Timeline for Lie algebra and base change
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 12, 2015 at 2:45 | vote | accept | Lisa S. | ||
| Oct 12, 2015 at 2:02 | answer | added | nfdc23 | timeline score: 10 | |
| Oct 12, 2015 at 1:53 | comment | added | Lisa S. | Thank you, t3suji, this is very helpful. Could you comment on what is the relationship between the two: is the $\mathbb{Z}_p$-module just the global sections of the scheme version? I am seeing in SGA 3, Expose II, 3.3 that the scheme version should maybe be $\mathbb{V}(e^*(\Omega^1_{\mu_p/\mathbb{Z}_p}))$, but I guess the $\mathbb{V}$ functor is parametrizing global sections of the dual of the coherent sheaf $e^*(\Omega^1_{\mu_p/\mathbb{Z}_p})$, so this must be the source of the bad behavior of global sections with respect to base change beyond the smooth case? | |
| Oct 12, 2015 at 1:04 | comment | added | t3suji | On the other hand, if you want your Lie algebra to be a scheme over $\mathbb Z_p$, it is supposed to commute with the base change, but there is no contradiction: the Lie algebra of $\mathbb{G}m$ is the constant scheme with one-dimensional fiber, and it contains a non-flat subscheme sitting over the closed point. | |
| Oct 12, 2015 at 1:02 | comment | added | t3suji | Well, what happens depends on the kind of object you want the Lie algebra to be. Most of your question is written for `naive' object, where you expect the Lie algebra to be a $\mathbb{Z}_p$-module. In this setup, the formation of Lie algebra does not commute with base change (unless the group is smooth). | |
| Oct 12, 2015 at 0:33 | history | asked | Lisa S. | CC BY-SA 3.0 |