Timeline for Rigidity, moduli space, and moduli field
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jul 26, 2013 at 8:57 | vote | accept | Colin McLarty | ||
| Jul 23, 2013 at 22:51 | comment | added | user36938 | @Colin: I don't wish to get into a discussion about this on "chat". I recommend that you email JSE and discuss it with him further. | |
| Jul 23, 2013 at 12:43 | comment | added | Colin McLarty | let us continue this discussion in chat | |
| Jul 23, 2013 at 9:10 | answer | added | Colin McLarty | timeline score: 0 | |
| Jul 23, 2013 at 0:10 | comment | added | user36938 | @Colin: Yes, the direct limit games to bring stuff over an algebraic closure down to some finite-degree extension are used all the time, much as results over a local ring $A_{\mathfrak{p}}$ at some prime tend to "spread out" to the coordinate ring $A[1/a]$ of a Zariski-open neighborhood of $\mathfrak{p}$ (for some $a \not\in \mathfrak{p}$). And properties attained over $\mathbf{C}$ descend to a finitely generated subfield (or even to a subalgebra finitely generated over $\mathbf{Q}$, or even over $\mathbf{Z}$). | |
| Jul 22, 2013 at 23:17 | comment | added | Colin McLarty | One major point I had not noticed until these comments is that (assuming each object is defined by finitely many equations) the issue of finiteness of field extensions is trivial. The issue is between the two algebraically closed fields $\mathbb{C}$ and $\overline{\mathbb{Q}}$. | |
| Jul 22, 2013 at 14:05 | comment | added | user36938 | It should be noted that for field-valued points one doesn't see non-reducedness, so yet a 3rd viewpoint is that if the finite type moduli scheme $M$ over $F$ is 0-dimensional then its $F'$-points are $F$-points since $M_{\rm{red}}$ is a finite disjoint union of copies of Spec($F$). But to make use of this one needs to have really defined a moduli scheme and proved it is 0-dimensional (perhaps not via dual-number considerations). For CM abelian varieties one can make such an argument, though its real content can be expressed without any mention of moduli schemes. | |
| Jul 22, 2013 at 11:49 | comment | added | Colin McLarty | @user36938 Thanks. I prefer the first, so far. | |
| Jul 22, 2013 at 11:15 | comment | added | user36938 | Another principle you (& JSE?) may prefer: if $F'/F$ is an extension of algebraically closed fields, $X$ is a "finite type algebro-geometric structure" over $F'$ with no nontrivial automorphisms, and $g^*(X)\simeq X$ for all $g\in G={\rm{Aut}}(F'/F)$ then $X$ descends to $F$. This is true if $X$ arises from an $F'$-point of a separated Artin stack $M$ of finite type over $F$. Indeed, it is a theorem that the locus of "rigid" geometric points of such an $M$ is Zariski-open and an algebraic space, and if $U$ is an algebraic space of finite type over $F$ then $U(F')^G=U(F)$ (via work of Knutson). | |
| Jul 22, 2013 at 10:39 | comment | added | user36938 | My first observation contains much of the content of his remark (it focuses real work into defining the appropriate moduli scheme). To prove it, triviality of dual-number deformations says (by the meaning of "moduli scheme" $M$) that the tangent space to $M$ at any $F$-point vanishes, so the local ring at such a point has vanishing maximal ideal (by Nakayama's Lemma), so the local ring is $F$. This gives etaleness of $M$ at such points, and since $F$ is algebraically closed it follows that $M$ is a disjoint union of copies of Spec($F$). Hence, for $F'/F$ we get $M(F)=M(F')$. QED | |
| Jul 22, 2013 at 10:05 | comment | added | Colin McLarty | @user36938 Of course what I wrote (like what JSE wrote) is not precise. But does your first observation capture all the scope of his remark? And is it somehow trivial from the definition of dual number deformation? I had guessed it was rather a Galois theoretic point. Roughly how is it proved? | |
| Jul 22, 2013 at 4:04 | comment | added | user36938 | JSE's statement means just that for a locally finite type moduli scheme over an algebraically closed field $F$, if there are no nontrivial dual-number deformations at $F$-points then for any $F'/F$ all $F'$-structures arise over $F$. But in the absence of precise hypotheses one can be led to nonsense. Consider an elliptic curve $E$ over $\mathbf{C}$ with $j(E)$ not algebraic. Pointed connected finite etale covers of $E$ are "rigid" but not defined over a number field as abstract curves. This is unsurprising because one hasn't even defined a moduli scheme over $\overline{\mathbf{Q}}$! | |
| Jul 21, 2013 at 23:25 | history | edited | Colin McLarty | CC BY-SA 3.0 |
Replaced question number by link
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| Jul 21, 2013 at 23:12 | comment | added | Asaf Karagila♦ | Also, following your question number, it shows that the number is actually the answer's id. The question's number is 137108. The comment can be found in this direct link. (Generally, the timestamp of a comment is a direct link to the comment.) | |
| Jul 21, 2013 at 23:09 | comment | added | Asaf Karagila♦ | Perhaps include a link alongside the question number? In the new software you can even get the link for particular comments if you'd like. | |
| Jul 21, 2013 at 22:41 | history | asked | Colin McLarty | CC BY-SA 3.0 |