Lemma 63.3.11. Let $f : X \to Y$ be a morphism of schemes which is separated and locally of finite type. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$. Then there is a canonical isomorphism
for any geometric point $\overline{y} : \mathop{\mathrm{Spec}}(k) \to Y$.
Proof. Recall that $(f_*\mathcal{F})_{\overline{y}} = \mathop{\mathrm{colim}}\nolimits f_*\mathcal{F}(V)$ where the colimit is over the étale neighbourhoods $(V, \overline{v})$ of $\overline{y}$. If $s \in f_*\mathcal{F}(V) = \mathcal{F}(X_ V)$, then we can pullback $s$ to a section of $\mathcal{F}$ over $(X_ V)_{\overline{v}} = X_{\overline{y}}$. Thus we obtain a canonical map
We claim that this map induces a bijection between the subgroups $(f_!\mathcal{F})_{\overline{y}}$ and $H^0_ c(X_{\overline{y}}, \mathcal{F}|_{X_{\overline{y}}})$. The claim implies the lemma, but is a little bit more precise in that it describes the identification of the lemma as given by pullbacks of sections of $\mathcal{F}$ to the geometric fibre of $f$.
Observe that any element $s \in (f_!\mathcal{F})_{\overline{y}} \subset (f_*\mathcal{F})_{\overline{y}}$ is mapped by $c_{\overline{y}}$ to an element of $H^0_ c(X_{\overline{y}}, \mathcal{F}|_{X_{\overline{y}}}) \subset H^0(X_{\overline{y}}, \mathcal{F}|_{X_{\overline{y}}})$. This is true because taking the support of a section commutes with pullback and because properness is preserved by base change. This at least produces the map in the statement of the lemma. To prove that it is an isomorphism we may work Zariski locally on $Y$ and hence we may and do assume $Y$ is affine.
An observation that we will use below is that given an open subscheme $X' \subset X$ and if $f' = f|_{X'}$, then we obtain a commutative diagram
where the horizontal arrows are the maps constructed above and the vertical arrows are given in Remarks 63.3.5 and 63.3.9. The reason is that given an étale neighbourhood $(V, \overline{v})$ of $\overline{y}$ and a section $s \in f_*\mathcal{F}(V) = \mathcal{F}(X_ V)$ whose support $Z$ happens to be contained in $X'_ V$ and is proper over $V$, so that $s$ gives rise to an element of both $(f'_!(\mathcal{F}|_{X'}))_{\overline{y}}$ and $(f_!\mathcal{F})_{\overline{y}}$ which correspond via the vertical arrow of the diagram, then these elements are mapped via the horizontal arrows to the pullback $s|_{X_{\overline{y}}}$ of $s$ to $X_{\overline{y}}$ whose support $Z_{\overline{y}}$ is contained in $X'_{\overline{y}}$ and hence this restriction gives rise to a compatible pair of elements of $H^0_ c(X'_{\overline{y}}, \mathcal{F}|_{X'_{\overline{y}}})$ and $H^0_ c(X_{\overline{y}}, \mathcal{F}|_{X_{\overline{y}}})$.
Suppose $s \in (f_!\mathcal{F})_{\overline{y}}$ maps to zero in $H^0_ c(X_{\overline{y}}, \mathcal{F}|_{X_{\overline{y}}})$. Say $s$ corresponds to $s \in f_*\mathcal{F}(V) = \mathcal{F}(X_ V)$ with support $Z$ proper over $V$. We may assume that $V$ is affine and hence $Z$ is quasi-compact. Then we may choose a quasi-compact open $X' \subset X$ containing the image of $Z$. Then $Z$ is contained in $X'_ V$ and hence $s$ is the image of an element $s' \in f'_!(\mathcal{F}|_{X'})(V)$ where $f' = f|_{X'}$ as in the previous paragraph. Then $s'$ maps to zero in $H^0_ c(X'_{\overline{y}}, \mathcal{F}|_{X'_{\overline{y}}})$. Hence in order to prove injectivity, we may replace $X$ by $X'$, i.e., we may assume $X$ is quasi-compact. We will prove this case below.
Suppose that $t \in H^0_ c(X_{\overline{y}}, \mathcal{F}|_{X_{\overline{y}}})$. Then the support of $t$ is contained in a quasi-compact open subscheme $W \subset X_{\overline{y}}$. Hence we can find a quasi-compact open subscheme $X' \subset X$ such that $X'_{\overline{y}}$ contains $W$. Then it is clear that $t$ is contained in the image of the injective map $H^0_ c(X'_{\overline{y}}, \mathcal{F}|_{X'_{\overline{y}}}) \to H^0_ c(X_{\overline{y}}, \mathcal{F}|_{X_{\overline{y}}})$. Hence in order to show surjectivity, we may replace $X$ by $X'$, i.e., we may assume $X$ is quasi-compact. We will prove this case below.
In this last paragraph of the proof we prove the lemma in case $X$ is quasi-compact and $Y$ is affine. By More on Flatness, Theorem 38.33.8 there exists a compactification $j : X \to \overline{X}$ over $Y$. Set $\mathcal{G} = j_!\mathcal{F}$ so that $\mathcal{F} = \mathcal{G}|_ X$ by Étale Cohomology, Lemma 59.70.4. By the disussion above we get a commutative diagram
By Lemmas 63.3.6 and 63.3.10 the vertical maps are isomorphisms. This reduces us to the case of the proper morphism $\overline{X} \to Y$. For a proper morphism our map is an isomorphism by Lemmas 63.3.4 and 63.3.8 and proper base change for pushforwards, see Étale Cohomology, Lemma 59.91.4. $\square$
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