Timeline for $\mathbb{G}_m$-torsors and line bundles
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 8, 2016 at 3:09 | vote | accept | stupid_question_bot | ||
| Nov 8, 2016 at 0:32 | comment | added | JJJ | I think you can show this via descent. It is discussed in slightly more general terms in this lecture youtube.com/watch?v=dY6-TenYUEo. | |
| Nov 7, 2016 at 8:52 | answer | added | HeinrichD | timeline score: 12 | |
| Nov 7, 2016 at 7:43 | comment | added | Niels | The first question is a general fact about diagonalizable groups and is extremely well documented. For the second point, being locally free of finite rank is even fpqc local, by descent theory, see mathoverflow.net/questions/155224/… | |
| Nov 7, 2016 at 7:26 | comment | added | Sasha | You can also define a line bundle geometrically as $G \times_{G_m} A^1$, where $G$ is a torsor. | |
| Nov 7, 2016 at 1:57 | comment | added | Theo Johnson-Freyd | Let $R_n$ be the kernel of the map $\mu - x^n \otimes$. Now use coassociativity to show that $R_n = R_1^{\otimes n}$. | |
| Nov 7, 2016 at 0:57 | comment | added | R. van Dobben de Bruyn | The fact that étale locally free modules are Zariski locally free is sometimes referred to as Hilbert's theorem 90 (after the analogous statement in Galois cohomology). See for example Tag 03P7 in the Stacks project. | |
| Nov 7, 2016 at 0:27 | history | asked | stupid_question_bot | CC BY-SA 3.0 |