Given a unital commutative ring A (not nec. noetherian) and a formally smooth morphism of rings f:A --> B, where B is not nec. noetherian, is (or when is) B a filtered inductive limit of smooth A-algebras?
There is the partial result of D. Popescu, Thm. 1.1, in
MARK SPIVAKOVSKY, A NEW PROOF OF D. POPESCU'S THEOREM ON SMOOTHING OF RING HOMOMORPHISMS, JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY, Volume 12, Number 2, April 1999, Pages 381-444,
where the rings are assumed to be noetherian.