# Maps between spectral sequences

I am trying to understand a subtle point about how Theorem 2.2.5 is used in Kedlaya, Abbott, and Roe's "Bounding Picard numbers of surfaces using p-adic cohomology". Below I've tried to pose the question in more generality, with $\mathcal{F}$ representing the log-de Rham complex of $X$ along a divisor $Z$, and $\mathcal{G}$ representing a twist of $\mathcal{F}$.

Suppose $X$ is a topological space and that $\mathcal{F}^0 \subset \mathcal{F}^1 \subset \cdots \subset \mathcal{F}^n$ and $\mathcal{G}^0 \subset \mathcal{G}^1 \subset \cdots \subset \mathcal{G}^n$ are cochain complexes of sheaves on $X$. Then there are spectral sequences $E_r$,$\widetilde{E}_r$ abutting to the hypercohomology of $\mathcal{F}^*$ and $\mathcal{G}^*$, respectively, such that $E_2^{a,b} = \text{H}^b(X,\text{H}^a(\mathcal{F}^*))$ and $\widetilde{E}_2^{a,b} = \text{H}^b(X,\text{H}^a(\mathcal{G}^*))$.

Suppose $m$ is a positive integer and $\varphi : \mathcal{F}^* \rightarrow \mathcal{G}^*$ is a morphism inducing maps on homology $\varphi_a: \text{H}^a(\mathcal{F}^*) \rightarrow \text{H}^a(\mathcal{G}^*)$ such that the following hold

• $\varphi_0$ is an isomorphism.
• For each point $P \in X$ there is an open set $U$ containing $P$ with the following property: For each $y \in \text{H}^a(\mathcal{G}^*)(U)$ one can write $$y=\varphi_a(x)+z$$ for some $x \in \text{H}^a(\mathcal{F}^*)(U)$ and $z \in \text{H}^a(\mathcal{G}^*)(U)$ with $mz=0$.

My question: Can we conclude (by looking at maps on the $E_\infty$ terms of the spectral sequences) that the cokernel of the induced map on hypercohomology $$\mathbb{H}^{n}(X,\mathcal{F}^*) \rightarrow \mathbb{H}^{n}(X,\mathcal{G}^*)$$ is killed by $m^n$? Could someone present a proof?