hcju/stacksqa huggingface dataset:
The result holds for $q = 0$ by Lemma \ref{lemma-h0-henselian-pair}. Let $q \geq 1$. Suppose the result has been shown in all degrees $< q$. Let $\mathcal{F}$ be a torsion abelian sheaf. Let $\mathcal{F} \to \mathcal{F}'$ be an injective map of torsion abelian sheaves (to be chosen later) with cokernel $\mathcal{Q}$ so that we have the short exact sequence $$ 0 \to \mathcal{F} \to \mathcal{F}' \to \mathcal{Q} \to 0 $$ of torsion abelian sheaves on $X_\etale$. This gives a map of long exact cohomology sequences over $X$ and $Z$ part of which looks like $$ \xymatrix{ H^{q - 1}_\etale(X, \mathcal{F}') \ar[d] \ar[r] & H^{q - 1}_\etale(X, \mathcal{Q}) \ar[d] \ar[r] & H^q_\etale(X, \mathcal{F}) \ar[d] \ar[r] & H^q_\etale(X, \mathcal{F}') \ar[d] \\ H^{q - 1}_\etale(Z, \mathcal{F}'|_Z) \ar[r] & H^{q - 1}_\etale(Z, \mathcal{Q}|_Z) \ar[r] & H^q_\etale(Z, \mathcal{F}|_Z) \ar[r] & H^q_\etale(Z, \mathcal{F}'|_Z) } $$ Using this commutative diagram of abelian groups with exact rows we will finish the proof. \medskip\noindent Injectivity for $\mathcal{F}$. Let $\xi$ be a nonzero element of $H^q_\etale(X, \mathcal{F})$. By Lemma \ref{lemma-efface-cohomology-on-closed-by-finite-cover} applied with $Z = X$ (!) we can find $\mathcal{F} \subset \mathcal{F}'$ such that $\xi$ maps to zero to the right. Then $\xi$ is the image of an element of $H^{q - 1}_\etale(X, \mathcal{Q})$ and bijectivity for $q - 1$ implies $\xi$ does not map to zero in $H^q_\etale(Z, \mathcal{F}|_Z)$. \medskip\noindent Surjectivity for $\mathcal{F}$. Let $\xi$ be an element of $H^q_\etale(Z, \mathcal{F}|_Z)$. By Lemma \ref{lemma-efface-cohomology-on-closed-by-finite-cover} applied with $Z = Z$ we can find $\mathcal{F} \subset \mathcal{F}'$ such that $\xi$ maps to zero to the right. Then $\xi$ is the image of an element of $H^{q - 1}_\etale(Z, \mathcal{Q}|_Z)$ and bijectivity for $q - 1$ implies $\xi$ is in the image of the vertical map.
The theorem says that in Situation \ref{situation-what-to-prove} statements (\ref{item-vanishing}) -- (\ref{item-surjective}) hold. Our first step is to replace $X$ by its reduction, which is permissible by Proposition \ref{proposition-topological-invariance}. By Lemma \ref{lemma-torsion-colimit-constructible} we can write $\mathcal{F}$ as a filtered colimit of constructible abelian sheaves. Taking cohomology commutes with colimits, see Lemma \ref{lemma-colimit}. Moreover, pullback via $X_{k'} \to X$ commutes with colimits as a left adjoint. Thus it suffices to prove the statements for a constructible sheaf. \medskip\noindent In this paragraph we use Lemma \ref{lemma-ses-statements} without further mention. Writing $\mathcal{F} = \mathcal{F}_1 \oplus \ldots \oplus \mathcal{F}_r$ where $\mathcal{F}_i$ is $\ell_i$-primary for some prime $\ell_i$, we may assume that $\ell^n$ kills $\mathcal{F}$ for some prime $\ell$. Now consider the exact sequence $$ 0 \to \mathcal{F}[\ell] \to \mathcal{F} \to \mathcal{F}/\mathcal{F}[\ell] \to 