Let $X$ be a scheme and $F$ a coherent sheaf on $X$. Does the etale cohomology of $F$ (i.e the cohomology of $F$ on the etale site of $X$) agree with the cohomology of $F$ on the Zariski site?
2025-01-13 02:59:05.1736737145
etale vs zariski cohomology for coherent sheaves
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They agree. In fact, you only need to assume quasicoherence; the key point is that the étale cohomology of a quasicoherent sheaf on an affine scheme vanishes in degrees $> 0$, just as for the Zariski topology.
For more details, see tag
03DWin the Stacks project.