Timeline for Cohomology ring of $\mathbb{P}(\mathcal{O}(-1)\oplus \mathcal{O})$
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Dec 26, 2023 at 11:32 | review | Close votes | |||
| Dec 31, 2023 at 3:04 | |||||
| Dec 26, 2023 at 11:06 | history | edited | asv | CC BY-SA 4.0 |
added 50 characters in body
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| Dec 26, 2023 at 11:05 | comment | added | asv | Yes, this is the trivial rank one bundle. Thank you. | |
| Dec 26, 2023 at 10:57 | comment | added | mme | Here $\mathcal O$ is the trivial rank 1 bundle? So your sum has Chern class 1-x. The projective bundle formula (en.wikipedia.org/wiki/…) gives you that your total space has cohomology ring $\Bbb Z[x, \zeta]/(\zeta^2-x\zeta)$. If you instead meant the projectivization of $\mathcal O(-1) \oplus \mathcal O(1)$, the formula instead gives $\Bbb Z[x, \zeta]/(\zeta^2-x^2)$. In any case, the Gysin sequence implies that $H^*(E)$ is a free module of rank 2 over $\Bbb Z[x]$. | |
| Dec 26, 2023 at 10:37 | history | asked | asv | CC BY-SA 4.0 |