# Math Genius: 2 out of 3 property for faithfully flat ring maps

Proposition 1: Let $$f : Ato B$$ and $$g : Bto C$$ be ring maps such that $$g$$ is faithfully flat. Then the composition $$gf$$ is flat (resp. faithfully flat) if and only if $$f$$ is flat (resp. faithfully flat).

Proof: Certainly flatness (resp. faithful flatness) of $$f$$ implies flatness (resp. faithful flatness) of $$gf.$$

Conversely, suppose that $$gf$$ is flat, and let $$M’to Mto M”$$ be an exact sequence of $$A$$-modules. Then flatness of $$gf$$ implies that $$M’otimes_A Cto Motimes_A Cto M”otimes_A C$$ is exact as well, whence by faithful flatness of $$g$$ we have that $$M’otimes_A Bto Motimes_A Bto M”otimes_A B$$ is exact.

If $$gf$$ is faithfully flat, and $$M’to Mto M”$$ is a sequence of $$A$$-modules such that the composition $$M’to M”$$ is $$0,$$ then faithful flatness of $$gf$$ implies that $$M’to Mto M”$$ is exact if and only if $$M’otimes_A Cto Motimes_A Cto M”otimes_A C$$ is. Faithful flatness of $$g$$ implies that this is exact if and only if $$M’otimes_A Bto Motimes_A Bto M”otimes_A B$$ is. $$square$$

This result has been surprisingly hard to track down in the literature, despite the simplicity of the proof. (The proof and statement of proposition 1 in the flat case can be found here; I do not know of anywhere that the version of proposition 1 for faithful flatness exists, although I’m sure I just haven’t searched carefully enough.) Note that this proposition is not true only assuming flatness of $$g,$$ as evidenced by the composition $$k[t^2, t^3]to k[t]to k(t).$$

I was originally interested in the situation where we assume that the composition is flat, and use that to deduce that $$f$$ is flat. However, I was wondering whether the following stronger proposition is true.

Proposition 2: Let $$f : Ato B$$ and $$g : Bto C$$ be ring maps such that the composition $$gf : Ato C$$ is faithfully flat. Then $$f$$ is faithfully flat.

Disclosure: I asked this question in the hopes that this 2 out of 3 property for faithfully flat morphisms will more easily searchable for anyone trying to find a result of this type in the future.

Indeed, Proposition 2 is true.

Proof: Recall that $$Ato B$$ is faithfully flat if and only if the canonical map $$Nto Notimes_A B$$ is injective for every $$A$$-module $$N.$$ To that end, let $$N$$ be an $$A$$-module. We need to prove that $$Nto Notimes_A B$$ is injective, but we know by faithful flatness of $$gf$$ that the composition $$Nto Notimes_A Bto Notimes_A C$$ is injective. Thus, it follows that $$Nto Notimes_A B$$ must be injective. $$square$$

Certainly, it need not be true that $$Bto C$$ be faithfully flat even if both $$Ato B$$ and $$Ato C$$ are (consider the composition $$kto k[X]to k(X),$$ for $$k$$ a field). So the best we can hope for is Proposition 2.