Three-dimensional standard quadric/Not affine/Example
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Let be a field and consider the ring
The ideal is a prime ideal in of height one. Hence the open subset is the complement of an irreducible hypersurface. However, is not affine. For this we consider the closed subscheme
and . If were affine, then also the closed subscheme would be affine, but this is not true, since the complement of the punctured plane has codimension .