Let $A$ be a commutative ring, and let $B,C$ be commutative $A$-algebras.

Let $M$ be a flat $B$-module and $N$ a flat $C$-module.

I want to show that $M\otimes_A N$ is a flat $B\otimes_A C$-module.

Here is my idea of the proof, I would like to know if it is correct:

Clearly, if $M$ and $N$ are finitely generated free then $M\otimes_A N$ is finitely generated free $B\otimes_A C$-module.

By Lazard Theorem, we may write $M = \varinjlim F_i$ and $N = \varinjlim G_i$ where $F_i$ and $G_i$ are finitely generated free.

Hence, since tensor product commutes with colimits,

$M \otimes_A N \cong \varinjlim F_i\otimes_A G_i$, so it is a direct limit of finitely generated free modules, so again by Lazard's theorem, it is a flat $B\otimes_A C$-module.

Is this reasoning correct? is there a simpler proof?


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