hcju/stacksqa huggingface dataset:
By Lemmas \ref{lemma-torsion-colimit-constructible} and \ref{lemma-colimit} we can find a map $\mathcal{G} \to \mathcal{F}$ with $\mathcal{G}$ a constructible abelian sheaf and $\xi$ coming from an element $\zeta$ of $H^q_\etale(Z, \mathcal{G}|_Z)$. Suppose we can find an injective map $\mathcal{G} \to \mathcal{G}'$ of torsion abelian sheaves on $X_\etale$ such that the image of $\zeta$ in $H^q_\etale(Z, \mathcal{G}'|_Z)$ is zero. Then we can take $\mathcal{F}'$ to be the pushout $$ \mathcal{F}' = \mathcal{G}' \amalg_{\mathcal{G}} \mathcal{F} $$ and we conclude the result of the lemma holds. (Observe that restriction to $Z$ is exact, so commutes with finite limits and colimits and moreover it commutes with arbitrary colimits as a left adjoint to pushforward.) Thus we may assume $\mathcal{F}$ is constructible. \medskip\noindent Assume $\mathcal{F}$ is constructible. By Lemma \ref{lemma-constructible-maps-into-constant-general} it suffices to prove the result when $\mathcal{F}$ is of the form $f_*\underline{M}$ where $M$ is a finite abelian group and $f : Y \to X$ is a finite morphism of finite presentation (such sheaves are still constructible by Lemma \ref{lemma-finite-pushforward-constructible} but we won't need this). Since formation of $f_*$ commutes with any base change (Lemma \ref{lemma-finite-pushforward-commutes-with-base-change}) we see that the restriction of $f_*\underline{M}$ to $Z$ is equal to the pushforward of $\underline{M}$ via $Y \times_X Z \to Z$. By the Leray spectral sequence (Proposition \ref{proposition-leray}) and vanishing of higher direct images (Proposition \ref{proposition-finite-higher-direct-image-zero}), we find $$ H^q_\etale(Z, f_*\underline{M}|_Z) = H^q_\etale(Y \times_X Z, \underline{M}). $$ By Lemma \ref{lemma-integral-cover-trivial-cohomology} we can find a finite surjective morphism $Y' \to Y$ of finite presentation such that $\xi$ maps to zero in $H^q(Y' \times_X Z, \underline{M})$. Denoting $f' : Y' \to X$ the composition $Y' \to Y \to X$ we claim the map $$ f_*\underline{M} \longrightarrow f'_*\underline{M} $$ is injective which finishes the proof by what was said above. To see the desired injectivity we can look at stalks. Namely, if $\overline{x} : \Spec(k) \to X$ is a geometric point, then $$ (f_*\underline{M})_{\overline{x}} = \bigoplus\nolimits_{f(\overline{y}) = \overline{x}} M $$ by Proposition \ref{proposition-finite-higher-direct-image-zero} and similarly for the other sheaf. Since $Y' \to Y$ is surjective and finite we see that the induced map on geometric points lifting $\overline{x}$ is surjective too and we conclude.
