Lemma 59.82.1. Let $X$ be an affine scheme. Let $\mathcal{F}$ be a torsion abelian sheaf on $X_{\acute{e}tale}$. Let $Z \subset X$ be a closed subscheme. Let $\xi \in H^ q_{\acute{e}tale}(Z, \mathcal{F}|_ Z)$ for some $q > 0$. Then there exists an injective map $\mathcal{F} \to \mathcal{F}'$ of torsion abelian sheaves on $X_{\acute{e}tale}$ such that the image of $\xi $ in $H^ q_{\acute{e}tale}(Z, \mathcal{F}'|_ Z)$ is zero.
Proof. By Lemmas 59.73.2 and 59.51.4 we can find a map $\mathcal{G} \to \mathcal{F}$ with $\mathcal{G}$ a constructible abelian sheaf and $\xi $ coming from an element $\zeta $ of $H^ q_{\acute{e}tale}(Z, \mathcal{G}|_ Z)$. Suppose we can find an injective map $\mathcal{G} \to \mathcal{G}'$ of torsion abelian sheaves on $X_{\acute{e}tale}$ such that the image of $\zeta $ in $H^ q_{\acute{e}tale}(Z, \mathcal{G}'|_ Z)$ is zero. Then we can take $\mathcal{F}'$ to be the pushout
\[ \mathcal{F}' = \mathcal{G}' \amalg _{\mathcal{G}} \mathcal{F} \]
and we conclude the result of the lemma holds. (Observe that restriction to $Z$ is exact, so commutes with finite limits and colimits and moreover it commutes with arbitrary colimits as a left adjoint to pushforward.) Thus we may assume $\mathcal{F}$ is constructible.
Assume $\mathcal{F}$ is constructible. By Lemma 59.74.4 it suffices to prove the result when $\mathcal{F}$ is of the form $f_*\underline{M}$ where $M$ is a finite abelian group and $f : Y \to X$ is a finite morphism of finite presentation (such sheaves are still constructible by Lemma 59.73.9 but we won't need this). Since formation of $f_*$ commutes with any base change (Lemma 59.55.3) we see that the restriction of $f_*\underline{M}$ to $Z$ is equal to the pushforward of $\underline{M}$ via $Y \times _ X Z \to Z$. By the Leray spectral sequence (Proposition 59.54.2) and vanishing of higher direct images (Proposition 59.55.2), we find
\[ H^ q_{\acute{e}tale}(Z, f_*\underline{M}|_ Z) = H^ q_{\acute{e}tale}(Y \times _ X Z, \underline{M}). \]
By Lemma 59.80.9 we can find a finite surjective morphism $Y' \to Y$ of finite presentation such that $\xi $ maps to zero in $H^ q(Y' \times _ X Z, \underline{M})$. Denoting $f' : Y' \to X$ the composition $Y' \to Y \to X$ we claim the map
\[ f_*\underline{M} \longrightarrow f'_*\underline{M} \]
is injective which finishes the proof by what was said above. To see the desired injectivity we can look at stalks. Namely, if $\overline{x} : \mathop{\mathrm{Spec}}(k) \to X$ is a geometric point, then
\[ (f_*\underline{M})_{\overline{x}} = \bigoplus \nolimits _{f(\overline{y}) = \overline{x}} M \]
by Proposition 59.55.2 and similarly for the other sheaf. Since $Y' \to Y$ is surjective and finite we see that the induced map on geometric points lifting $\overline{x}$ is surjective too and we conclude. $\square$
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