Lemma 10.133.3. Let $k$ be a field. Let $S$ be a finite type $k$-algebra. If $S$ is a local complete intersection, then $S$ is a Cohen-Macaulay ring.

**Proof.**
Choose a maximal prime $\mathfrak m$ of $S$. We have to show that $S_\mathfrak m$ is Cohen-Macaulay. By assumption we may assume $S = k[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ with $\dim (S) = n - c$. Let $\mathfrak m' \subset k[x_1, \ldots , x_ n]$ be the maximal ideal corresponding to $\mathfrak m$. According to Proposition 10.113.2 the local ring $k[x_1, \ldots , x_ n]_{\mathfrak m'}$ is regular local of dimension $n$. In particular it is Cohen-Macaulay by Lemma 10.105.3. By Lemma 10.59.12 applied $c$ times the local ring $S_{\mathfrak m} = k[x_1, \ldots , x_ n]_{\mathfrak m'}/(f_1, \ldots , f_ c)$ has dimension $\geq n - c$. By assumption $\dim (S_{\mathfrak m}) \leq n - c$. Thus we get equality. This implies that $f_1, \ldots , f_ c$ is a regular sequence in $k[x_1, \ldots , x_ n]_{\mathfrak m'}$ and that $S_{\mathfrak m}$ is Cohen-Macaulay, see Proposition 10.102.4.
$\square$

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