Proposition

If $M$ and $N$ are flat $A$-modules, then so is $M \otimes_A N$.
If $B$ is a flat $A$-algebra and $N$ is a flat $B$-module, then $N$ is flat as an $A$-module.

Solution

1

Let $f: L’ \rightarrow L$ be an injective $A$-module homomorphism. Since $M$ is flat, $f \otimes 1:L’ \otimes_A M \rightarrow L \otimes_A M$ is injective by Proposition 2.19[Atiyah]. Since $N$ is flat, $(f \otimes 1):(L’ \otimes_A M) \otimes_A N \rightarrow (L \otimes_A M) \otimes_A N$ is injective by Proposition 2.19[Atiyah]. By Proposition 2.14, we obtain isomorphism $\phi: L’ \otimes_A (M \otimes_A N) \rightarrow (L’ \otimes_A M) \otimes_A N$ and $\psi: (L \otimes_A M) \otimes_A N \rightarrow L \otimes_A (M \otimes_A N)$. Then the composition $\psi \circ ((f \otimes 1) \otimes 1) \circ \phi$ is an injection. Moreover, because $\phi$ and $\psi$ are the canonical maps, $\psi \circ ((f \otimes 1) \otimes 1) \circ \phi = f \otimes 1$ where $(f \otimes 1) \otimes 1$ on the left side is the map we constructed above and $f \otimes 1$ on the right side is the map $L’ \otimes_A (M \otimes_A N) \rightarrow L \otimes_A (M \otimes_A N)$. By Proposition 2.19[Atiyah], $M \otimes_A N$ is flat.

2

TODO(Finish this!)