Lemma 10.128.5. Suppose that $R \to S$ is a local ring homomorphism of local rings. Denote $\mathfrak m$ the maximal ideal of $R$. Suppose

$S$ is essentially of finite presentation over $R$,

$S$ is flat over $R$, and

$f \in S$ is a nonzerodivisor in $S/{\mathfrak m}S$.

Then $S/fS$ is flat over $R$, and $f$ is a nonzerodivisor in $S$.

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