About de Smit's question on flatness
Let \(\phi : (A, \mathfrak{m}_A) \to (B, \mathfrak{m}_B)\) denote a local homomorphism of local rings. A conjecture - attributed by the authors to Bart de Smit - says the following: Assume that \(A\) and \(B\) are of the same embedding dimension and both Artinian. Then any finitely generated \(B\)-module that is \(A\)-flat is \(B\)-flat. A consequence of the assumptions on \(\phi\) in de Smits conjecture is that \(\phi\) is a Gorenstein homomorphism. The conjecture is proved by the authors' in embedding dimension 1 or 2 and in arbitrary embedding dimension under some additional assumptions on the fibre ring \(B/\mathfrak{m}_A B.\) To be more precise: The authors call \(\phi\) a de Smit homomorphism if every \(B\)-module \(M\) that is flat and of finite type over \(A\) is flat over \(B.\) They introduce a technical condition called niceness for \(\phi\) and prove that a nice and Gorenstein homomorphism of Artinian local rings is de Smit. E.g. niceness is fulfilled in embedding dimension 1 and 2 and several other cases.
scientific article; zbMATH DE number 5859320