The Stacks project

Remark 62.3.5 (Covariance with respect to open embeddings). Let $f : X \to Y$ be morphism of schemes which is separated and locally of finite type. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$. Let $X' \subset X$ be an open subscheme. Denote $f' : X' \to Y$ the restriction of $f$. There is a canonical injective map

\[ f'_!(\mathcal{F}|_{X'}) \longrightarrow f_!\mathcal{F} \]

Namely, let $V \in Y_{\acute{e}tale}$ and consider a section $s' \in f'_*(\mathcal{F}|_{X'})(V) = \mathcal{F}(X' \times _ Y V)$ with support $Z'$ proper over $V$. Then $Z'$ is closed in $X \times _ Y V$ as well, see Cohomology of Schemes, Lemma 30.26.5. Thus there is a unique section $s \in \mathcal{F}(X \times _ Y V) = f_*\mathcal{F}(V)$ whose restriction to $X' \times _ Y V$ is $s'$ and whose restriction to $X \times _ Y V \setminus Z'$ is zero, see Lemma 62.2.2. This construction is compatible with restriction maps and hence induces the desired map of sheaves $f'_!(\mathcal{F}|_{X'}) \to f_!\mathcal{F}$ which is clearly injective. By construction we obtain a commutative diagram

\[ \xymatrix{ f'_!(\mathcal{F}|_{X'}) \ar[r] \ar[d] & f_!\mathcal{F} \ar[d] \\ f'_*(\mathcal{F}|_{X'}) & f_*\mathcal{F} \ar[l] } \]

functorial in $\mathcal{F}$. It is clear that for $X'' \subset X'$ open with $f'' = f|_{X''} : X'' \to Y$ the composition of the canonical maps $f''_!\mathcal{F}|_{X''} \to f'_!\mathcal{F}|_{X'} \to f_!\mathcal{F}$ just constructed is the canonical map $f''_!\mathcal{F}|_{X''} \to f_!\mathcal{F}$.


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