Timeline for Is it always possible to write a scheme as a colimit of affine schemes?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 17, 2016 at 17:36 | comment | added | Marc Hoyois | @user40276 I don't see what could go wrong. | |
| Sep 17, 2016 at 17:10 | comment | added | user40276 | @MarcHoyois Just out of curiosity. Is it possible to use the same argument for derived schemes if the intersection of affines are not affines (that is using the Cech cover together with this $V_{ij}^k$)? | |
| Sep 17, 2016 at 16:52 | comment | added | Marc Hoyois | In fact, if you want a hypercover by affines, there is no need to refine your intersections after the first stage. Once you've chosen the $U_{i}$s and the $V^{ij}_k$s, you can form a 1-coskeletal hypercover and all the pieces will be affine, because the $U_i$s have affine diagonal. This is useful because coskeletal hypercovers are colimits in the ∞-category of Zariski sheaves, but arbitrary hypercovers need not be. | |
| Sep 17, 2016 at 3:49 | comment | added | Yosemite Sam | my bad, I got carried away | |
| Sep 17, 2016 at 2:49 | comment | added | Anette | You beat me to it Eric! :) (But thank you Yosemite Sam for pointing me towards hypercovers, it was something anyway that I needed to look at) | |
| Sep 17, 2016 at 2:25 | comment | added | Eric Wofsey | You don't actually need to look at triple intersections; the diagram with just the $V^{ij}_k$ and the $U_i$ already has colimit $X$. | |
| Sep 17, 2016 at 2:10 | history | answered | Yosemite Sam | CC BY-SA 3.0 |