Lemma 10.129.1. Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Let $f_1, \ldots , f_ i$ be elements of $S$. Assume that $S$ is Cohen-Macaulay and equidimensional of dimension $d$, and that $\dim V(f_1, \ldots , f_ i) \leq d - i$. Then equality holds and $f_1, \ldots , f_ i$ forms a regular sequence in $S_{\mathfrak q}$ for every prime $\mathfrak q$ of $V(f_1, \ldots , f_ i)$.
Proof. If $S$ is Cohen-Macaulay and equidimensional of dimension $d$, then we have $\dim (S_{\mathfrak m}) = d$ for all maximal ideals $\mathfrak m$ of $S$, see Lemma 10.114.7. By Proposition 10.103.4 we see that for all maximal ideals $\mathfrak m \in V(f_1, \ldots , f_ i)$ the sequence is a regular sequence in $S_{\mathfrak m}$ and the local ring $S_{\mathfrak m}/(f_1, \ldots , f_ i)$ is Cohen-Macaulay of dimension $d - i$. This actually means that $S/(f_1, \ldots , f_ i)$ is Cohen-Macaulay and equidimensional of dimension $d - i$. $\square$
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