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 27.26.4. Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $s \in \Gamma (X, \mathcal{L})$. For any affine $U \subset X$ the intersection $U \cap X_ s$ is affine.

Proof. This translates into the following algebra problem. Let $R$ be a ring. Let $N$ be an invertible $R$-module (i.e., locally free of rank 1). Let $s \in N$ be an element. Then $U = \{ \mathfrak p \mid s \not\in \mathfrak p N\} $ is an affine open subset of $\mathop{\mathrm{Spec}}(R)$. This you can see as follows. Think of $s$ as an $R$-module map $R \to N$. This gives rise to $R$-module maps $N^{\otimes k} \to N^{\otimes k + 1}$. Consider

\[ R' = \mathop{\mathrm{colim}}\nolimits _ n N^{\otimes n} \]

with transition maps as above. Define an $R$-algebra structure on $R'$ by the rule $x \cdot y = x \otimes y \in N^{\otimes n + m}$ if $x \in N^{\otimes n}$ and $y \in N^{\otimes m}$. We claim that $\mathop{\mathrm{Spec}}(R') \to \mathop{\mathrm{Spec}}(R)$ is an open immersion with image $U$.

To prove this is a local question on $\mathop{\mathrm{Spec}}(R)$. Let $\mathfrak p \in \mathop{\mathrm{Spec}}(R)$. Pick $f \in R$, $f \not\in \mathfrak p$ such that $N_ f \cong R_ f$ as a module. Replacing $R$ by $R_ f$, $N$ by $N_ f$ and $R'$ by $R'_ f = \mathop{\mathrm{colim}}\nolimits N_ f^{\otimes n}$ we may assume that $N \cong R$. Say $N = R$. In this case $s$ is an element of $R$ and it is easy to see that $R' \cong R_ s$. Thus the lemma follows. $\square$


Comments (1)

Comment #3508 by Yicheng Zhou on

By using direction (2) to (1) of Lemma 28.11.3, one can give a more schematic proof as follows. By hypothesis we have locally , therefore the open immersion is locally given by a principal open set (in an affine neighborhood). By the lemma cited, is affine, therfore is affine whenever is affine.


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