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
9 events
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| Nov 29, 2013 at 14:46 | comment | added | Olivier Benoist | I also believe that the answer is not correct as it stands : one should not be able to reconstruct a vector bundle from its endomorphism bundle. | |
| Oct 12, 2013 at 11:28 | comment | added | Martin Brandenburg | At which section(s)? Locally it is a localization, but not globally. | |
| Oct 12, 2013 at 10:15 | comment | added | S. Carnahan♦ | @MartinBrandenburg I'm pretty sure it is a localization of $\mathcal{O}_Y$-algebras. | |
| Oct 12, 2013 at 8:25 | comment | added | Martin Brandenburg | But still this cannot work, since for $n=1$ say $\oplus_{d \in \mathbb{Z}} V^{\otimes d}$ is not a localization of $\oplus_{d \in \mathbb{N}} V^{\otimes d}$. | |
| Oct 11, 2013 at 16:34 | comment | added | Martin Brandenburg | This cannot be true. If $n=1$, then $R = \oplus_{n \in \mathbb{Z}} V^{\otimes n}$. However, your algebra is $\mathcal{O}[T,T^{-1}]$, which doesn't depend on $V$. I think instead of $\underline{\mathrm{End}}(V) = V^* \otimes V$, we have to consider $\underline{\mathrm{Hom}}(\mathcal{O}^n,V) \cong V^n$, as in Mattia's answer. | |
| Jun 9, 2010 at 15:36 | comment | added | S. Carnahan♦ | This clears up a lot. I think the jet bundles come from an associated bundle construction using automorphism groups of infinitesimal neighborhoods, so first order uses $GL_n$, but higher order uses nilpotent extensions by $1 + M_n[t]/t^k$. | |
| Jun 9, 2010 at 7:36 | comment | added | Michael Thaddeus | My understanding is that the 1-jets lie in an extension 0 --> T* --> J(1) --> O --> 0, which is split by the functions vanishing at the given point and the constant functions. So J(1) = T* + O. On the other hand, the extensions defining higher jets 0 --> Sym^k T* --> J(k) --> J(k+1) --> 0 need not split. Since representations of GL(n) are completely reducible, this means you won't obtain the jet bundles from any associated bundle construction starting from the tangent bundle. | |
| Jun 9, 2010 at 1:19 | comment | added | S. Carnahan♦ | Thank you, Michael. Now I think I have mixed up two constructions, and that the torsor P with the associated bundle property I wanted is not the same as the adjoint $GL_n$-bundle of automorphisms. Is that the case? | |
| Jun 8, 2010 at 23:16 | history | answered | Michael Thaddeus | CC BY-SA 2.5 |