A freeness criterion without patching for modules over local rings

 Let A->B be a local homomorphism of (commutative) Noetherian local rings. Bart de Smit conjectured in the late 1990's that if A and B have the same embedding dimension, then any finitely generated B-module that is flat over A, is flat over B. This conjecture was proved in 2017 and allows one in some situations to dispense with patching in the techniques à la Wiles to prove modularity lifting theorems (e.g. in the proof of FLT). We generalize this result as follows: if M is a finitely generated B-module whose flat dimension over A satisfies flat dim_A(M) \leq edim(A)-edim(B), then M is free as a B-module. If moreover M is nonzero this forces the morphism A->B to be a special type of complete intersection. This provides by the way a new and simpler proof of de Smit's conjecture. This is joint work with Srikanth Iyengar and Chandrashekhar Khare.