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We generally consider vector bundles (locally free sheaves) in the Zariski site.

Is the category of vector bundles (locally free sheaves) different if instead of Zariski site we look at the flat site or etale site? In other words, is there an example of a vector bundle in flat site or etale site but no longer a vector bundle when restricted to Zariski site?

grok
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    https://math.stackexchange.com/questions/127089/principal-g-bundles-in-zariski-vs-etale-topology – Tsemo Aristide Jan 31 '18 at 02:36
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    No it is the same thing, this is essentially Hilbert 90 : vector bundles of rank $n$ for the topology $\tau$ are classified by $H^1_\tau(X,\operatorname{GL}n)$, but you have isomorphism $H^1{fppf}(X,\operatorname{GL}n)\simeq H^1{ét}(X,\operatorname{GL}n)\simeq H^1{Zar}(X,\operatorname{GL}_n)$. – Roland Jan 31 '18 at 10:25
  • @grok See https://stacks.math.columbia.edu/tag/03O6 for a discussion of descent – peter a g Jan 31 '18 at 14:20

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