Lemma 10.135.9. Let $k$ be a field. Let $S$ be a finite type $k$-algebra. The following are equivalent:

1. The ring $S$ is a local complete intersection over $k$.

2. All local rings of $S$ are complete intersection rings over $k$.

3. All localizations of $S$ at maximal ideals are complete intersection rings over $k$.

Proof. This follows from Lemma 10.135.8, the fact that $\mathop{\mathrm{Spec}}(S)$ is quasi-compact and the definitions. $\square$

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