The Stacks project

Flat local ring homomorphisms preserve and reflect regular sequences.

Lemma 10.68.5. Let $R, S$ be local rings. Let $R \to S$ be a flat local ring homomorphism. Let $x_1, \ldots , x_ r$ be a sequence in $R$. Let $M$ be an $R$-module. The following are equivalent

  1. $x_1, \ldots , x_ r$ is an $M$-regular sequence in $R$, and

  2. the images of $x_1, \ldots , x_ r$ in $S$ form a $M \otimes _ R S$-regular sequence.

Proof. This is so because $R \to S$ is faithfully flat by Lemma 10.39.17. $\square$

Comments (2)

Comment #6836 by Herman Rohrbach on

Suggested slogan: "Flat local ring homomorphisms preserve and reflect regular sequences."

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