I am learning local cohomology from Hartshorne’s book Local Cohomology. My question is about understanding a line in the proof of proposition 1.11 in this book. The set-up for proposition 1.11 is that a topological space $X$, a closed subset $Y$ of $X$ and an abelian sheaf $F$ are given, and that $V=X-Y$, $n$ is an integer. Proposition 1.11 states that conditions (1) and (2) are equivalent: (1) For all $i\leq n$, $\underline{H^i_Y}(F)=0$. (2) For all open subsets $U$ of $X$,the map $\alpha_i:H^i(U,F)\rightarrow H^i(U\cap V,F)$ is injective for $i=0$, and an isomorphism for all $i<n$. My question is about a line in the proof of (2) implies (1), which states that “But the sheaf associated to the presheaf $U\rightarrow H^n(U,F)$ is zero, since $n>0$.” I don’t see why this statement is true. Could someone please explain why this statement holds? Thank you so much for your kind help.
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1$\begingroup$ Your functors (sheafification of cohomology, in all degrees) form a universal delta functor from the category of sheaves on X to itself, and so is the system consisting of identity in degree zero and zero in positive degrees. More generally, for a map from X to Y, sheafifying cohomology of F on preimages of opens on Y computes the higher direct images. $\endgroup$– Piotr AchingerCommented Feb 2, 2023 at 16:49
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$\begingroup$ Thank you so much for your kind guidance. Your explanations, including the generalization to a map of topological spaces, are very insightful. I understand the reasoning now. $\endgroup$– BorisCommented Feb 2, 2023 at 16:58
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