The Stacks project

Lemma 10.104.10. Let $\varphi : A \to B$ be a map of local rings. Assume that $B$ is Noetherian and Cohen-Macaulay and that $\mathfrak m_ B = \sqrt{\varphi (\mathfrak m_ A) B}$. Then there exists a sequence of elements $f_1, \ldots , f_{\dim (B)}$ in $A$ such that $\varphi (f_1), \ldots , \varphi (f_{\dim (B)})$ is a regular sequence in $B$.

Proof. By induction on $\dim (B)$ it suffices to prove: If $\dim (B) \geq 1$, then we can find an element $f$ of $A$ which maps to a nonzerodivisor in $B$. By Lemma 10.104.2 it suffices to find $f \in A$ whose image in $B$ is not contained in any of the finitely many minimal primes $\mathfrak q_1, \ldots , \mathfrak q_ r$ of $B$. By the assumption that $\mathfrak m_ B = \sqrt{\varphi (\mathfrak m_ A) B}$ we see that $\mathfrak m_ A \not\subset \varphi ^{-1}(\mathfrak q_ i)$. Hence we can find $f$ by Lemma 10.15.2. $\square$

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