0. $$ Thus we see that it suffices to assume that $\mathcal{F}$ is $\ell$-torsion. This means that $\mathcal{F}$ is a constructible sheaf of $\mathbf{F}_\ell$-vector spaces for some prime number $\ell$. \medskip\noindent By definition this means there is a dense open $U \subset X$ such that $\mathcal{F}|_U$ is finite locally constant sheaf of $\mathbf{F}_\ell$-vector spaces. Since $\dim(X) \leq 1$ we may assume, after shrinking $U$, that $U = U_1 \amalg \ldots \amalg U_n$ is a disjoint union of irreducible schemes (just remove the closed points which lie in the intersections of $\geq 2$ components of $U$). Consider the short exact sequence $$ 0 \to j_!j^{-1}\mathcal{F} \to \mathcal{F} \to \bigoplus\nolimits_{x \in Z} i_{x *}M_x \to 0 $$ where $Z = X \setminus U$ and $M_x$ is a finite dimensional $\mathbf{F}_\ell$ vector space, see Lemma \ref{lemma-ses-associated-to-open}. Since the \'etale cohomology of $i_{x *}M_x$ vanishes in degrees $\geq 1$ and is equal to $M_x$ in degree $0$ it suffices to prove the theorem for $j_!j^{-1}\mathcal{F}$ (argue exactly as in the proof of Lemma \ref{lemma-even-easier}). Thus we reduce to the case $\mathcal{F} = j_!\mathcal{G}$ where $\mathcal{G}$ is a finite locally constant sheaf of $\mathbf{F}_\ell$-vector spaces on $U$. \medskip\noindent Since we chose $U = U_1 \amalg \ldots \amalg U_n$ with $U_i$ irreducible we have $$ j_!\mathcal{G} = j_{1!}(\mathcal{G}|_{U_1}) \oplus \ldots \oplus j_{n!}(\mathcal{G}|_{U_n}) $$ where $j_i : U_i \to X$ is the inclusion morphism. The case of $j_{i!}(\mathcal{G}|_{U_i})$ is handled in Lemma \ref{lemma-vanishing-easier}.
If $X$ is separated, this follows immediately from the more precise Theorem \ref{theorem-vanishing-affine-curves}. If $X$ is nonseparated, choose an affine open covering $X = X_1 \cup \ldots \cup X_n$. By induction on $n$ we may assume the vanishing holds over $U = X_1 \cup \ldots \cup X_{n - 1}$. Then Mayer-Vietoris (Lemma \ref{lemma-mayer-vietoris}) gives $$ H^2_\etale(U, \mathcal{F}) \oplus H^2_\etale(X_n, \mathcal{F}) \to H^2_\etale(U \cap X_n, \mathcal{F}) \to H^3_\etale(X, \mathcal{F}) \to 0 $$ However, since $U \cap X_n$ is an open of an affine scheme and hence affine by our dimension assumption, the group $H^2_\etale(U \cap X_n, \mathcal{F})$ vanishes by Theorem \ref{theorem-vanishing-affine-curves}.
This follows at once from the definition of a short exact sequence of presheaves. Namely, as the category of abelian presheaves is the category of functors on some category with values in $\textit{Ab}$, it is automatically an abelian category: a sequence $\mathcal{F}_1\to \mathcal{F}_2\to \mathcal{F}_3$ is exact in $\textit{PAb}$ if and only if for all $U \in \Ob(\mathcal{C})$, the sequence $\mathcal{F}_1(U) \to \mathcal{F}_2(U) \to \mathcal{F}_3(U)$ is exact in $\textit{Ab}$. So the complex above is merely a product of short exact sequences in each degree. See also Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cech-exact-presheaves}.
