My inquisitive search for a nice geometric interpretation for flat morphism between schemes $f:X \to Y$ lastly almost succeeds in following MO-thread presenting an interesting approach:
https://mathoverflow.net/questions/6789/why-are-flat-morphisms-flat
@Marc Nieper-Wißkirchen described the often mentioned interpretation that flat morphisms "vary parametrized families continuously" in affine case as follows:
As others have stated above, flatness of a family should mean that the fibres of the family vary somehow continuously. Let state this in terms of a module M over a ring R. Here a fibre of M over a prime P of R is M(P), the k(P)-vector space MP/PAP, where k(P) denotes the quotient field of R/P. If the fibres vary continuously, it should be possible to extend a basis of M(P) to nearby fibres, i.e. that the lift of a k(P)-basis wrt. the canonical map MP -> M(P) should yield a basis of MP over AP, i.e. that the stalk MP is a free module.
And in fact: If M is a finitely presented R-module it is flat if and only if M is locally free, i.e. that stalks are free.
(And that a notion may become less geometric when we turn to non finitely presented modules is something which one may expect anyway.)
One point irritates me: Why the fact that the fibers (in affine case the fibers are the $k(P)$ vector spaces $M(P) = M \otimes_A A_P/P A_P$) vary continuously "translates" to the statement that they can be lifted free $A_P$-modules $M_P$. Especially what is here the meaning of "continuous" and why it provides this "lifting"?
Remark: I know the criterion for Noetherian modules that locally freeness is equivalent to flatness, but the true reason of this question how to interpret to "continuous variation" of the fibers such that this "continuity" along nearby fibers allows to pull the generators to a free basis.
Does anybody have an idea how this meaning/ intuition of the author of the comment above for "continuous fiber variation" translates into freeness of the stalk?