Why is $\mathbb{P}^{1} \times \mathbb{A}^{1}$ not isomorphic to an affine variety?
Product of projective and affine line is not affine
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algebraic-geometry
1 Answers
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Every closed subset of an affine variety is affine.
But here the closed subset $\mathbb P^1\times \lbrace 0 \rbrace \subset \mathbb{P}^{1} \times \mathbb{A}^{1}$ (why is it closed?) is isomorphic to $\mathbb P^1$ and thus not affine ( why?)
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0Dear @GeorgesElencwajg I am looking at Proposition 3.1.9 of Qing Liu which says $\text{Proj} (B\otimes_A C) \cong \text{Proj} B\times_{\text{Spec} A } \text{Spec} C$. Does this mean $\Bbb{P}^2 \cong \Bbb{P}^1 \times_k \Bbb{A}^1$? – 2013-10-21