Lemma 10.99.2. Suppose that $R \to S$ is a flat and local ring homomorphism of Noetherian local rings. Denote $\mathfrak m$ the maximal ideal of $R$. Suppose $f \in S$ is a nonzerodivisor in $S/{\mathfrak m}S$. Then $S/fS$ is flat over $R$, and $f$ is a nonzerodivisor in $S$.

Proof. Follows directly from Lemma 10.99.1. $\square$

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