Let $\mathcal{F}$ be a sheaf on $X_\etale$. There is a canonical (base change) map $$ i^{-1}(\mathcal{F}|_{X_{Zar}}) \longrightarrow (i_{small}^{-1}\mathcal{F})|_{Z_{Zar}} $$ of sheaves on $Z_{Zar}$. We will show this map is injective by looking at stalks. The stalk on the left hand side at $z \in Z$ is the stalk of $\mathcal{F}|_{X_{Zar}}$ at $z$. The stalk on the right hand side is the colimit over all elementary \'etale neighbourhoods $(U, u) \to (X, z)$ such that $U \times_X Z \to Z$ has a section over a neighbourhood of $z$. As \'etale morphisms are open, the image of $U \to X$ is an open neighbourhood $U_0$ of $z$ in $X$. The map $\mathcal{F}(U_0) \to \mathcal{F}(U)$ is injective by the sheaf condition for $\mathcal{F}$ with respect to the \'etale covering $U \to U_0$. Taking the colimit over all $U$ and $U_0$ we obtain injectivity on stalks. \medskip\noindent It follows from this and assumption (1) that the map $\Gamma(X, \mathcal{F}) \to \Gamma(Z, i^{-1}_{small}\mathcal{F})$ is injective. By (2) the same thing is true on all $X'$ finite over $X$. \medskip\noindent Let $s \in \Gamma(Z, i^{-1}_{small}\mathcal{F})$. By construction of $i^{-1}_{small}\mathcal{F}$ there exists an \'etale covering $\{V_j \to Z\}$, \'etale morphisms $U_j \to X$, sections $s_j \in \mathcal{F}(U_j)$ and morphisms $V_j \to U_j$ over $X$ such that $s|_{V_j}$ is the pullback of $s_j$. Observe that every nonempty closed subscheme $T \subset X$ meets $Z$ by assumption (1) applied to the sheaf $(T \to X)_*\underline{\mathbf{Z}}$ for example. Thus we see that $\coprod U_j \to X$ is surjective. By More on Morphisms, Lemma \ref{more-morphisms-lemma-there-is-a-scheme-integral-over} we can find a finite surjective morphism $X' \to X$ such that $X' \to X$ Zariski locally factors through $\coprod U_j \to X$. It follows that $s|_{Z'}$ Zariski locally comes from a section of $\mathcal{F}|_{X'}$. In other words, $s|_{Z'}$ comes from $t' \in \Gamma(X', \mathcal{F}|_{X'})$ by assumption (2). By injectivity we conclude that the two pullbacks of $t'$ to $X' \times_X X'$ are the same (after all this is true for the pullbacks of $s$ to $Z' \times_Z Z'$). Hence we conclude $t'$ comes from a section of $\mathcal{F}$ over $X$ by Remark \ref{remark-cohomological-descent-finite}.
Observe that $Z_i$ is quasi-compact. Hence the set of points $W_i$ specializing to $Z_i$ is closed in the constructible topology by Topology, Lemma \ref{topology-lemma-make-spectral-space}. Assumption (2) implies that $X = W_1 \amalg W_2$. Let $x \in \overline{W_1}$. By Topology, Lemma \ref{topology-lemma-constructible-stable-specialization-closed} part (1) there exists a specialization $x_1 \leadsto x$ with $x_1 \in W_1$. Thus $\overline{\{x\}} \subset \overline{\{x_1\}}$ and we see that $x \in W_1$. In other words, setting $X_i = W_i$ does the job.
If $x \leadsto x'$ is a specialization of points, then there is a canonical map $\mathcal{F}_{x'} \to \mathcal{F}_x$ compatible with sections over opens and functorial in $\mathcal{F}$. Since every point of $X$ specializes to a point of $Z$ it follows that $\Gamma(X, \mathcal{F}) \to \Gamma(Z, \mathcal{F}|_Z)$ is injective. The difficult part is to show that it is surjective. \medskip\noindent Denote $\mathcal{B}$ be the set of all quasi-compact opens of $X$. Write $\mathcal{F}$ as a filtered colimit $\mathcal{F} = \colim \mathcal{F}_i$ where each $\mathcal{F}_i$ is as in Modules, Equation (\ref{modules-equation-towards-constructible-sets}). See Modules, Lemma \ref{modules-lemma-filtered-colimit-constructibles}. Then $\mathcal{F}|_Z = \colim \mathcal{F}_i|_Z$ as restriction to $Z$ is a left adjoint (Categories, Lemma \ref{categories-lemma-adjoint-exact} and Sheaves, Lemma \ref{sheaves-lemma-f-map}). By Sheaves, Lemma \ref{sheaves-lemma-directed-colimits-sections} the