Today I encountered the following lemma:
Let A be a ring. Let u : M —> N be an A-module map. Let R = colim_i R_i be a directed colimit of A-algebras. Assume that M is a finite A-module, N is a finitely presented A-module, and u ⊗ 1 : M ⊗ R —> N ⊗ R is an isomorphism. Then there exists an i ∈ I such that u ⊗ 1 : M ⊗ R_i —> N ⊗ R_i is an isomorphism. (All tensor products over A.)
What I like about this statement is that M only needs to be a finite A-module. This is similar to what happened in this post.