Let A —> B be a finitely presented ring map. Let M be a finitely presented B-module flat over A. Then the projective dimension of M as an A-module is at most 1.

This follows from a result of Jensen which you can find as Theorem 3.2 in Lazard’s paper “Autour de la platitude” which says that a countably presented flat A-module has projective dimension at most 1.

In Raynaud-Gruson they prove that “locally” M is actually projective. Namely, suppose that q is a prime of B lying over a prime p of A. Then there exists an \’etale ring map (A, p) —> (A’, p’) inducing a trivial residue field extension k(p) = k(p’) and an element h ∈ B’ = B ⊗_A A’ not contained in the unique prime q’ ⊂ B’ lying over q and p’ such that M’_h is a projective A’-module, where M’ = M ⊗_A A’.

See Lemma 05ME. The key case is where A and B are of finite type over Z.

Challenge: Find a simple proof of Lemma 05ME.

Naively, you might think there is a chance as we only need to reduce the projective dimension from 1 to 0… Of course, it probably is hopeless to find an elementary proof, perhaps even more hopeless than the challenge in the previous blog post.