Spectra of rings that are projective module over a subring - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T00:45:31Z http://mathoverflow.net/feeds/question/5841 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/5841/spectra-of-rings-that-are-projective-module-over-a-subring Spectra of rings that are projective module over a subring Jose Capco 2009-11-17T17:49:57Z 2009-11-17T19:23:11Z <p>This question is motivated from my <a href="http://mathoverflow.net/questions/4317/integrally-closed-factor-rings-and-projective-modules" rel="nofollow">last question</a> 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?</p> <p>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?</p> http://mathoverflow.net/questions/5841/spectra-of-rings-that-are-projective-module-over-a-subring/5858#5858 Answer by Mark Hovey for Spectra of rings that are projective module over a subring Mark Hovey 2009-11-17T19:23:11Z 2009-11-17T19:23:11Z <p>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.</p>