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)


 * 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)
 * 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)