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 10.107.5. Let $R$ be a ring. Let $I \subset R$ be an ideal. The following are equivalent

  1. $I$ is pure and finitely generated,

  2. $I$ is generated by an idempotent,

  3. $I$ is pure and $V(I)$ is open, and

  4. $R/I$ is a projective $R$-module.

Proof. If (1) holds, then $I = I \cap I = I^2$ by Lemma 10.107.2. Hence $I$ is generated by an idempotent by Lemma 10.20.5. Thus (1) $\Rightarrow $ (2). If (2) holds, then $I = (e)$ and $R = (1 - e) \oplus (e)$ as an $R$-module hence $R/I$ is flat and $I$ is pure and $V(I) = D(1 - e)$ is open. Thus (2) $\Rightarrow $ (1) $+$ (3). Finally, assume (3). Then $V(I)$ is open and closed, hence $V(I) = D(1 - e)$ for some idempotent $e$ of $R$, see Lemma 10.20.3. The ideal $J = (e)$ is a pure ideal such that $V(J) = V(I)$ hence $I = J$ by Lemma 10.107.3. In this way we see that (3) $\Rightarrow $ (2). By Lemma 10.77.2 we see that (4) is equivalent to the assertion that $I$ is pure and $R/I$ finitely presented. Moreover, $R/I$ is finitely presented if and only if $I$ is finitely generated, see Lemma 10.5.3. Hence (4) is equivalent to (1). $\square$


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