The Stacks project

Proof. Recall that the stalk at $\overline{y}$ of a presheaf is defined by the usual colimit over étale neighbourhoods $(V, \overline{v})$ of $\overline{y}$, see Étale Cohomology, Definition 59.29.6. Accordingly suppose $s = \sum _{i = 1, \ldots , n} (Z_ i, s_ i)$ as in (63.4.0.1) is an element of $f_{p!}\mathcal{F}(V)$ where $(V, \overline{v})$ is an étale neighbourhood of $\overline{y}$. Then since

and since $s_ i$ is a section of $\mathcal{F}$ on an open neighbourhood of $Z_ i$ in $X_ V$ we can send $s$ to

We omit the verification that this is compatible with restriction maps and that the relations (1) $(Z, s) + (Z, s') - (Z, s + s')$ and (2) $(Z, s) - (Z', s)$ if $Z \subset Z'$ are sent to zero. Thus we obtain a map

Let us prove this arrow is surjective. For this it suffices to pick an $\overline{x}$ with $f(\overline{x}) = \overline{y}$ and prove that an element $s$ in the summand $\mathcal{F}_{\overline{x}}$ is in the image. Let $s$ correspond to the element $s \in \mathcal{F}(U)$ where $(U, \overline{u})$ is an étale neighbourhood of $\overline{x}$. Since $f$ is locally quasi-finite, the morphism $U \to Y$ is locally quasi-finite too. By More on Morphisms, Lemma 37.41.3 we can find an étale neighbourhood $(V, \overline{v})$ of $\overline{y}$, an open subscheme

and a geometric point $\overline{w}$ mapping to $\overline{u}$ and $\overline{v}$ such that $W \to V$ is finite and $\overline{w}$ is the only geometric point of $W$ mapping to $\overline{v}$. (We omit the translation between the language of geometric points we are currently using and the language of points and residue field extensions used in the statement of the lemma.) Observe that $W \to X_ V = X \times _ Y V$ is étale. Choose an affine open neighbourhood $W' \subset X_ V$ of the image $\overline{w}'$ of $\overline{w}$. Since $\overline{w}$ is the only point of $W$ over $\overline{v}$ and since $W \to V$ is closed, after replacing $V$ by an open neighbourhood of $\overline{v}$, we may assume $W \to X_ V$ maps into $W'$. Then $W \to W'$ is finite and étale and there is a unique geometric point $\overline{w}$ of $W$ lying over $\overline{w}'$. It follows that $W \to W'$ is an open immersion over an open neighbourhood of $\overline{w}'$ in $W'$, see Étale Morphisms, Lemma 41.14.2. Shrinking $V$ and $W'$ we may assume $W \to W'$ is an isomorphism. Thus $s$ may be viewed as a section $s'$ of $\mathcal{F}$ over the open subscheme $W' \subset X_ V$ which is finite over $V$. Hence by definition $(W', s')$ defines an element of $j_{p!}\mathcal{F}(V)$ which maps to $s$ as desired.

Let us prove the arrow is injective. To do this, let $s = \sum _{i = 1, \ldots , n} (Z_ i, s_ i)$ as in (63.4.0.1) be an element of $f_{p!}\mathcal{F}(V)$ where $(V, \overline{v})$ is an étale neighbourhood of $\overline{y}$. Assume $s$ maps to zero under the map constructed above. First, after replacing $(V, \overline{v})$ by an étale neighbourhood of itself, we may assume there exist decompositions $Z_ i = Z_{i, 1} \amalg \ldots \amalg Z_{i, m_ i}$ into open and closed subschemes such that each $Z_{i, j}$ has exactly one geometric point over $\overline{v}$. Say under the obvious direct sum decomposition

the element $s_ i$ corresponds to $\sum s_{i, j}$. We may use relations (1) and (2) to replace $s$ by $\sum _{i = 1, \ldots , n} \sum _{j = 1, \ldots , m_ i} (Z_{i, j}, s_{i, j})$. In other words, we may assume $Z_ i$ has a unique geometric point lying over $\overline{v}$. Let $\overline{x}_1, \ldots , \overline{x}_ m$ be the geometric points of $X$ over $\overline{y}$ corresponding to the geometric points of our $Z_ i$ over $\overline{v}$; note that for one $j \in \{ 1, \ldots , m\}$ there may be multiple indices $i$ for which $\overline{x}_ j$ corresponds to a point of $Z_ i$. By More on Morphisms, Lemma 37.41.3 applied to both $X_ V \to V$ after replacing $(V, \overline{v})$ by an étale neighbourhood of itself we may assume there exist open subschemes

and a geometric point $\overline{w}_ j$ of $W_ j$ mapping to $\overline{x}_ j$ and $\overline{v}$ such that $W_ j \to V$ is finite and $\overline{w}_ j$ is the only geometric point of $W_ j$ mapping to $\overline{v}$. After shrinking $V$ we may assume $Z_ i \subset W_ j$ for some $j$ and we have the map $H_{Z_ i}(\mathcal{F}) \to H_{W_ j}(\mathcal{F})$. Thus by the relation (2) we see that our element is equivalent to an element of the form

for some $t_ j \in H_{W_ j}(\mathcal{F})$. Clearly, this element is mapped simply to the class of $t_ j$ in the summand $\mathcal{F}_{\overline{x}_ j}$. Since $s$ maps to zero, we find that $t_ j$ maps to zero in $\mathcal{F}_{\overline{x}_ j}$. This implies that $t_ j$ restricts to zero on an open neighbourhood of $\overline{w}_ j$ in $W_ j$, see Étale Cohomology, Lemma 59.31.2. Shrinking $V$ once more we obtain $t_ j = 0$ for all $j$ as desired. $\square$