Let $B$ be an $A$-algebra, and let $E$ be a faithfully flat $B$-module. How can I show that $E$ is flat over $A$ if and only if $B$ is flat over $A$? (Liu, Algebraic Geometry and Arithmetic Curves, Exercise 2.18.)
2026-03-26 17:34:56.1774546496
Faitfully flatness over $B$ and flatness over $A$ equivalence
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Note that $E \otimes_A - = E \otimes_B B \otimes_A - $ and due to $E$ being faithfully flat this functor is exact if and only $B \otimes_A -$ is exact, that is, if and only if $B$ is flat over $A$.