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*}

The spectrum of a ring is quasi-compact

Lemma 10.16.10. Let $R$ be a ring. The space $\mathop{\mathrm{Spec}}(R)$ is quasi-compact.

Proof. It suffices to prove that any covering of $\mathop{\mathrm{Spec}}(R)$ by standard opens can be refined by a finite covering. Thus suppose that $\mathop{\mathrm{Spec}}(R) = \cup D(f_ i)$ for a set of elements $\{ f_ i\} _{i\in I}$ of $R$. This means that $\cap V(f_ i) = \emptyset $. According to Lemma 10.16.2 this means that $V(\{ f_ i \} ) = \emptyset $. According to the same lemma this means that the ideal generated by the $f_ i$ is the unit ideal of $R$. This means that we can write $1$ as a finite sum: $1 = \sum _{i \in J} r_ i f_ i$ with $J \subset I$ finite. And then it follows that $\mathop{\mathrm{Spec}}(R) = \cup _{i \in J} D(f_ i)$. $\square$


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