Lemma 60.25.5. With notations and assumptions as in Lemma 60.25.4 the map

has kernel and cokernel annihilated by $q^{i + 1}$.

Lemma 60.25.5. With notations and assumptions as in Lemma 60.25.4 the map

\[ f^* : H^ i(\text{Cris}(X/S), \mathcal{F}) \longrightarrow H^ i(\text{Cris}(X'/S), \mathcal{F}') \]

has kernel and cokernel annihilated by $q^{i + 1}$.

**Proof.**
This follows from the fact that $E$ has nonzero cohomology sheaves in degrees $-1$ and up, so that the spectral sequence $H^ a(\mathcal{H}^ b(E)) \Rightarrow H^{a + b}(E)$ converges. This combined with the long exact cohomology sequence associated to a distinguished triangle gives the bound.
$\square$

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