functors $\Gamma(X, -)$ and $\Gamma(Z, -)$ commute with filtered colimits. Hence we may assume our sheaf $\mathcal{F}$ is as in Modules, Equation (\ref{modules-equation-towards-constructible-sets}). \medskip\noindent Suppose that we have an embedding $\mathcal{F} \subset \mathcal{G}$. Then we have $$ \Gamma(X, \mathcal{F}) = \Gamma(Z, \mathcal{F}|_Z) \cap \Gamma(X, \mathcal{G}) $$ where the intersection takes place in $\Gamma(Z, \mathcal{G}|_Z)$. This follows from the first remark of the proof because we can check whether a global section of $\mathcal{G}$ is in $\mathcal{F}$ by looking at the stalks and because every point of $X$ specializes to a point of $Z$. \medskip\noindent By Modules, Lemma \ref{modules-lemma-constructible-in-constant} there is an injection $\mathcal{F} \to \prod (Z_i \to X)_*\underline{S_i}$ where the product is finite, $Z_i \subset X$ is closed, and $S_i$ is finite. Thus it suffices to prove surjectivity for the sheaves $(Z_i \to X)_*\underline{S_i}$. Observe that $$ \Gamma(X, (Z_i \to X)_*\underline{S_i}) = \Gamma(Z_i, \underline{S_i}) \quad\text{and}\quad \Gamma(X, (Z_i \to X)_*\underline{S_i}|_Z) = \Gamma(Z \cap Z_i, \underline{S_i}) $$ Moreover, conditions (1) and (2) are inherited by $Z_i$; this is clear for (2) and follows from Topology, Lemma \ref{topology-lemma-spectral-sub} for (1). Thus it suffices to prove the lemma in the case of a (finite) constant sheaf. This case is a restatement of Lemma \ref{lemma-connected-topological} which finishes the proof.
Recall that the spectrum of any ring is a spectral space, see Algebra, Lemma \ref{algebra-lemma-spec-spectral}. By More on Algebra, Lemma \ref{more-algebra-lemma-irreducible-henselian-pair-connected} we see that $\overline{\{x\}} \cap Z$ is connected for every $x \in X$. By Lemma \ref{lemma-h0-topological} we see that the statement is true for sheaves on $X_{Zar}$. For any finite morphism $X' \to X$ we have $X' = \Spec(A')$ and $Z \times_X X' = \Spec(A'/IA')$ with $(A', IA')$ a henselian pair, see More on Algebra, Lemma \ref{more-algebra-lemma-integral-over-henselian-pair} and we get the same statement for sheaves on $(X')_{Zar}$. Thus we can apply Lemma \ref{lemma-gabber-h0} to conclude.
We will prove this by induction on $n$. If $n = 0$, then $X$ is affine. Say $X = \Spec(A)$ and $Z = \Spec(A/I)$. Let $A^h$ be the filtered colimit of \'etale $A$-algebras $B$ such that $A/I \to B/IB$ is an isomorphism. Then $(A^h, IA^h)$ is a henselian pair and $A/I = A^h/IA^h$, see More on Algebra, Lemma \ref{more-algebra-lemma-henselization} and its proof. Set $X^h = \Spec(A^h)$. By Theorem \ref{theorem-gabber} we see that $$ H^q_\etale(Z, \mathcal{I}|_Z) = H^q_\etale(X^h, \mathcal{I}|_{X^h}) $$ By Theorem \ref{theorem-colimit} we have $$ H^q_\etale(X^h, \mathcal{I}|_{X^h}) = \colim_{A \to B} H^q_\etale(\Spec(B), \mathcal{I}|_{\Spec(B)}) $$ where the colimit is over the $A$-algebras $B$ as above. Since the morphisms $\Spec(B) \to \Spec(A)$ are \'etale, the restriction $\mathcal{I}|_{\Spec(B)}$ is an injective sheaf of $\mathcal{A}|_{\Spec(B)}$-modules (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-of-open}). Thus the cohomology groups on the right are zero and we get the result in this case. \medskip\noindent Induction step. We can use Mayer-Vietoris to do the induction step. Namely, suppose that $X = U \cup V$ where $U$ is a union of $n$ affine opens and $V$ is affine. Then, using that the diagonal of $X$ is affine, we see that $U \cap V$ is the union of $n$ affine opens. Mayer-Vietoris gives an exact sequence $$ H^{q - 1}_\etale(U \cap V \cap Z, \mathcal{I}|_Z) \to H^q_\etale(Z, \mathcal{I}|_Z) \to H^q_\etale(U \cap Z, \mathcal{I}|_Z) \oplus H^q_\etale(V \cap Z, \mathcal{I}|_Z) $$ and by our induction hypothesis we obtain vanishing for $q > n$ as desired.
