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:
The local ring S_{\mathfrak q} is a complete intersection ring (Definition 10.135.5).
There exists a g \in S, g \not\in \mathfrak q such that S_ g is a local complete intersection over k.
There exists a g \in S, g \not\in \mathfrak q such that S_ g is a global complete intersection over k.
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.
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