Lemma 10.135.8. Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Let $\mathfrak q$ be a prime of $S$. The following are equivalent:

1. The local ring $S_{\mathfrak q}$ is a complete intersection ring (Definition 10.135.5).

2. There exists a $g \in S$, $g \not\in \mathfrak q$ such that $S_ g$ is a local complete intersection over $k$.

3. There exists a $g \in S$, $g \not\in \mathfrak q$ such that $S_ g$ is a global complete intersection over $k$.

4. For any presentation $S = k[x_1, \ldots , x_ n]/I$ with $\mathfrak q' \subset k[x_1, \ldots , x_ n]$ corresponding to $\mathfrak q$ any of the equivalent conditions (1) – (5) of Lemma 10.135.4 hold.

Proof. This is a combination of Lemmas 10.135.4 and 10.135.7 and the definitions. $\square$

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