Since $X$ is smooth, we see that $X$ is a finite disjoint union of smooth curves. Hence we may assume $X$ is a smooth curve. \medskip\noindent Case I: $\ell$ different from the characteristic of $k$. This case follows from Lemma \ref{lemma-cohomology-smooth-projective-curve} (projective case) and Lemma \ref{lemma-vanishing-cohomology-mu-smooth-curve} (affine case). Statement (\ref{item-base-change-prime-to-p}) on cohomology and extension of algebraically closed ground field follows from the fact that the genus $g$ and the number of ``punctures'' $r$ do not change when passing from $k$ to $k'$. Statement (\ref{item-surjective}) follows as $H^2_\etale(U, \mathcal{F})$ is zero as soon as $U \not = X$, because then $U$ is affine (Varieties, Lemmas \ref{varieties-lemma-proper-minus-point} and \ref{varieties-lemma-curve-affine-projective}). \medskip\noindent Case II: $\ell$ is equal to the characteristic of $k$. Vanishing by Lemma \ref{lemma-vanishing-variety-char-p-p}. Statements (\ref{item-finite-proper}) and (\ref{item-base-change-proper}) follow from Lemma \ref{lemma-finiteness-proper-variety-char-p-p}.
This is mostly immediate from the definitions and the long exact sequence of cohomology. Also observe that $\mathcal{F}$ is constructible (resp.\ of torsion prime to the characteristic of $k$) if and only if both $\mathcal{F}_1$ and $\mathcal{F}_2$ are constructible (resp.\ of torsion prime to the characteristic of $k$). See Proposition \ref{proposition-constructible-over-noetherian}. Some details omitted.
Consider the short exact sequence $$ 0 \longrightarrow j_!\underline{\mathbf{Z}/\ell\mathbf{Z}}_U \longrightarrow \underline{\mathbf{Z}/\ell\mathbf{Z}}_X \longrightarrow \bigoplus\nolimits_{x \in X \setminus U} {i_x}_*(\underline{\mathbf{Z}/\ell\mathbf{Z}}) \longrightarrow 0. $$ Statements (\ref{item-vanishing}) -- (\ref{item-surjective}) hold for $\underline{\mathbf{Z}/\ell\mathbf{Z}}$ by Lemma \ref{lemma-constant-smooth-statements}. Since the inclusion morphisms $i_x : x \to X$ are finite and since $x$ is the spectrum of an algebraically closed field, we see that $H^q_\etale(X, i_{x*}\underline{\mathbf{Z}/\ell\mathbf{Z}})$ is zero for $q > 0$ and equal to $\mathbf{Z}/\ell\mathbf{Z}$ for $q = 0$. Thus we get from the long exact cohomology sequence $$ \xymatrix{ 0 \ar[r] & H^0_\etale(X, \mathcal{F}) \ar[r] & H^0(X, \underline{\mathbf{Z}/\ell\mathbf{Z}}_X) \ar[r] & \bigoplus\nolimits_{x \in X \setminus U} \mathbf{Z}/\ell\mathbf{Z} \ar@(rd, ul)[rdllllr] \\  & H^1_\etale(X, \mathcal{F}) \ar[r] & H^1_\etale(X, \underline{\mathbf{Z}/\ell\mathbf{Z}}_X) \ar[r] & 0 } $$ and $H^q_\etale(X, \mathcal{F}) = H^q_\etale(X, \underline{\mathbf{Z}/\ell\mathbf{Z}}_X)$ for $q \geq 2$. Each of the statements  (\ref{item-vanishing}) -- (\ref{item-surjective}) follows by inspection.
