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Suppose $F$ is a coherent sheaf on a smooth (algebraic or complex) variety $X$. Then we can consider the cohomology groups $$H^p(X,F)$$ for all $i$. Now, let we consider the sheaf $$\mathcal{H}^p(X,F)$$ defined as the sheaf associated to the presheaf $$U\mapsto H^p(U,F|_U).$$ Of course $\mathcal{H}^0(X,F)=F$, so $H^p(X,\mathcal{H}^0(X,F))=H^{p+0}(X,F)$. Is true in general that $$H^p(X,\mathcal{H}^p(X,F))=H^{p+q}(X,F)?$$ I think that the answer is yes but I cannot prove it, so in the affermative case can you give me a skecth (or a suggestion) of proof?

Thank you!

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    $\begingroup$ Actually $\mathcal{H}^p(X,\mathcal{F})=0$ for $p\geq 1$, because $H^p(U,\mathcal{F}_{|U})=0$ for every affine open subset $U\subset X$. $\endgroup$
    – abx
    Commented Jul 25, 2014 at 15:34
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    $\begingroup$ This is probably the fanciest version of the "Law of universal linearity" that I've seen yet. $\endgroup$
    – Ryan Reich
    Commented Jul 25, 2014 at 15:42
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    $\begingroup$ More generally, even without coherence or schemes, for any abelian sheaf $F$ on any topological space $X$ the sheafification of $U \mapsto H^p(U,F|_U)$ vanishes for $p > 0$ literally from the derived-functor definition of sheaf cohomology. This vanishing is of course very important in the proof of "Cartan's Lemma" relating derived functor cohomology and Cech theory (which in turn underlies the higher-degree vanishing for quasi-coherent sheaves on affines which abx mentions). $\endgroup$
    – user27920
    Commented Jul 25, 2014 at 16:48
  • $\begingroup$ Even more generally still, this is true for arbitrary sites... $\endgroup$
    – Zhen Lin
    Commented Jul 25, 2014 at 23:41

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