The Stacks project

Remark 62.4.6 (Covariance with respect to open embeddings). Let $f : X \to Y$ be locally quasi-finite morphism of schemes. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$. Let $X' \subset X$ be an open subscheme and denote $f' : X' \to Y$ the restriction of $f$. We claim there is a canonical map

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

Namely, this map will be the sheafification of a canonical map

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

constructed as follows. Let $V \in Y_{\acute{e}tale}$ and consider a section $s' = \sum _{i = 1, \ldots , n} (Z'_ i, s'_ i)$ as in (62.4.0.1) defining an element of $f'_{p!}(\mathcal{F}|_{X'})(V)$. Then $Z'_ i \subset X'_ V$ may also be viewed as a locally closed subscheme of $X_ V$ and we have $H_{Z'_ i}(\mathcal{F}|_{X'}) = H_{Z'_ i}(\mathcal{F})$. We will map $s'$ to the exact same sum $s = \sum _{i = 1, \ldots , n} (Z'_ i, s'_ i)$ but now viewed as an element of $f_{p!}\mathcal{F}(V)$. We omit the verification that this construction is compatible with restriction mappings and functorial in $\mathcal{F}$. This construction has the following properties:

  1. The maps $f'_{p!}\mathcal{F}' \to f_{p!}\mathcal{F}$ and $f'_!\mathcal{F}' \to f_!\mathcal{F}$ are compatible with the description of stalks given in Lemmas 62.4.2 and 62.4.5.

  2. If $f$ is separated, then the map $f'_{p!}\mathcal{F}' \to f_{p!}\mathcal{F}$ is the same as the map constructed in Remark 62.3.5 via the isomorphism in Lemma 62.4.1.

  3. If $X'' \subset X'$ is another open, then the composition of $f''_{p!}(\mathcal{F}|_{X''}) \to f'_{p!}(\mathcal{F}|_{X'}) \to f_{p!}\mathcal{F}$ is the map $f''_{p!}(\mathcal{F}|_{X''}) \to f_{p!}\mathcal{F}$ for the inclusion $X'' \subset X$. Sheafifying we conclude the same holds true for $f''_!(\mathcal{F}|_{X''}) \to f'_!(\mathcal{F}|_{X'}) \to f_!\mathcal{F}$.

  4. The map $f'_!\mathcal{F}' \to f_!\mathcal{F}$ is injective because we can check this on stalks.

All of these statements are easily proven by representing elements as finite sums as above and considering what happens to these elements.


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