The difference with Lemma \ref{lemma-even-easier} is that here we do not assume $X$ is smooth. Let $\nu : X' \to X$ be the normalization morphism which is finite as varieties are Nagata schemes. Let $j' : U' \to X'$ be the inverse image of $U$. By Lemma \ref{lemma-even-easier} the result holds for $j'_!\underline{\mathbf{Z}/\ell\mathbf{Z}}$. By Lemma \ref{lemma-finite-pushforward-statements} the result holds for $\nu_*j'_!\underline{\mathbf{Z}/\ell\mathbf{Z}}$. In general it won't be true that $\nu_*j'_!\underline{\mathbf{Z}/\ell\mathbf{Z}}$ is equal to $j_!\underline{\mathbf{Z}/\ell\mathbf{Z}}$, but there will be a canonical injective map $$ j_!\underline{\mathbf{Z}/\ell\mathbf{Z}} \longrightarrow \nu_*j'_!\underline{\mathbf{Z}/\ell\mathbf{Z}} $$ whose cokernel is of the form $\bigoplus_{x \in Z} i_{x *}M_x$ where $Z \subset X$ is a finite set of closed points and $M_x$ is a finite dimensional $\mathbf{F}_\ell$-vector space for each $x \in Z$. We obtain a short exact sequence $$ 0 \to j_!\underline{\mathbf{Z}/\ell\mathbf{Z}} \to \nu_*j'_!\underline{\mathbf{Z}/\ell\mathbf{Z}} \to \bigoplus\nolimits_{x \in Z} i_{x *}M_x \to 0 $$ and we can argue exactly as in the proof of Lemma \ref{lemma-even-easier} to finish the argument. Some details omitted.
Let $f : V \to U$ be a finite \'etale morphism of degree prime to $\ell$ as in Lemma \ref{lemma-pullback-filtered}. The discussion in Section \ref{section-trace-method} gives maps $$ \mathcal{G} \to f_*f^{-1}\mathcal{G} \to \mathcal{G} $$ whose composition is an isomorphism. Hence it suffices to prove the lemma with $\mathcal{F} = j_!f_*f^{-1}\mathcal{G}$. By Zariski's Main theorem (More on Morphisms, Lemma \ref{more-morphisms-lemma-quasi-finite-separated-pass-through-finite}) we can choose a diagram $$ \xymatrix{ V \ar[r]_{j'} \ar[d]_f & Y \ar[d]^{\overline{f}} \\ U \ar[r]^j & X } $$ with $\overline{f} : Y \to X$ finite and $j'$ an open immersion with dense image. We may replace $Y$ by its reduction (this does not change $V$ as $V$ is reduced being \'etale over $U$). Since $f$ is finite we have $V = U \times_X Y$. By Lemma \ref{lemma-finite-pushforward-commutes-with-base-change} we have $$ j^{-1}\overline{f}_*j'_!f^{-1}\mathcal{G} = f_*(j')^{-1}j'_!f^{-1}\mathcal{G} = f_*f^{-1}\mathcal{G} $$ because $(j')^{-1}j'_! = \text{id}$ by Lemma \ref{lemma-jshriek-open}. Hence there is a canonical map $j_!f_*f^{-1}\mathcal{G} \to \overline{f}_*j'_!f^{-1}\mathcal{G}$ which is an isomorphism over $U$ (by the same lemma). However, the stalk of $\overline{f}_*j'_!f^{-1}\mathcal{G}$ at a geometric point $\overline{x}$ of $X \setminus U$ are zero by Proposition \ref{proposition-finite-higher-direct-image-zero} and the fact that the stalks of $j'_!f^{-1}\mathcal{F}$ are zero at all geometric points $\overline{y}$ of $Y$ lying over $\overline{x}$ (by Lemma \ref{lemma-jshriek-open}). Hence $j_!f_*f^{-1}\mathcal{G} \to \overline{f}_*j'_!f^{-1}\mathcal{G}$ is an isomorphism, again by Lemma \ref{lemma-jshriek-open}. By Lemma \ref{lemma-finite-pushforward-statements} it suffices to prove the lemma for $j'_!f^{-1}\mathcal{G}$. The existence of the filtration given by Lemma \ref{lemma-pullback-filtered}, the fact that $j'_!$ is exact, and Lemma \ref{lemma-ses-statements} reduces us to the case $\mathcal{F} = j'_!\underline{\mathbf{Z}/\ell\mathbf{Z}}$ which is Lemma \ref{lemma-somewhat-easier}.