For any $V\in \Ob(\mathcal{C})$ the complex of abelian groups $\mathbf{Z}_\mathcal{U}^\bullet(V)$ is $$ \begin{matrix} \mathbf{Z}\left[ \coprod_{i_0\in I} \Mor_\mathcal{C}(V, U_{i_0})\right] \leftarrow \mathbf{Z}\left[ \coprod_{i_0, i_1 \in I} \Mor_\mathcal{C}(V, U_{i_0} \times_U U_{i_1})\right] \leftarrow \ldots = \\ \bigoplus_{\varphi : V \to U} \left( \mathbf{Z}\left[\coprod_{i_0 \in I} \Mor_\varphi(V, U_{i_0})\right] \leftarrow \mathbf{Z}\left[\coprod_{i_0, i_1\in I} \Mor_\varphi(V, U_{i_0}) \times \Mor_\varphi(V, U_{i_1})\right] \leftarrow \ldots \right) \end{matrix} $$ where $$ \Mor_{\varphi}(V, U_i) = \{ V \to U_i \text{ such that } V \to U_i \to U \text{ equals } \varphi \}. $$ Set $S_\varphi = \coprod_{i\in I} \Mor_\varphi(V, U_i)$, so that $$ \mathbf{Z}_\mathcal{U}^\bullet(V) = \bigoplus_{\varphi : V \to U} \left( \mathbf{Z}[S_\varphi] \leftarrow \mathbf{Z}[S_\varphi \times S_\varphi] \leftarrow \mathbf{Z}[S_\varphi \times S_\varphi \times S_\varphi] \leftarrow \ldots \right). $$ Thus it suffices to show that for each $S = S_\varphi$, the complex \begin{align*} \mathbf{Z}[S] \leftarrow \mathbf{Z}[S \times S] \leftarrow \mathbf{Z}[S \times S \times S] \leftarrow \ldots \end{align*} is exact in negative degrees. To see this, we can give an explicit homotopy. Fix $s\in S$ and define $K: n_{(s_0, \ldots, s_p)} \mapsto n_{(s, s_0, \ldots, s_p)}.$ One easily checks that $K$ is a nullhomotopy for the operator $$ \delta : \eta_{(s_0, \ldots, s_p)} \mapsto \sum\nolimits_{i = 0}^p (-1)^p \eta_{(s_0, \ldots, \hat s_i, \ldots, s_p)}. $$ See Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-homology-complex} for more details.
By Topologies, Lemmas \ref{topologies-lemma-at-the-bottom} and \ref{topologies-lemma-at-the-bottom-etale} the functors $S_\tau \to (\Sch/S)_\tau$ satisfy the hypotheses of Sites, Lemma \ref{sites-lemma-bigger-site}. Hence our lemma follows from Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-bigger-site}.
By Topologies, Lemma \ref{topologies-lemma-change-alpha} the inclusion functor $(\Sch/S)_\tau \to (\Sch'/S)_\tau$ satisfies the assumptions of Sites, Lemma \ref{sites-lemma-bigger-site}. This implies (2) and (1) follows from Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-bigger-site}.
We have to show, given two big \'etale sites $\Sch_\etale$ and $\Sch_\etale'$ containing $S$, then $\Sh(S_\etale) \cong \Sh(S_\etale')$ with obvious notation. By Topologies, Lemma \ref{topologies-lemma-contained-in} we may assume $\Sch_\etale \subset \Sch_\etale'$. By Sets, Lemma \ref{sets-lemma-what-is-in-it} any affine scheme \'etale over $S$ is isomorphic to an object of both $\Sch_\etale$ and $\Sch_\etale'$. Thus the induced functor $S_{affine, \etale} \to S_{affine, \etale}'$ is an equivalence. Moreover, it is clear that both this functor and a quasi-inverse map transform standard \'etale coverings into standard \'etale coverings. Hence the result follows from Lemma \ref{lemma-alternative}.
The defintion of a standard $\tau$-covering is given in Topologies, Definition \ref{topologies-definition-standard-Zariski}, \ref{topologies-definition-standard-etale}, \ref{topologies-definition-standard-smooth}, \ref{topologies-definition-standard-syntomic}, and \ref{topologies-definition-standard-fppf}. By definition each of the schemes $U_i$ is affine and $I$ is a finite set. Hence $V$ is an affine scheme. It is clear that $V \to U$ is flat and surjective, hence $\mathcal{V}$ is an fpqc covering, see Example \ref{example-fpqc-coverings}. Excepting the Zariski case, the covering $\mathcal{V}$ is also a $\tau$-covering, see Topologies, Definition \ref{topologies-definition-etale-covering}, \ref{topologies-definition-smooth-covering}, \ref{topologies-definition-syntomic-covering}, and \ref{topologies-definition-fppf-covering}. \medskip\noindent Note that $\mathcal{U}$ is a refinement of $\mathcal{V}$ and hence there is a map of {\v C}ech complexes $\check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{F}) \to \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$, see Cohomology on Sites, Equation (\ref{sites-cohomology-equation-map-cech-complexes}). Next, we observe that if $T = \coprod_{j \in J} T_j$ is a disjoint union of schemes in the site on which $\mathcal{F}$ is defined then the family of morphisms with fixed target $\{T_j \to T\}_{j \in J}$ is a Zariski covering, and so \begin{equation} \label{equation-sheaf-coprod} \mathcal{F}(T) = \mathcal{F}(\coprod\nolimits_{j \in J} T_j) = \prod\nolimits_{j \in J} \mathcal{F}(T_j) \end{equation} by the sheaf condition of $\mathcal{F}$. This implies the map of {\v C}ech complexes above is an isomorphism in each degree because $$ V \times_U \ldots \times_U V = \coprod\nolimits_{i_0, \ldots i_p} U_{i_0} \times_U \ldots \times_U U_{i_p} $$ as schemes.
Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$. Then $\xi$ is represented by a cocycle $\tilde{\xi} \in \mathcal{I}^p(U)$ with $d^p(\tilde{\xi}) = 0$. By assumption, the sequence $\mathcal{I}^{p - 1} \to \mathcal{I}^p \to \mathcal{I}^{p + 1}$ in exact in $\textit{Ab}(\mathcal{C})$, which means that there exists a covering $\mathcal{U} = \{U_i \to U\}_{i \in I}$ such that $\tilde{\xi}|_{U_i} = d^{p - 1}(\xi_i)$ for some $\xi_i \in \mathcal{I}^{p-1}(U_i)$. Since the cohomology class $\xi|_{U_i}$ is represented by the cocycle $\tilde{\xi}|_{U_i}$ which is a coboundary, it vanishes. For more details see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-kill-cohomology-class-on-covering}.
(See also Topologies, Lemma \ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc}.) Let $\{\varphi_i : U_i \to U\}_{i \in I}$ be an \'etale covering. Since an \'etale morphism is flat, and the elements of the covering should cover its target, the property fp (faithfully flat) is satisfied. To check the property qc (quasi-compact), let $V \subset U$ be an affine open, and write $\varphi_i^{-1}(V) = \bigcup_{j \in J_i} V_{ij}$ for some affine opens $V_{ij} \subset U_i$. Since $\varphi_i$ is open (as \'etale morphisms are open), we see that $V = \bigcup_{i\in I} \bigcup_{j \in J_i} \varphi_i(V_{ij})$ is an open covering of $V$. Further, since $V$ is quasi-compact, this covering has a finite refinement.
By definition the sheaf $\mu_{n, S}$ is the kernel of the map $(\cdot)^n$. Hence it suffices to show that the last map is surjective. Since the syntomic topology is weaker than the fppf topology, see Topologies, Lemma \ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf}, it suffices to prove this for the syntomic topology. Let $U$ be a scheme over $S$. Let $f \in \mathbf{G}_m(U) = \Gamma(U, \mathcal{O}_U^*)$. We need to show that we can find a syntomic cover of $U$ over the members of which the restriction of $f$ is an $n$th power. Set $$ U' = \underline{\Spec}_U(\mathcal{O}_U[T]/(T^n-f)) \xrightarrow{\pi} U. $$ (See Constructions, Section \ref{constructions-section-spec-via-glueing} or \ref{constructions-section-spec} for a discussion of the relative spectrum.) Let $\Spec(A) \subset U$ be an affine open, and say $f|_{\Spec(A)}$ corresponds to the unit $a \in A^*$. Then $\pi^{-1}(\Spec(A)) = \Spec(B)$ with $B = A[T]/(T^n - a)$. The ring map $A \to B$ is finite free of rank $n$, hence it is faithfully flat, and hence we conclude that $\Spec(B) \to \Spec(A)$ is surjective. Since this holds for every affine open in $U$ we conclude that $\pi$ is surjective. In addition, $B$ is a global relative complete intersection over $A$, so the ring map $A \to B$ is standard syntomic, in particular syntomic. Since this holds for every affine open of $U$ we conclude that $\pi$ is syntomic. Hence $\mathcal{U} = \{\pi : U' \to U\}$ is a syntomic covering. Moreover, $f|_{U'} = (f')^n$ where $f'$ is the class of $T$ in $\Gamma(U', \mathcal{O}_{U'}^*)$, so $\mathcal{U}$ has the desired property.
