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 10.20.1. Let $R$ be a ring. Let $e \in R$ be an idempotent. In this case

\[ \mathop{\mathrm{Spec}}(R) = D(e) \amalg D(1-e). \]

Proof. Note that an idempotent $e$ of a domain is either $1$ or $0$. Hence we see that

\begin{eqnarray*} D(e) & = & \{ \mathfrak p \in \mathop{\mathrm{Spec}}(R) \mid e \not\in \mathfrak p \} \\ & = & \{ \mathfrak p \in \mathop{\mathrm{Spec}}(R) \mid e \not= 0\text{ in }\kappa (\mathfrak p) \} \\ & = & \{ \mathfrak p \in \mathop{\mathrm{Spec}}(R) \mid e = 1\text{ in }\kappa (\mathfrak p) \} \end{eqnarray*}

Similarly we have

\begin{eqnarray*} D(1-e) & = & \{ \mathfrak p \in \mathop{\mathrm{Spec}}(R) \mid 1 - e \not\in \mathfrak p \} \\ & = & \{ \mathfrak p \in \mathop{\mathrm{Spec}}(R) \mid e \not= 1\text{ in }\kappa (\mathfrak p) \} \\ & = & \{ \mathfrak p \in \mathop{\mathrm{Spec}}(R) \mid e = 0\text{ in }\kappa (\mathfrak p) \} \end{eqnarray*}

Since the image of $e$ in any residue field is either $1$ or $0$ we deduce that $D(e)$ and $D(1-e)$ cover all of $\mathop{\mathrm{Spec}}(R)$. $\square$


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