Lemma 10.133.2. Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Let $g \in S$.

If $S$ is a global complete intersection so is $S_ g$.

If $S$ is a local complete intersection so is $S_ g$.

Lemma 10.133.2. Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Let $g \in S$.

If $S$ is a global complete intersection so is $S_ g$.

If $S$ is a local complete intersection so is $S_ g$.

**Proof.**
The second statement follows immediately from the first. Proof of the first statement. If $S_ g$ is the zero ring, then it is true. Assume $S_ g$ is nonzero. Write $S = k[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ with $n - c = \dim (S)$ as in Definition 10.133.1. By the remarks following the definition $S$ is equidimensional of dimension $n - c$, so $\dim (S_ g) = n - c$ as well. Let $g' \in k[x_1, \ldots , x_ n]$ be an element whose residue class corresponds to $g$. Then $S_ g = k[x_1, \ldots , x_ n, x_{n + 1}]/(f_1, \ldots , f_ c, x_{n + 1}g' - 1)$ as desired.
$\square$

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## Comments (2)

Comment #2913 by Dario Weißmann on

Comment #2942 by Johan on