Lemma 10.128.2. Let $R \to S$ be a homomorphism of Noetherian local rings. Assume that $R$ is a regular local ring and that a regular system of parameters maps to a regular sequence in $S$. Then $R \to S$ is flat.
Proof. Suppose that $x_1, \ldots , x_ d$ are a system of parameters of $R$ which map to a regular sequence in $S$. Note that $S/(x_1, \ldots , x_ d)S$ is flat over $R/(x_1, \ldots , x_ d)$ as the latter is a field. Then $x_ d$ is a nonzerodivisor in $S/(x_1, \ldots , x_{d - 1})S$ hence $S/(x_1, \ldots , x_{d - 1})S$ is flat over $R/(x_1, \ldots , x_{d - 1})$ by the local criterion of flatness (see Lemma 10.99.10 and remarks following). Then $x_{d - 1}$ is a nonzerodivisor in $S/(x_1, \ldots , x_{d - 2})S$ hence $S/(x_1, \ldots , x_{d - 2})S$ is flat over $R/(x_1, \ldots , x_{d - 2})$ by the local criterion of flatness (see Lemma 10.99.10 and remarks following). Continue till one reaches the conclusion that $S$ is flat over $R$. $\square$
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