This question is motivated from my [last question](https://mathoverflow.net/questions/4317/integrally-closed-factor-rings-and-projective-modules) here. I wonder if one has a ring A and an over-ring of this ring say B, and if we know that B is a projective A-module can we have a particular idea of how Spec B would look like if we know how Spec A would look like?


This question does make sense to me. Because for instance given a local ring A, then B's are some form of copies of A. If A were zero dimensional reduced and commutative then Spec B would look like copies of clopen subsets of Spec A (because projective modules over von Neumann regular rings are isomorphic to direct sum of principal ideals). So what else do we know?



Spectra of rings that are projective module over a subring
So, for example, A could be a field k, and B could be any k-algebra whatsoever. Basically you would be trying to recover all of classical algebraic geometry from Spec k. It does not seem likely.


