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Let $\pi:X\rightarrow S$ be a smooth scheme over $S$ and let's assume for simplicity that $S=\mathrm{Spec} R$ is affine, Noetherian and regular. Let $\mathcal F$ be a coherent sheaf on $X$ and let's ...

I have been trying to understand the proof of the following theorem for the last month, I read some basics of sheaves theory and their cohomology, but still can't get the idea of this important ...