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The Stacks project

Example 26.9.3. Let k be a field. An example of a scheme which is not affine is given by the open subspace U = \mathop{\mathrm{Spec}}(k[x, y]) \setminus \{ (x, y)\} of the affine scheme X =\mathop{\mathrm{Spec}}(k[x, y]). It is covered by two affines, namely D(x) = \mathop{\mathrm{Spec}}(k[x, y, 1/x]) and D(y) = \mathop{\mathrm{Spec}}(k[x, y, 1/y]) whose intersection is D(xy) = \mathop{\mathrm{Spec}}(k[x, y, 1/xy]). By the sheaf property for \mathcal{O}_ U there is an exact sequence

0 \to \Gamma (U, \mathcal{O}_ U) \to k[x, y, 1/x] \times k[x, y, 1/y] \to k[x, y, 1/xy]

We conclude that the map k[x, y] \to \Gamma (U, \mathcal{O}_ U) (coming from the morphism U \to X) is an isomorphism. Therefore U cannot be affine since if it was then by Lemma 26.6.5 we would have U \cong X.


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