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.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).