Lemma 63.5.1. Let $f : X \to Y$ be a locally quasi-finite morphism of schemes. Let $\Lambda$ be a ring. Let $\mathcal{F}$ be a sheaf of $\Lambda$-modules on $X_{\acute{e}tale}$ and let $\mathcal{G}$ be a sheaf of $\Lambda$-modues on $Y_{\acute{e}tale}$. There is a canonical isomorphism

$can : f_!\mathcal{F} \otimes _\Lambda \mathcal{G} \longrightarrow f_!(\mathcal{F} \otimes _\Lambda f^{-1}\mathcal{G})$

of sheaves of $\Lambda$-modules on $Y_{\acute{e}tale}$.

Proof. Recall that $f_!\mathcal{F} = (f_{p!}\mathcal{F})^\#$ by Definition 63.4.4 where $f_{p!}\mathcal{F}$ is the presheaf constructed in Section 63.4. Thus in order to construct the arrow it suffices to construct a map

$f_{p!}\mathcal{F} \otimes _{p, \Lambda } \mathcal{G} \longrightarrow f_{p!}(\mathcal{F} \otimes _\Lambda f^{-1}\mathcal{G})$

of presheaves on $Y_{\acute{e}tale}$. Here the symbol $\otimes _{p, \Lambda }$ denotes the presheaf tensor product, see Modules on Sites, Section 18.26. Let $V$ be an object of $Y_{\acute{e}tale}$. Recall that

$f_{p!}\mathcal{F}(V) = \mathop{\mathrm{colim}}\nolimits _ Z H_ Z(\mathcal{F}) \quad \text{and}\quad f_{p!}(\mathcal{F} \otimes _\Lambda f^{-1}\mathcal{G})(V) = \mathop{\mathrm{colim}}\nolimits _ Z H_ Z(\mathcal{F} \otimes _\Lambda f^{-1}\mathcal{G})$

See Section 63.4. Our map will be defined on pure tensors by the rule

$(Z, s) \otimes t \longmapsto (Z, s \otimes f^{-1}t)$

(for notation see below) and extended by linearity to all of $(f_{p!}\mathcal{F} \otimes _{p, \Lambda } \mathcal{G})(V) = f_{p!}\mathcal{F}(V) \otimes _\Lambda \mathcal{G}(V)$. Here the notation used is as follows

1. $Z \subset X_ V$ is a locally closed subscheme finite over $V$,

2. $s \in H_ Z(\mathcal{F})$ which means that $s \in \mathcal{F}(U)$ with $\text{Supp}(s) \subset Z$ for some $U \subset X_ V$ open such that $Z \subset U$ is closed, and

3. $t \in \mathcal{G}(V)$ with image $f^{-1}t \in f^{-1}\mathcal{G}(U)$.

Since the support of $s \in \mathcal{F}(U)$ is contained in $Z$ it is clear that the support of $s \otimes f^{-1}t$ is contained in $Z$ as well. Thus considering the pair $(Z, s \otimes f^{-1}t)$ makes sense. It is immediate that the construction commutes with the transition maps in the colimit $\mathop{\mathrm{colim}}\nolimits _ Z H_ Z(\mathcal{F})$ and that it is compatible with restriction mappings. Finally, it is equally clear that the construction is compatible with the identifications of stalks of $f_!$ in Lemma 63.4.5. In other words, the map $can$ we've produced on stalks at a geometric point $\overline{y}$ fits into a commutative diagram

$\xymatrix{ (f_!\mathcal{F} \otimes _\Lambda \mathcal{G})_{\overline{y}} \ar[r]_-{can_{\overline{y}}} \ar[d] & f_!(\mathcal{F} \otimes _\Lambda f^{-1}\mathcal{G})_{\overline{y}} \ar[d] \\ (\bigoplus \mathcal{F}_{\overline{x}}) \otimes _\Lambda \mathcal{G}_{\overline{y}} \ar[r] & \bigoplus (\mathcal{F}_{\overline{x}} \otimes _\Lambda \mathcal{G}_{\overline{y}}) }$

where the direct sums are over the geometric points $\overline{x}$ lying over $\overline{y}$, where the vertical arrows are the identifications of Lemma 63.4.5, and where the lower horizontal arrow is the obvious isomorphism. We conclude that $can$ is an isomorphism as desired. $\square$

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