Timeline for Is there something "Koszul dual" to formal groups?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 14, 2019 at 11:48 | comment | added | Adrien | Now in char 0, the Hopf algebra corresponding to a formal group $G$ under this identification is the enveloping algebra $U(g)$ of its Lie algebra, and indeed the BCH formula tells you the full dual of $U(g)$ is the algebra of functions on $G$. I guess in positive characteristic this Koszul duality picture should still hold, except probably that Dunn theorem fails (I'm not so sure), so that formal groups are Koszul dual of algebras in $E_\infty$-algebras ? I'm clearly out of my depth here. | |
| Dec 14, 2019 at 11:48 | comment | added | Adrien | Well, in general the category of formal schemes is opposite to that of profinite commutative algebras, which itself is opposite to that of cocommutative coalgebras. I think it's now customary to actually define a formal scheme as the "cospectrum" of a cocommutative coalgebra, and it's connected iff the coalgebra is conilpotent I think. Hence formal group schemes are group objects in there, ie conilpotent cocommutative Hopf algebras. This should work in any characteristic I think ? (continued..) | |
| Dec 13, 2019 at 17:15 | comment | added | Tim Campion | I'm not familiar with how to regard a formal group as a conilpotent cocommutative Hopf algebra. Where can I read about this? | |
| Dec 13, 2019 at 17:12 | comment | added | Tim Campion | Wow, I just noticed how helpful my phone's autocorrect was in that last comment! | |
| Dec 13, 2019 at 14:15 | comment | added | Adrien | Right, sorry, positive characteristic is way too scary for my taste. | |
| Dec 13, 2019 at 14:13 | comment | added | Tim Campion | Thanks! When you say that Lie algebras and football groups are equivalent, I assume you mean in characteristic zero, right? | |
| Dec 13, 2019 at 14:10 | history | answered | Adrien | CC BY-SA 4.0 |