The following statement appears in these notes I'm trying to read: $\mathcal{F}$ be a sheaf of $R-$modules on a topological space $X$. Corresponding to $\mathcal{F}$ we can construct a presheaf $H^j(\mathcal{F})$ of cohomology groups of $F$ by $U \mapsto H^j(U,\mathcal{F})$. Consider $x \in X$. Then,
If $H^j(U,\mathcal{F})$ vanishes for some neighborhood $U$ of $x$, then the stalk $H^j(\mathcal{F})_x$ vanishes.
It will be enough to show that the restriction maps $H^j(\mathcal{F})(U) \rightarrow H^j(\mathcal{F})(V)$ for open sets $U,V$ of $X, V \subset U$ i.e given any closed section in $\mathcal{I}^j(V)$ I need to lift it to a closed section in $\mathcal{I}^j(U)$, where $0\rightarrow\mathcal{F}\rightarrow\mathcal{I}^0\rightarrow\mathcal{I}^1...$ is an injective resolution of $\mathcal{F}$. There always exists a lift by flabiness of injective sheaves, but how do I ensure that the lift be closed ?
Any help would be greatly appreciated.