# Local cohomology with disjoint support

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 $$X$$.

• 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 $i=1,2$. – abx Sep 29 '18 at 18:14
• @abx: Thank you. Is there any known condition under which this could be possible (for eg. if Z_i's have higher codimension)? – Jana Sep 29 '18 at 18:31