example of quasi-affine variety that is not affine
The ring of regular functions of is the same as the ring of regular functions of . To see this, first observe that the two varieties are clearly birational, so they have the same function field. Clearly also any function regular on is regular on . So let be regular on . Then it is a rational function on , and its poles (if any) have codimension one, which means they will have support on . Thus it must have no poles, and therefore it is regular on .
We know that the morphisms are in natural bijection with the morphisms from the coordinate ring of to the coordinate ring of ; so isomorphisms would have to correspond to automorphisms of , but this is just the set of invertible linear transformations of and ; none of these yield an isomorphism .
For further information on this sort of subject, see Chapter I of Hartshorne’s (which lists this as exercise I.3.6). See the bibliography for algebraic geometry for this and other books.
|Title||example of quasi-affine variety that is not affine|
|Date of creation||2013-03-22 14:16:39|
|Last modified on||2013-03-22 14:16:39|
|Last modified by||Mathprof (13753)|