We have seen this for algebraically closed fields in Theorem \ref{theorem-vanishing-affine-curves}. Given $k \subset k'$ as in the statement of the lemma we can choose a diagram $$ \xymatrix{ k' \ar[r] & \overline{k}' \\ k \ar[u] \ar[r] & \overline{k} \ar[u] } $$ where $k \subset \overline{k}$ and $k' \subset \overline{k}'$ are the algebraic closures. Since $k$ and $k'$ are separably closed the field extensions $k \subset \overline{k}$ and $k' \subset \overline{k}'$ are algebraic and purely inseparable. In this case the morphisms $X_{\overline{k}} \to X$ and $X_{\overline{k}'} \to X_{k'}$ are universal homeomorphisms. Thus the cohomology of $\mathcal{F}$ may be computed on $X_{\overline{k}}$ and the cohomology of $\mathcal{F}|_{X_{k'}}$ may be computed on $X_{\overline{k}'}$, see Proposition \ref{proposition-topological-invariance}. Hence we deduce the general case from the case of algebraically closed fields.
By Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-torsors-h1} an element of $H^1_\etale(X, \underline{M})$ corresponds to a $\underline{M}$-torsor $\mathcal{F}$ on $X_\etale$. Such a torsor is clearly a finite locally constant sheaf. Hence $\mathcal{F}$ is representable by a scheme $V$ finite \'etale over $X$, Lemma \ref{lemma-characterize-finite-locally-constant}. Conversely, a scheme $V$ finite \'etale over $X$ with an $M$-action which turns it into an $M$-torsor over $X$ gives rise to a cohomology class. The same translation between cohomology classes over $X_0$ and torsors finite \'etale over $X_0$ holds. Thus the lemma is a consequence of the equivalence of categories of Fundamental Groups, Lemma \ref{pione-lemma-finite-etale-on-proper-over-henselian}.
Observe that $X$ is a separated scheme which can be covered by $2$ affine opens. Hence for $q > 1$ this follows from Gabber's affine variant of the proper base change theorem, see Lemma \ref{lemma-vanishing-restriction-injective}. Thus we may assume $q = 1$. Let $\xi \in H^1_\etale(X_0, \mathcal{I}|_{X_0})$. Goal: show that $\xi$ is $0$. By Lemmas \ref{lemma-torsion-colimit-constructible} and \ref{lemma-colimit} we can find a map $\mathcal{F} \to \mathcal{I}$ with $\mathcal{F}$ a constructible sheaf of $\mathbf{Z}/\ell\mathbf{Z}$-modules and $\xi$ coming from an element $\zeta$ of $H^1_\etale(X_0, \mathcal{F}|_{X_0})$. Suppose we have an injective map $\mathcal{F} \to \mathcal{F}'$ of sheaves of $\mathbf{Z}/\ell\mathbf{Z}$-modules on $X_\etale$. Since $\mathcal{I}$ is injective we can extend the given map $\mathcal{F} \to \mathcal{I}$ to a map $\mathcal{F}' \to \mathcal{I}$. In this situation we may replace $\mathcal{F}$ by $\mathcal{F}'$ and $\zeta$ by the image of $\zeta$ in $H^1_\etale(X_0, \mathcal{F}'|_{X_0})$. Also, if $\mathcal{F} = \mathcal{F}_1 \oplus \mathcal{F}_2$ is a direct sum, then we may replace $\mathcal{F}$ by $\mathcal{F}_i$ and $\zeta$ by the image of $\zeta$ in $H^1_\etale(X_0, \mathcal{F}_i|_{X_0})$. \medskip\noindent By Lemma \ref{lemma-constructible-maps-into-constant-general} and the remarks above