Lemma 51.17.5 (0EBZ)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 51: Local Cohomology
Section 51.17: Frobenius action
Lemma 51.17.5 (cite)

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Lemma 51.17.5. Let $\varphi : A \to B$ be a flat ring map. If $f_1, \ldots , f_ r \in A$ are independent, then $\varphi (f_1), \ldots , \varphi (f_ r) \in B$ are independent.

Proof. Let $I = (f_1, \ldots , f_ r)$ and $J = \varphi (I)B$ . By flatness we have $I/I^2 \otimes _ A B = J/J^2$ . Hence freeness of $I/I^2$ over $A/I$ implies freeness of $J/J^2$ over $B/J$ . $\square$

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