Here is a challenge to an commutative algebraist out there. Give a direct algebraic proof of the following statement (see Lemma Tag 05U9):

Let A —> B be a local ring homomorphism which is essentially of finite type. Let N be a finite type B-module. Let M be a flat A-module. Let u : N —> N be an A-module map such that N/m_AN —> M/m_AM is injective. Then u is A-universally injective, N is a B-module of finite presentation, and N is flat as an A-module.

To my mind it is at least conceivable that there is a direct proof of this (not using the currently used technology). It wouldn’t directly imply all the wonderful things proved by Raynaud and Gruson but it would go a long way towards verifying some of them. In particular, it would give an independent proof of the following result (see Theorem Tag 05UA):

Let f : X —> S be a finite type morphism of schemes. Let x ∈ X with s = f(x) ∈ S. Suppose that X is flat over S at all points x’ ∈ Ass(X_s) which specialize to x. Then X is flat over S at x.

This result is used in an essential way in the main result on universal flattening which I will explain in the next blog post.