The Stacks project

Lemma 61.26.5. Let $X$ be a scheme. Let $Z \subset X$ be a closed subscheme and let $U \subset X$ be the complement. Denote $i : Z \to X$ and $j : U \to X$ the inclusion morphisms. Assume that $j$ is a quasi-compact morphism. For every abelian sheaf on $X_{pro\text{-}\acute{e}tale}$ there is a canonical short exact sequence

\[ 0 \to j_!j^{-1}\mathcal{F} \to \mathcal{F} \to i_*i^{-1}\mathcal{F} \to 0 \]

on $X_{pro\text{-}\acute{e}tale}$ where all the functors are for the pro-├ętale topology.

Proof. We obtain the maps by the adjointness properties of the functors involved. It suffices to show that $X_{pro\text{-}\acute{e}tale}$ has enough objects (Sites, Definition 7.40.2) on which the sequence evaluates to a short exact sequence. Let $V = \mathop{\mathrm{Spec}}(A)$ be an affine object of $X_{pro\text{-}\acute{e}tale}$ such that $A$ is w-contractible (there are enough objects of this type). Then $V \times _ X Z$ is cut out by an ideal $I \subset A$. The assumption that $j$ is quasi-compact implies there exist $f_1, \ldots , f_ r \in I$ such that $V(I) = V(f_1, \ldots , f_ r)$. We obtain a faithfully flat, ind-Zariski ring map

\[ A \longrightarrow A_{f_1} \times \ldots \times A_{f_ r} \times A_{V(I)}^\sim \]

with $A_{V(I)}^\sim $ as in Lemma 61.5.1. Since $V_ i = \mathop{\mathrm{Spec}}(A_{f_ i}) \to X$ factors through $U$ we have

\[ j_!j^{-1}\mathcal{F}(V_ i) = \mathcal{F}(V_ i) \quad \text{and}\quad i_*i^{-1}\mathcal{F}(V_ i) = 0 \]

On the other hand, for the scheme $V^\sim = \mathop{\mathrm{Spec}}(A_{V(I)}^\sim )$ we have

\[ j_!j^{-1}\mathcal{F}(V^\sim ) = 0 \quad \text{and}\quad \mathcal{F}(V^\sim ) = i_*i^{-1}\mathcal{F}(V^\sim ) \]

the first equality by Lemma 61.26.3 and the second by Lemmas 61.25.5 and 61.11.7. Thus the sequence evaluates to an exact sequence on $\mathop{\mathrm{Spec}}(A_{f_1} \times \ldots \times A_{f_ r} \times A_{V(I)}^\sim )$ and the lemma is proved. $\square$

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