Timeline for What is integration along the fibers in D-module theory?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 8, 2017 at 13:13 | answer | added | Will Sawin | timeline score: 8 | |
| Oct 8, 2017 at 11:50 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Sep 8, 2017 at 11:35 | answer | added | Saal Hardali | timeline score: 1 | |
| Oct 12, 2016 at 5:15 | comment | added | user40276 | My knowledge about D-modules is very poor. I just tried to clarify what you said that was written on Wikipedia. For the case you mention you should try computing the cohomology with compact support of the fibers of your local system given by the connection corresponding to your diff. equation. Then the pushfoward will be integration of the derived functors of these solutions (the covariant global sections). I'm not sure if the de Rham cohomology of the flat bundle is the same as the singular cohomology of the local system, but if this is true, this map will be integration of these diff. forms. | |
| Oct 8, 2016 at 20:31 | comment | added | 54321user | For example, if I consider the projective $f:\mathbb{A}^2 \to \mathbb{A}^1$, and have the distribution giving a differential equation $\int (\partial^2_x + \partial^2_y)(-)dx\wedge dy = 0$, what would its pushforward mean? | |
| Oct 8, 2016 at 20:27 | comment | added | 54321user | @user40276 But what does this mean in terms of a differential equation? | |
| Oct 7, 2016 at 0:41 | comment | added | user40276 | So, at least for locally constant sheaves, this map is integration along the fibers. Now for more general sheaves or complexes of sheaves we just generalize this procedure, which a priori has nothing to do with integration. | |
| Oct 7, 2016 at 0:35 | comment | added | user40276 | At least for de Rham cohomology (with compact support in the vertical), it's integration along the fibers. For ordinary cohomology theories it's the inverse of the Thom isomorphism. For a sheaf recall that the stalks of $R^i f_{!} F$ are the cohomology with compact support, so when you have a locally constant sheaf the stalks are exactly ordinary cohomology with values in this local system $H_c^i (X_y, L)$ and by applying integration along the fiber (or composing with Poincaré duality and it's inverse), you get $H^{i - ({d_X - d_Y})} (Y, L) $. | |
| Oct 6, 2016 at 19:05 | history | edited | 54321user | CC BY-SA 3.0 |
added 153 characters in body
|
| Oct 6, 2016 at 18:33 | history | asked | 54321user | CC BY-SA 3.0 |