Characterization of faithfully flat homomorphisms

Because $\rm A \to B$ is injective you have the following sequence $$ 0 \to \rm A \to B \to B/A \to 0.$$

Then because $\rm B/A$ is flat, we have for any $\rm A$-module $\rm M$, $$ 0 \to \rm M \to B \otimes M \to B/A \otimes M \to 0.$$

Hence $ \rm B \otimes M =0 \Rightarrow M = 0$ and so $\rm A \to \rm B$ is faithfully flat.

That's one part.

I seem to get old because I had already come across this result three years ago. It is Lemma 5.5. in Lurie's paper on Tannaka duality. It even works in arbitrary tame abelian $\otimes$-categories.

To go the other way, suppose $f\colon A\to B$ is faithfully flat. The map $f\otimes B\colon A\otimes_AB \to B\otimes_AB$ is injective since it has a section, namely the multiplication map from $B\otimes_AB\to B$ given by $b\otimes b'\mapsto bb'$. Which means it is injective. The original map must therefore be injective, since it is so after a faithfully flat base change.