For part (1), consider the fibre product $U = U_1 \times_S U_2$. It is \'etale over both $U_1$ and $U_2$ because \'etale morphisms are preserved under base change, see Proposition \ref{proposition-etale-morphisms}. The map $\overline{s} \to U$ defined by $(\overline{u}_1, \overline{u}_2)$ gives it the structure of an \'etale neighborhood mapping to both $U_1$ and $U_2$. For part (2), define $U''$ as the fibre product $$ \xymatrix{ U'' \ar[r] \ar[d] & U \ar[d]^{(h_1, h_2)} \\ U' \ar[r]^-\Delta & U' \times_S U'. } $$ Since $\overline{u}$ and $\overline{u}'$ agree over $S$ with $\overline{s}$, we see that $\overline{u}'' = (\overline{u}, \overline{u}')$ is a geometric point of $U''$. In particular $U'' \not = \emptyset$. Moreover, since $U'$ is \'etale over $S$, so is the fibre product $U'\times_S U'$ (see Proposition \ref{proposition-etale-morphisms}). Hence the vertical arrow $(h_1, h_2)$ is \'etale by Remark \ref{remark-etale-between-etale} above. Therefore $U''$ is \'etale over $U'$ by base change, and hence also \'etale over $S$ (because compositions of \'etale morphisms are \'etale). Thus $(U'', \overline{u}'')$ is a solution to the problem.
As $U = \bigcup_{i\in I} \varphi_i(U_i)$, the fibre product $\overline{s} \times_{\overline{u}, U, \varphi_i} U_i$ is not empty for some $i$. Then look at the cartesian diagram $$ \xymatrix{ \overline{s} \times_{\overline{u}, U, \varphi_i} U_i \ar[d]^{\text{pr}_1} \ar[r]_-{\text{pr}_2} & U_i \ar[d]^{\varphi_i} \\ \Spec(k) = \overline{s} \ar@/^1pc/[u]^\sigma \ar[r]^-{\overline{u}} & U } $$ The projection $\text{pr}_1$ is the base change of an \'etale morphisms so it is \'etale, see Proposition \ref{proposition-etale-morphisms}. Therefore, $\overline{s} \times_{\overline{u}, U, \varphi_i} U_i$ is a disjoint union of finite separable extensions of $k$, by Proposition \ref{proposition-etale-morphisms}. Here $\overline{s} = \Spec(k)$. But $k$ is algebraically closed, so all these extensions are trivial, and there exists a section $\sigma$ of $\text{pr}_1$. The composition $\text{pr}_2 \circ \sigma$ gives a map compatible with $\overline{u}$.
In the proof of Lemma \ref{lemma-geometric-lift-to-cover} we have seen that the scheme $U_{\overline{s}}$ is a disjoint union of schemes isomorphic to $\overline{s}$. Thus we can also think of $|U_{\overline{s}}|$ as the set of geometric points of $U$ lying over $\overline{s}$, i.e., as the collection of morphisms $\overline{u} : \overline{s} \to U$ fitting into the diagram of Definition \ref{definition-geometric-point}. From this it follows that $u(S)$ is a singleton, and that $u(U \times_V W) = u(U) \times_{u(V)} u(W)$ whenever $U \to V$ and $W \to V$ are morphisms in $S_\etale$. And, given a covering $\{U_i \to U\}_{i \in I}$ in $S_\etale$ we see that $\coprod u(U_i) \to u(U)$ is surjective by Lemma \ref{lemma-geometric-lift-to-cover}. Hence Sites, Proposition \ref{sites-proposition-point-limits} applies, so $p$ is a point of the site $S_\etale$. Finally, our functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$ is given by exactly the same colimit as the functor $\mathcal{F} \mapsto \mathcal{F}_p$ associated to $p$ in Sites, Equation \ref{sites-equation-stalk} which proves the final assertion.
