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This question is motivated from my last question 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?

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  • $\begingroup$ Is B finitely generated over A? $\endgroup$ – Tyler Lawson Nov 17 '09 at 18:08
  • $\begingroup$ Initially we could assume this to be so. But for the results i mentioned they need not be (modules over local rings being free or over von Neumann regular rings being direct sums of principal ideals is a general case)..except the word "copies" may not be appropriate for infinitely generated case $\endgroup$ – Jose Capco Nov 17 '09 at 18:39
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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.

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  • $\begingroup$ Well, if we have Spec k as a scheme, we still have k by taking global sections of the structure sheaf, so we at least know what a K-algebra looks like, but "B is a k-algebra" doesn't give us much information about spec B. Agreeing with you, but clarifying for the questioner. $\endgroup$ – Harry Gindi Nov 18 '09 at 1:39
  • $\begingroup$ Yep, I agree too. Thanks. That pretty much wraps it! :) $\endgroup$ – Jose Capco Nov 18 '09 at 18:45

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