Timeline for How do I describe the GL_n torsor attached to a smooth morphism of relative dimension n?
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
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| Oct 10, 2014 at 1:54 | vote | accept | S. Carnahan♦ | ||
| Oct 9, 2014 at 17:04 | answer | added | Martin Brandenburg | timeline score: 5 | |
| Oct 12, 2013 at 11:29 | comment | added | Martin Brandenburg | Meanwhile I think I can write down $A$ explicitly as some quotient of $\mathrm{Sym}(\mathcal{E}^n) \otimes \mathrm{Sym}((\mathcal{E}^*)^n)$. If you are interested ... | |
| Oct 12, 2013 at 10:12 | comment | added | S. Carnahan♦ | @MartinBrandenburg Mattia's answer was close enough for me to accept. Instead of $\mathcal{O}_Y$-module Isom (as you suggest), you can also replace Mattia's Isom of $Y$-schemes with Isom of $\mathbb{O}$-modules (as in SGA3 Exp 1) which encodes the vector structure. | |
| Oct 12, 2013 at 8:57 | comment | added | Martin Brandenburg | The two given answers are not correct and don't give a (global) description of A (the local one is clear anyway). | |
| Jun 9, 2010 at 20:26 | history | edited | S. Carnahan♦ | CC BY-SA 2.5 |
Remarks about my misconceptions
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| Jun 9, 2010 at 17:52 | comment | added | S. Carnahan♦ | Thank you, unknown(google). I had not seen the notation $Frame_Y(E)$ before, but it makes a lot of sense. I realized that there was some kind of difference after reading Michael Thaddeus's answer. Perhaps I should edit the question to make it clear where my misconceptions lay. | |
| Jun 9, 2010 at 16:04 | comment | added | Qfwfq | About 3, pay attention that for a vector bundle $E$ on $Y$, the principal bundle $Frame_Y(E)$ is not the same as the (non principal) $GL_n$-bundle $\operatorname{Aut}_Y(E)$. | |
| Jun 9, 2010 at 15:39 | vote | accept | S. Carnahan♦ | ||
| Oct 10, 2014 at 1:54 | |||||
| Jun 9, 2010 at 8:31 | answer | added | Mattia Talpo | timeline score: 8 | |
| Jun 8, 2010 at 23:16 | answer | added | Michael Thaddeus | timeline score: 6 | |
| Jun 8, 2010 at 23:03 | comment | added | David Treumann | $V$ is the space of linear functions on $V^*$, and Sym($V$) is the space of polynomial functions on $V^*$, so that Spec(Sym($V$)) is naturally identified with the dual vector space (or dual vector bundle) to $V$. This explains your tangent/cotangent concern. After replacing $\Omega_{Y/X}$ by its dual, the answer to your auxiliary question 1. is "yes," and to answer question 2. using this description note that the S-points of GL_n,Y are the S-automorphisms of O_S^{\oplus n}. | |
| Jun 8, 2010 at 22:12 | history | asked | S. Carnahan♦ | CC BY-SA 2.5 |