Talk:Projective bundle

Should the scheme-theoretic definition of a projective bundle be more specific about the topology being used?

It is unclear to me whether the definition is adopting Zariski topology or étale topology - namey, a projective bundle is locally a product with projective space over a Zariski open cover or an étale open cover. The statement "Over a regular scheme S such as a smooth variety, every projective bundle is of the form $\mathbb {P} (E)$ for some vector bundle (locally free sheaf) E", which is an exercise in Hartshorne's Algebraic Geometry book, is based on the Zariski topology version - the statement is not true if we use the étale topology version. In fact, the "regular" condition can be dropped - see Eisenbud and Harris' 3264 and All That book Proposition 9.4.

On the other hand, the statement "Every vector bundle over a variety X gives a projective bundle by taking the projective spaces of the fibers, but not all projective bundles arise in this way" is actually only valid under the étale topology version.— Preceding unsigned comment added by Jingtaisong (talk • contribs) 01:08, 22 April 2018 (UTC)[reply]

I actually had a vague awareness of this issue; the problem I had was I didn’t have a ref discussing the étale case. Do you know any ref? But in any case, I’m in a complete agreement that the article needs to be specific about topology. —- Taku (talk) 21:56, 22 April 2018 (UTC)[reply]

I also find the statement "In mathematics, a projective bundle is a fiber bundle whose fibers are projective spaces" problematic. This is true only in Zariski topology, and "fiber bundle" is more commonly used in analytic context where non-trivial Brauer-Saveri varities exist. 79.191.64.164 (talk) 15:01, 30 April 2024 (UTC)[reply]