The Stacks project

Lemma 10.108.5. 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 10.108.2. Hence $I$ is generated by an idempotent by Lemma 10.21.5. 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 10.21.3. The ideal $J = (e)$ is a pure ideal such that $V(J) = V(I)$ hence $I = J$ by Lemma 10.108.3. In this way we see that (3) $\Rightarrow $ (2). By Lemma 10.78.2 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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