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The Stacks project

Lemma 63.4.3. Let f = j : U \to X be an étale of schemes. Denote j_{p!} the construction of Étale Cohomology, Equation (59.70.1.1) and denote f_{p!} the construction above. Functorially in \mathcal{F} \in \textit{Ab}(X_{\acute{e}tale}) there is a canonical map

j_{p!}\mathcal{F} \longrightarrow f_{p!}\mathcal{F}

of abelian presheaves which identifies the sheaf j_!\mathcal{F} = (j_{p!}\mathcal{F})^\# of Étale Cohomology, Definition 59.70.1 with (f_{p!}\mathcal{F})^\# .

Proof. Please read the proof of Étale Cohomology, Lemma 59.70.6 before reading the proof of this lemma. Let V be an object of X_{\acute{e}tale}. Recall that

j_{p!}\mathcal{F}(V) = \bigoplus \nolimits _{\varphi : V \to U} \mathcal{F}(V \xrightarrow {\varphi } U)

Given \varphi we obtain an open subscheme Z_\varphi \subset U_ V = U \times _ X V, namely, the image of the graph of \varphi . Via \varphi we obtain an isomorphism V \to Z_\varphi over U and we can think of an element

s_\varphi \in \mathcal{F}(V \xrightarrow {\varphi } U) = \mathcal{F}(Z_\varphi ) = H_{Z_\varphi }(\mathcal{F})

as a section of \mathcal{F} over Z_{\varphi }. Since Z_\varphi \subset U_ V is open, we actually have H_{Z_\varphi }(\mathcal{F}) = \mathcal{F}(Z_\varphi ) and we can think of s_\varphi as an element of H_{Z_\varphi }(\mathcal{F}). Having said this, our map j_{p!}\mathcal{F} \to f_{p!}\mathcal{F} is defined by the rule

\sum \nolimits _{i = 1, \ldots , n} s_{\varphi _ i} \longmapsto \sum \nolimits _{i = 1, \ldots , n} (Z_{\varphi _ i}, s_{\varphi _ i})

with right hand side a sum as in (63.4.0.1). We omit the verification that this is compatible with restriction mappings and functorial in \mathcal{F}.

To finish the proof, we claim that given a geometric point \overline{y} : \mathop{\mathrm{Spec}}(k) \to Y there is a commutative diagram

\xymatrix{ (j_{p!}\mathcal{F})_{\overline{y}} \ar[r] \ar[d] & \bigoplus _{j(\overline{x}) = \overline{y}} \mathcal{F}_{\overline{x}} \ar@{=}[d] \\ (f_{p!}\mathcal{F})_{\overline{y}} \ar[r] & \bigoplus _{f(\overline{x}) = \overline{y}} \mathcal{F}_{\overline{x}} }

where the top horizontal arrow is constructed in the proof of Étale Cohomology, Proposition 59.70.3, the bottom horizontal arrow is constructed in the proof of Lemma 63.4.2, the right vertical arrow is the obvious equality, and the left vertical arrow is the map defined in the previous paragraph on stalks. The claim follows in a straightforward manner from the explicit description of all of the arrows involved here and in the references given. Since the horizontal arrows are isomorphisms we conclude so is the left vertical arrow. Hence we find that our map induces an isomorphism on sheafifications by Étale Cohomology, Theorem 59.29.10. \square


Comments (1)

Comment #10126 by Winand Schaap on

Small typo, Let be an étale morphism of schemes.


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