This is an exercise from Atiyah.
Let $N$ be a flat $B$-module, and $B$ a flat $A$-algebra where $A$ is a commutative ring with unit. Then $N$ is flat as $A$-module
Any hint ?
2026-04-02 12:55:52.1775134552
Exercise from Atiyah about flatness
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Take a short exact sequence of $A$-modules. If you tensor with $B$, then by flatness you will have a short exact sequence of $B$-modules. If you tensor with $N$, then you will again have a short exact sequence of $B$ modules. But $M\otimes_A N\cong M\otimes_A(B\otimes_B N)\cong (M\otimes_A B)\otimes_B N$, and so the two step process is the same as tensoring with $N$ as an $A$-module.