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
Comments (0)
There are also: