The Stacks project

Example 26.21.17. Consider the nonaffine scheme $U = \mathop{\mathrm{Spec}}(k[x, y]) \setminus \{ (x, y)\} $ of Example 26.9.3. On the other hand, consider the scheme

\[ \mathbf{GL}_{2, k} = \mathop{\mathrm{Spec}}(k[a, b, c, d, 1/ad - bc]). \]

There is a morphism $\mathbf{GL}_{2, k} \to U$ corresponding to the ring map $x \mapsto a$, $y \mapsto b$. It is easy to see that this is a surjective morphism, and hence the image is not contained in any affine open of $U$. In fact, the affine scheme $\mathbf{GL}_{2, k}$ also surjects onto $\mathbf{P}^1_ k$, and $\mathbf{P}^1_ k$ does not even have an immersion into any affine scheme.

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