example of quasi-affine variety that is not affine

Let $k$ be an algebraically closed field. Then the affine plane $\\mathbb{A}^{2}$ 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 $\\mathbb{A}^{2}$ . 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 $\\mathbb{A}^{2}$ is regular on $A$ . So let $f$ be regular on $A$ . Then it is a rational function on $\\mathbb{A}^{2}$ , and its poles (if any) have codimension one, which means they will have support on $A$ . Thus it must have no poles, and therefore it is regular on $\\mathbb{A}^{2}$ .

We know that the morphisms $A\\to\\mathbb{A}^{2}$ are in natural bijection with the morphisms from the coordinate ring of $\\mathbb{A}^{2}$ to the coordinate ring of $A$ ; so isomorphisms would have to correspond to automorphisms of $k[X,Y]$ , but this is just the set of invertible linear transformations of $X$ and $Y$ ; none of these yield an isomorphism $A\\to\\mathbb{A}^{2}$ .

Alternatively, one can use Čech cohomology to show that $H^{1}(A,\\mathcal{O}_{A})$ 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.

单词	ExampleOfQuasiaffineVarietyThatIsNotAffine
释义	example of quasi-affine variety that is not affine Let $k$ be an algebraically closed field. Then the affine plane $\\mathbb{A}^{2}$ 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 $\\mathbb{A}^{2}$ . 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 $\\mathbb{A}^{2}$ is regular on $A$ . So let $f$ be regular on $A$ . Then it is a rational function on $\\mathbb{A}^{2}$ , and its poles (if any) have codimension one, which means they will have support on $A$ . Thus it must have no poles, and therefore it is regular on $\\mathbb{A}^{2}$ . We know that the morphisms $A\\to\\mathbb{A}^{2}$ are in natural bijection with the morphisms from the coordinate ring of $\\mathbb{A}^{2}$ to the coordinate ring of $A$ ; so isomorphisms would have to correspond to automorphisms of $k[X,Y]$ , but this is just the set of invertible linear transformations of $X$ and $Y$ ; none of these yield an isomorphism $A\\to\\mathbb{A}^{2}$ . Alternatively, one can use Čech cohomology to show that $H^{1}(A,\\mathcal{O}_{A})$ 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.
随便看	WeakAIThesis weak AI thesis WeakApproximationTheorem weak approximation theorem WeakBisimulation weak bisimulation WeakConvergence weak convergence WeakConvergenceInNormedLinearSpace weak* convergence in normed linear space WeakDerivative weak derivative WeakDimensionOfAModule weak dimension of a module WeakerVersionOfStirlingsApproximation weaker version of Stirling’s approximation WeakestExtensionOfAPartialOrdering weakest extension of a partial ordering WeakGlobalDimension weak global dimension WeakHomotopyDoubleGroupoid weak homotopy double groupoid WeakHomotopyEquivalence weak homotopy equivalence WeakHopfAlgebra