Timeline for Sheaf cohomology and double covers
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Jul 6, 2013 at 15:03 | comment | added | Donu Arapura | I suppose what you really want to know is why is $H^p(X,\mathcal{O}_X)\cong H^p(Y,\mathcal{O}_Y)\oplus H^p(Y, L)$ for some line bundle $L$? Answer: because of (1), (2) and (3) $\pi_*\mathcal{O}_X = \mathcal{O}_Y\oplus L$, where $L$ is the (-1)-eigenbundle of the leftside under the Galois action. | |
| Jul 6, 2013 at 5:39 | review | Close votes | |||
| Jul 7, 2013 at 3:50 | |||||
| Jul 6, 2013 at 5:28 | comment | added | Charles Staats | Additional note: Since $\pi$ is affine, a cover of $Y$ by open affines pulls back to a cover of $X$ by open affines. Consequently, the Cech complex of $\pi_* \mathcal O_X$ with respect to a cover by open affines pulls back to a Cech complex of $\mathcal O_X$. This can be used for the first isomorphism. | |
| Jul 6, 2013 at 5:24 | history | edited | Charles Staats | CC BY-SA 3.0 |
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| Jul 6, 2013 at 5:10 | answer | added | Marci | timeline score: 1 | |
| Jul 6, 2013 at 5:08 | comment | added | Sam Gunningham | The first question follows from the exactness of $\pi_\ast$ (this would be true e.g. for any affine map $\pi$). The second is just because the functors $H^p(Y,-)$ are additive. This is very general. | |
| Jul 6, 2013 at 1:51 | history | asked | Tom Fellmann | CC BY-SA 3.0 |