Timeline for Is the theory of vector bundles just linear algebra done in a suitable topos?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 4, 2017 at 15:25 | vote | accept | Qfwfq | ||
| May 4, 2017 at 12:12 | answer | added | Ingo Blechschmidt | timeline score: 18 | |
| May 3, 2017 at 12:31 | comment | added | Simon Henry | There is one important difficulty: in general vector bundle corresponds (internally) to locally free module over a local ring and not over a field (topological vector bundle are modules over the sheaf of continuous functions, which is only a local ring, the structure sheaf of a scheme or of the etale topos are locale ring and so one) So it can be seen as 'internal' linear algebra but not over a field, over a local ring. The work of Ingo Blechschmidt mentioned above is definitely the place to look for an extensive acount on this perspective in algebraic geometry. | |
| May 2, 2017 at 22:42 | comment | added | Denis Nardin | Every time you have a sheaf of rings you can talk about the category of locally free modules, which behaves kind of like vector bundles when your ring is an internal field in the topos (roughly: every locally nonzero section is invertible). For the topos associated to a (nice) topological space there is a canonical choice of internal field: the real numbers (credit to @მამუკა ჯიბლაძე), which gives the classical notion of vector bundle. Different choices of structure sheaf give you different notions (smooth/holomorphic v. bundles). I don't know enough to say more. | |
| May 2, 2017 at 22:38 | comment | added | Qfwfq | @DenisNardin: Yea, fair point! Maybe to get the internal equivalent of "regular" vector bundles needs to fix an internal ring $\mathcal O_X$ and talk about modules? Otherwise -if I understood your comment- one gets all the continuous vector bundles (or, rather, sheaves of vector spaces?) on the underlying topological space. | |
| May 2, 2017 at 22:31 | comment | added | Denis Nardin | @Qfwfq You'll have to be more precise though, since the vector bundles over a scheme do not depend only on the underlying topos (both if you mean the Zariski and the étale topos: think of different algebraically closed fields). | |
| May 2, 2017 at 21:54 | comment | added | Qfwfq | Just a comment: I have been intentionally vague about what I mean by "space", so the question is intended to depend on the geometry you're working in, be it topological spaces, complex manifolds, schemes, stacks,... | |
| May 2, 2017 at 20:27 | comment | added | მამუკა ჯიბლაძე | @DenisNardin The sheaf of continuous functions is the object of (Dedekind) real numbers, it does have internal construction | |
| May 2, 2017 at 20:23 | comment | added | Qiaochu Yuan | Vector bundles should correspond to dualizable internal vector spaces. | |
| May 2, 2017 at 20:22 | comment | added | Ben | This is exactly what happens in the sheaf topos on a scheme: if the scheme is reduced, the structure sheaf internally is a field (in an appropriate sense) and in any case, the internal finitely free modules are the finite locally free sheaves. I don't think the latter restricts to schemes—it should hold on any ringed space. Have a look at Ingo Blechschmidt's expository notes. | |
| May 2, 2017 at 19:40 | comment | added | Denis Nardin | If you can characterize the sheaf of continous functions then vector bundles are just locally free modules over it. | |
| May 2, 2017 at 19:31 | history | asked | Qfwfq | CC BY-SA 3.0 |