Let $X$ be a topological space, $Z_1,Z_2 \subset X$ closed subsets and $\mathcal{F} \in Sh(X)$.
Then we have, for example by Hartshorne Excercise III 2.4, the Mayer Vietoris sequence for local cohomology \begin{equation*} \ldots \rightarrow H^i_{Z_1 \cap Z_2}(X,\mathcal{F}) \rightarrow H^i_{Z_1}(X,\mathcal{F}) \oplus H^i_{Z_2}(X,\mathcal{F}) \rightarrow H^i_{Z_1 \cup Z_2}(X,\mathcal{F}) \stackrel{\delta}{\rightarrow} H^{i+1}_{Z_1 \cap Z_2}(X,\mathcal{F})\rightarrow\ldots \end{equation*}
Additionally we have the long exact sequence
\begin{equation*} \ldots \rightarrow H^i_{Z_1 \cap Z_2}(X,\mathcal{F}) \rightarrow H^i_{Z_1 \cup Z_2 }(X,\mathcal{F}) \stackrel{f}{\rightarrow} H^i_{{(Z_1 \cup Z_2) \backslash (Z_1 \cap Z_2)}}(X,\mathcal{F}) \stackrel{g}{\rightarrow} H^{i+1}_{Z_1 \cap Z_2}(X,\mathcal{F})\rightarrow\ldots \end{equation*}
Does then by any chance $\delta=g\circ f=0 $ hold?
