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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