Timeline for Are $\overline{\mathbb{Q}}$-models of smooth affine curves over $\mathbb{C}$ with finitely many automorphisms unique?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Dec 17, 2017 at 0:22 | answer | added | Lilith | timeline score: 3 | |
| Oct 16, 2017 at 22:14 | comment | added | Will Chen | @nfdc23 Ah, okay I was being dumb. Sorry. | |
| Oct 16, 2017 at 21:48 | comment | added | nfdc23 | It completely solves your question in the affirmative in view of my first sentence. Say $A, A'$ are coordinate rings of smooth connected affine curves over $k$ and we're given isomorphisms $f:A\otimes_k K \simeq B$ and $f':A'\otimes_k K \simeq B$ for some $K$-algebra $B$. We want the $K$-algebra isomorphism $f^{-1} \circ f'$ to carry $A'$ to $A$. But abstractly there exists $i:A \simeq A'$ as $k$-algebras (by my first sentence), so harmlessly composing with $i_K$ reduces you to exactly what I do in my first comment. OK? | |
| Oct 16, 2017 at 17:45 | comment | added | Will Chen | @nfdc23 You seem to be proving that every automorphism of $B$ as a $\mathbb{C}$ algebra is the base change of an automorphism of $A$ as a $\overline{\mathbb{Q}}$-algebra. I agree with this statement, though I don't see how it relates to my question... | |
| Oct 14, 2017 at 13:09 | comment | added | nfdc23 | Uniqueness up to isomorphism follows from "spreading-out and specialization", so it remains to show if $K/k$ is an extension of algebraically closed fields, $X$ is smooth proper connected curve over $k$, and $U\subset X$ is a non-empty affine open with ${\rm{Aut}}_k(U)$ finite, then ${\rm{Aut}}_k(U)\to {\rm{Aut}}_K(U_K)$ is bijective. For $D=X-U\hookrightarrow X$ with reduced structure, you're studying the preimage $G$ of ${\rm{Aut}}_{D/k}$ under the map ${\rm{Aut}}_{X/k}\to {\rm{Hom}}(D,X)$ of locally finite type $k$-schemes. Since $G(k)$ is finite, so $G$ is $k$-finite, clearly $G(K)=G(k)$. | |
| Oct 14, 2017 at 6:20 | history | asked | Will Chen | CC BY-SA 3.0 |