Timeline for Motives (and examples) of projective bundles over projective spaces
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Sep 3, 2021 at 7:13 | history | edited | Mikhail Bondarko | CC BY-SA 4.0 |
added 358 characters in body; edited title
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| Sep 2, 2021 at 23:53 | comment | added | Jason Starr | To continue the remark of @WillSawin over any field $k$, the pullback map on Brauer groups from $\text{Br}(k)$ to $\text{Br}(\mathbb{P}^n_k)$ is an isomorphism. So any Brauer-Severi scheme over $\mathbb{P}^n_k$ is associated to an Azumaya algebra that is the tensor product of the pullback of a division algebra from $\text{Spec}(k)$ and End of a locally free sheaf on $\mathbb{P}^n_k$. | |
| Sep 2, 2021 at 20:30 | comment | added | Will Sawin | What field is $X$ over? Over an algebraically closed field, I think the Brauer group of a projective space is trivial and thus every projective bundle arises from a vector bundle. For the second problem, it matters greatly which fiber you assume has a Tate motive. Is it the generic fiber, the geometric generic fiber, or all fibers defined over the base field? | |
| Sep 2, 2021 at 19:31 | history | asked | Mikhail Bondarko | CC BY-SA 4.0 |