isomorphic to $U$ by Lemma 25.3.4. $\square$


Comments (1)

Comment #677 by Anfang Zhou on

Typo. We should replace by .


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