Lemma 10.60.14. Let $(R, \mathfrak m)$ be a Noetherian local ring. Suppose $x_1, \ldots , x_ d \in \mathfrak m$ generate an ideal of definition and $d = \dim (R)$. Then $\dim (R/(x_1, \ldots , x_ i)) = d - i$ for all $i = 1, \ldots , d$.

Proof. Follows either from the proof of Proposition 10.60.9, or by using induction on $d$ and Lemma 10.60.13. $\square$

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