Before we indicate how to prove this by direct arguments we note that the result follows from the general material in Modules on Sites, Section \ref{sites-modules-section-stalks}. This is true because $\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$ comes from a point of the small \'etale site of $S$, see Lemma \ref{lemma-stalk-gives-point}. We will only give a direct proof of (1), (2) and (3), and omit a direct proof of (4). \medskip\noindent Exactness as a functor on $\textit{PAb}(S_\etale)$ is formal from the fact that directed colimits commute with all colimits and with finite limits. The identification of the stalks in (2) is via the map $$ \kappa : \mathcal{F}_{\overline{s}} \longrightarrow (\mathcal{F}^\#)_{\overline{s}} $$ induced by the natural morphism $\mathcal{F}\to \mathcal{F}^\#$, see Theorem \ref{theorem-sheafification}. We claim that this map is an isomorphism of abelian groups. We will show injectivity and omit the proof of surjectivity. \medskip\noindent Let $\sigma\in \mathcal{F}_{\overline{s}}$. There exists an \'etale neighborhood $(U, \overline{u})\to (S, \overline{s})$ such that $\sigma$ is the image of some section $s \in \mathcal{F}(U)$. If $\kappa(\sigma) = 0$ in $(\mathcal{F}^\#)_{\overline{s}}$ then there exists a morphism of \'etale neighborhoods $(U', \overline{u}')\to (U, \overline{u})$ such that $s|_{U'}$ is zero in $\mathcal{F}^\#(U')$. It follows there exists an \'etale covering $\{U_i'\to U'\}_{i\in I}$ such that $s|_{U_i'}=0$ in $\mathcal{F}(U_i')$ for all $i$. By Lemma \ref{lemma-geometric-lift-to-cover} there exist $i \in I$ and a morphism $\overline{u}_i': \overline{s} \to U_i'$ such that $(U_i', \overline{u}_i') \to (U', \overline{u}')\to (U, \overline{u})$ are morphisms of \'etale neighborhoods. Hence $\sigma = 0$ since $(U_i', \overline{u}_i') \to (U, \overline{u})$ is a morphism of \'etale neighbourhoods such that we have $s|_{U'_i}=0$. This proves $\kappa$ is injective. \medskip\noindent To show that the functor $\textit{Ab}(S_\etale) \to \textit{Ab}$ is exact, consider any short exact sequence in $\textit{Ab}(S_\etale)$: $ 0\to \mathcal{F}\to \mathcal{G}\to \mathcal H \to 0. $ This gives us the exact sequence of presheaves $$ 0 \to \mathcal{F} \to \mathcal{G} \to \mathcal H \to \mathcal H/^p\mathcal{G} \to 0, $$ where $/^p$ denotes the quotient in $\textit{PAb}(S_\etale)$. Taking stalks at $\overline{s}$, we see that $(\mathcal H /^p\mathcal{G})_{\bar{s}} = (\mathcal H /\mathcal{G})_{\bar{s}} = 0$, since the sheafification of $\mathcal H/^p\mathcal{G}$ is $0$. Therefore, $$ 0\to \mathcal{F}_{\overline{s}} \to \mathcal{G}_{\overline{s}} \to \mathcal{H}_{\overline{s}} \to 0 = (\mathcal H/^p\mathcal{G})_{\overline{s}} $$ is exact, since taking stalks is exact as a functor from presheaves.
For each of the big sites the associated topos is equivalent to the topos defined by the site $(\textit{Aff}/S)_\tau$, see Topologies, Lemmas \ref{topologies-lemma-affine-big-site-Zariski}, \ref{topologies-lemma-affine-big-site-etale}, \ref{topologies-lemma-affine-big-site-smooth}, \ref{topologies-lemma-affine-big-site-syntomic}, and \ref{topologies-lemma-affine-big-site-fppf}. The result for the sites $(\textit{Aff}/S)_\tau$ follows immediately from Deligne's result Sites, Lemma \ref{sites-lemma-criterion-points}. \medskip\noindent The result for $S_{Zar}$ is clear. The result for $S_\etale$ either follows from (the proof of) Theorem \ref{theorem-exactness-stalks} or from Lemma \ref{lemma-alternative} and Deligne's result applied to $S_{affine, \etale}$.
