Lemma 21.30.5. In Situation 21.30.1 if $(V_ n)$ holds, then for $X$ in $\mathcal{C}$ and $\mathcal{F}$ in $\mathcal{A}_ X$ the map $H^{n + 1}_{\tau '}(X, \epsilon _{X, *}\mathcal{F}) \to H^{n + 1}_\tau (X, \mathcal{F})$ is injective with image those classes which become trivial on a $\tau '$-covering of $X$.
Proof. Recall that $\epsilon _ X^{-1}\epsilon _{X, *}\mathcal{F} = \mathcal{F}$ hence the map is given by pulling back cohomology classes by $\epsilon _ X$. The Leray spectral sequence (Lemma 21.14.5)
\[ E_2^{p, q} = H^ p_{\tau '}(X, R^ q\epsilon _{X, *}\mathcal{F}) \Rightarrow H^{p + q}_\tau (X, \mathcal{F}) \]
combined with the assumed vanishing gives an exact sequence
\[ 0 \to H^{n + 1}_{\tau '}(X, \epsilon _{X, *}\mathcal{F}) \to H^{n + 1}_\tau (X, \mathcal{F}) \to H^0_{\tau '}(X, R^{n + 1}\epsilon _{X, *}\mathcal{F}) \]
This is a restatement of the lemma. $\square$
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