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Let $X$ be a topological space, $Z_1, Z_2$ two disjoint subspaces of $X$. Let $F$ be a sheaf of abelian groups on $X$. Is it true that for any $i \ge 0$,

$$\mathrm{Im}(H^i_{Z_1}(X,F) \to H^i(X,F)) \cap \mathrm{Im}(H^i_{Z_2}(X,F) \to H^i(X,F))=0?$$

If necessary, assume $X$ is a quasi-projective variety and $F$ a locally-free sheaf on $X$.

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    $\begingroup$ No. Take for $X$ a curve of genus $\geq 1$, $F=\mathcal{O}_X$, $Z_1$ and $Z_2$ two distinct points. Then $X\smallsetminus Z_i$ is affine, so $H^1_{Z_i}(X,\mathcal{O}_X)\rightarrow H^1(X,\mathcal{O}_X)$ is surjective for $i=1,2$. $\endgroup$ – abx Sep 29 '18 at 18:14
  • $\begingroup$ @abx: Thank you. Is there any known condition under which this could be possible (for eg. if Z_i's have higher codimension)? $\endgroup$ – Jana Sep 29 '18 at 18:31

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