Let $\Spec(A) \subset S$ be an affine neighbourhood of $s$. Let $\mathfrak p \subset A$ be the prime ideal corresponding to $s$. With these choices we have canonical isomorphisms $\mathcal{O}_{S, s} = A_{\mathfrak p}$ and $\kappa(s) = \kappa(\mathfrak p)$. Thus we have $\kappa(\mathfrak p) \subset \kappa^{sep} \subset \kappa(\overline{s})$. Recall that $$ \mathcal{O}_{S, \overline{s}} = \colim_{(U, \overline{u})} \mathcal{O}(U) $$ where the limit is over the \'etale neighbourhoods of $(S, \overline{s})$. A cofinal system is given by those \'etale neighbourhoods $(U, \overline{u})$ such that $U$ is affine and $U \to S$ factors through $\Spec(A)$. In other words, we see that $$ \mathcal{O}_{S, \overline{s}} = \colim_{(B, \mathfrak q, \phi)} B $$ where the colimit is over \'etale $A$-algebras $B$ endowed with a prime $\mathfrak q$ lying over $\mathfrak p$ and a $\kappa(\mathfrak p)$-algebra map $\phi : \kappa(\mathfrak q) \to \kappa(\overline{s})$. Note that since $\kappa(\mathfrak q)$ is finite separable over $\kappa(\mathfrak p)$ the image of $\phi$ is contained in $\kappa^{sep}$. Via these translations the result of the lemma is equivalent to the result of Algebra, Lemma \ref{algebra-lemma-strict-henselization-different}.
This follows because the stalks $\mathcal{O}_{S, s}^{sh} = \mathcal{O}_{S, \overline{s}}$ are local, and because $S_\etale$ has enough points, see Lemma \ref{lemma-describe-etale-local-ring}, Theorem \ref{theorem-exactness-stalks}, and Remarks \ref{remarks-enough-points}. See Modules on Sites, Lemmas \ref{sites-modules-lemma-locally-ringed-stalk} and \ref{sites-modules-lemma-ringed-stalk-not-zero} for the fact that this implies the small \'etale site is locally ringed.
The exactness of $f^{-1}$ on sheaves of sets is a consequence of Sites, Proposition \ref{sites-proposition-get-morphism} applied to our functor $u$ of Equation (\ref{equation-functorial}). In fact the exactness of pullback is part of the definition of a morphism of topoi (or sites if you like). Thus we see (2) holds. It implies part (1) since given an abelian sheaf $\mathcal{G}$ on $Y_\etale$ the underlying sheaf of sets of $f^{-1}\mathcal{F}$ is the same as $f^{-1}$ of the underlying sheaf of sets of $\mathcal{F}$, see Sites, Section \ref{sites-section-sheaves-algebraic-structures}. See also Modules on Sites, Lemma \ref{sites-modules-lemma-flat-pullback-exact}. In the literature (1) and (2) are sometimes deduced from (3) via Theorem \ref{theorem-exactness-stalks}. \medskip\noindent Part (3) is a general fact about stalks of pullbacks, see Sites, Lemma \ref{sites-lemma-point-morphism-sites}. We will also prove (3) directly as follows. Note that by Lemma \ref{lemma-stalk-exact} taking stalks commutes with sheafification. Now recall that $f^{-1}\mathcal{G}$ is the sheaf associated to the presheaf $$ U \longrightarrow \colim_{U \to X \times_Y V} \mathcal{G}(V), $$ see Equation (\ref{equation-pullback}). Thus we have \begin{align*} (f^{-1}\mathcal{G})_{\overline{x}} & = \colim_{(U, \overline{u})} f^{-1}\mathcal{G}(U) \\ & = \colim_{(U, \overline{u})} \colim_{a : U \to X \times_Y V} \mathcal{G}(V) \\ & = \colim_{(V, \overline{v})} \mathcal{G}(V) \\ & = \mathcal{G}_{\overline{y}} \end{align*} in the third equality the pair $(U, \overline{u})$ and the map $a : U \to X \times_Y V$ corresponds to the pair $(V, a \circ \overline{u})$. \medskip\noindent Part (4) can be proved in a similar manner by identifying the colimits which define $f^{-1}h_V$. Or you can use Yoneda's lemma (Categories, Lemma \ref{categories-lemma-yoneda}) and the functorial equalities $$ \Mor_{\Sh(X_\etale)}(f^{-1}h_V, \mathcal{F}) = \Mor_{\Sh(Y_\etale)}(h_V, f_*\mathcal{F}) = f_*\mathcal{F}(V) = \mathcal{F}(X \times_Y V) $$ combined with the fact that representable presheaves are sheaves. See also Sites, Lemma \ref{sites-lemma-pullback-representable-sheaf} for a completely general result.