Timeline for Gysin map and $B\mathbf{G}_m$, confusion
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 24, 2020 at 12:17 | comment | added | Pulcinella | It seems my confusion was just down to confusing which direction $u$ went in! Normally I would delete this question in such a case, but I think Dylan and Will's comments make it worth keeping, thank you. | |
| Sep 22, 2020 at 13:27 | comment | added | Dylan Wilson | is what builds this module. Said slightly differently: on homotopy groups the action of B looks trivial, but there is a nontrivial massey product like <u,u, ..., u, x>=y which relates the two generators in $k\oplus k[-2r+1]$. | |
| Sep 22, 2020 at 13:26 | comment | added | Dylan Wilson | (iii) maybe the confusion stems from the following line of thought: I want to define an algebra map B-->End(M) for some k-module, but in the example at hand it looks like nothing is in degree -1 (for large r), so how could there possibly be an interesting map here? The key is that we are in derived land! B is not free as an associative algebra: we have to choose a homotopy (or cycle) killing u^2, and then we have to choose something killing the resulting massey product <u,u,u>, etc. In the example at hand, it turns out that choosing an interesting Massey product at the 2rth-ish step | |
| Sep 22, 2020 at 13:24 | comment | added | Dylan Wilson | I think there's lots of nice points of view here. (i) this is a prototypical case of Koszul duality; you could write down an explicit Koszul complex implementing the equivalence between (finite dimensional) B-modules and (perfect) k[t]-module, and then apply it k[t]/t^{2r}; (ii) if you do this you'll basically see the following play out: t corresponds to the extension with B in the middle and k's on each side, so then t^{2r} corresponds to splicing together 2r-copies of this extension, and this gives Will's answer; (contd) | |
| Sep 22, 2020 at 13:17 | comment | added | Will Sawin | I think $du$ and $ud$ are both $0$ on the complex I wrote down, so they commute. This is because $d$ and $u$ go different ways because $d$ is a cohomological differential and $u$ is in degree $-1$. | |
| Sep 22, 2020 at 13:10 | comment | added | Pulcinella | @WillSawin Thanks for the reply. Doesn't $u$ have to commute with the differential? I admit to not having checked 7.2 of [DG12] in detail but I thought that was what was meant by $B$-algebra. | |
| Sep 22, 2020 at 11:56 | comment | added | Will Sawin | Can't it have generators in degrees $0$ through $-2r+1$, differentials from degree $-2r+2$ to $2r+3$, $-2r+4$ to $-2r+5, \dots, -2$ to $-1$, and $u$ sends $0$ to $-1$, $-2$ to $-3$, $\dots, -2r+2$ to $-2r+1$? | |
| Sep 22, 2020 at 11:33 | history | edited | Pulcinella | CC BY-SA 4.0 |
added 16 characters in body
|
| Sep 22, 2020 at 11:22 | history | asked | Pulcinella | CC BY-SA 4.0 |