example of quasi-affine variety that is not affine

Let k be an algebraically closed field. Then the affine planeMathworldPlanetmath 𝔸2 is certainly affine. If we remove the point (0,0), then we obtain a quasi-affine variety A.

The ring of regular functions of A is the same as the ring of regular functions of 𝔸2. To see this, first observe that the two varieties are clearly birational, so they have the same function field. Clearly also any function regularPlanetmathPlanetmath on 𝔸2 is regular on A. So let f be regular on A. Then it is a rational function on 𝔸2, and its poles (if any) have codimension one, which means they will have supportPlanetmathPlanetmath on A. Thus it must have no poles, and therefore it is regular on 𝔸2.

We know that the morphismsMathworldPlanetmath A𝔸2 are in natural bijection with the morphisms from the coordinate ring of 𝔸2 to the coordinate ring of A; so isomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmath would have to correspond to automorphismsPlanetmathPlanetmathPlanetmath of k[X,Y], but this is just the set of invertible linear transformations of X and Y; none of these yield an isomorphism A𝔸2.

Alternatively, one can use Čech cohomology to show that H1(A,𝒪A) is nonzero (in fact, it is infinite-dimensional), while every affine variety has zero higher cohomology groups.

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
Canonical name ExampleOfQuasiaffineVarietyThatIsNotAffine
Date of creation 2013-03-22 14:16:39
Last modified on 2013-03-22 14:16:39
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 7
Author Mathprof (13753)
Entry type Example
Classification msc 14-00