I have to prove that $\Bbb{A}^1$ is not isomorphic to $\Bbb{A}^1-\{0\}$ . Apparently one does this by showing that the corresponding coordinate rings are not isomorphic, but I have $I(\Bbb{A}^1-\{0\})= I(\Bbb{A}^1)=\{0\}$, so I get that the coordinate rings are isomorphic...
2026-02-22 21:12:22.1771794742
The 1-affine space is not isomorphic to the 1-affine space minus one point
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What you can do is the following:
Show first that $\mathbb A^1-\{0\}\cong V:=\{(x,y)\in \mathbb A^2:xy=1\}$ where the isomorphism is given by $x\mapsto(x,\frac 1x)$ with inverse $(x,y)\mapsto x$. Now $V$ is an irreducible algebraic set with coordinate ring $k[x,y]/(xy-1)$ which is not isomorphic to $k[x]$ as k-algebra (look at the units!). Hence $\mathbb A^1$ is not isomorphic to $V$ and so not isomorphic to $\mathbb A^1-\{0\}$.