I know someone already asked about flatness over non-reduced schemes flatness over non-reduced schemes, but I think my question is different.
I'm reading Bosch, Lütkebohmert and Raynaud's "Néron models", and in the second chapter, they swiftly discuss some background material of algebraic geometry, notably flatness in section 2.4.
They mention first on page 52 that when the base has nilpotents : "there exists no criterion to test flatness by geometric properties".
On page 53, again they tease : "It is impossible to characterize the flatness of an S-scheme X of finite type by geometric properties when the base S is not reduced." Then they go on pointing EGA IV(2) 6.9.1 for the reduced case.
What had they in mind exactly?
