etale vs zariski cohomology for coherent sheaves

Question

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?

Answer 1

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 03DW in the Stacks project.