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6

The main difference between formally smooth/smooth and formally ├ętale/├ętale is that the formal versions don't need to be open maps. Chevalley's theorem on constructible topology says that a locally finite presentation map is an open map for the constructible topology, and if in addition you include flatness, you can show that it is an open map for the ...


3

As for your first question, concerning nonaffine R-varieties as you call them, yes, there are nonaffine R-varieties. However, they are considered pathological. Example 12.1.5 on page 301 of Bochnak-Coste-Roy, Real algebraic geometry, constructs an R-line bundle over $\mathbf R^2$ whose total space is not affine. In fact, it is not affine since it does not ...


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