we may assume $\mathcal{F}$ is of the form $f_*\underline{M}$ where $M$ is a finite $\mathbf{Z}/\ell\mathbf{Z}$-module and $f : Y \to X$ is a finite morphism of finite presentation (such sheaves are still constructible by Lemma \ref{lemma-finite-pushforward-constructible} but we won't need this). Since formation of $f_*$ commutes with any base change (Lemma \ref{lemma-finite-pushforward-commutes-with-base-change}) we see that the restriction of $f_*\underline{M}$ to $X_0$ is equal to the pushforward of $\underline{M}$ via the induced morphism $Y_0 \to X_0$ of special fibres. By the Leray spectral sequence (Proposition \ref{proposition-leray}) and vanishing of higher direct images (Proposition \ref{proposition-finite-higher-direct-image-zero}), we find $$ H^1_\etale(X_0, f_*\underline{M}|_{X_0}) = H^1_\etale(Y_0, \underline{M}). $$ Since $Y \to \Spec(A)$ is proper we can use Lemma \ref{lemma-proper-over-henselian-and-h1} to see that the $H^1_\etale(Y_0, \underline{M})$ is equal to $H^1_\etale(Y, \underline{M})$. Thus we see that our cohomology class $\zeta$ lifts to a cohomology class $$ \tilde\zeta \in H^1_\etale(Y, \underline{M}) = H^1_\etale(X, f_*\underline{M}) $$ However, $\tilde \zeta$ maps to zero in $H^1_\etale(X, \mathcal{I})$ as $\mathcal{I}$ is injective and by commutativity of $$ \xymatrix{ H^1_\etale(X, f_*\underline{M}) \ar[r] \ar[d] & H^1_\etale(X, \mathcal{I}) \ar[d] \\ H^1_\etale(X_0, (f_*\underline{M})|_{X_0}) \ar[r] & H^1_\etale(X_0, \mathcal{I}|_{X_0}) } $$ we conclude that the image $\xi$ of $\zeta$ is zero as well.
We have $f^{-1}R^qg_*\mathcal{F}|_{U_i} = f_i^{-1}R^qg_{i, *}\mathcal{F}_i$ and $R^qh_*e^{-1}\mathcal{F}|_{U_i} = R^qh_{i, *}e_i^{-1}\mathcal{F}_i$ as follows from the compatibility of localization with morphisms of sites, see Sites, Lemma \ref{sites-lemma-localize-morphism-strong} and Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-restrict-direct-image-open}.
We prove (2) and we omit the proof of (1). We will use without further mention that pullback of sheaves commutes with colimits as it is a left adjoint. Observe that $h_i$ is quasi-compact and quasi-separated as a base change of $g_i$. Denoting $q_i : Y \to Y_i$ the projections, observe that $e^{-1}\mathcal{F} = \colim e^{-1}p_i^{-1}\mathcal{F}_i = \colim q_i^{-1}e_i^{-1}\mathcal{F}_i$. By Lemma \ref{lemma-relative-colimit-general} this gives $$ R^qh_*e^{-1}\mathcal{F} = \colim r_i^{-1}R^qh_{i, *}e_i^{-1}\mathcal{F}_i $$ where $r_i : X \to X_i$ is the projection. Similarly, we have $$ f^{-1}Rg_*\mathcal{F} = f^{-1}\colim s_i^{-1}R^qg_{i, *}\mathcal{F}_i = \colim r_i^{-1}f_i^{-1}R^qg_{i, *}\mathcal{F}_i $$ where $s_i : S \to S_i$ is the projection. The lemma follows.
Since $Y = X \times_S T$ we have $\Spec(\mathcal{O}^{sh}_{X, \overline{x}}) \times_X Y = \Spec(\mathcal{O}^{sh}_{X, \overline{x}}) \times_S T$. Thus the map in (1) is the map of stalks at $\overline{x}$ for the map in (2) by Theorem \ref{theorem-higher-direct-images} (and Lemma \ref{lemma-stalk-pullback}). Thus the result by Theorem \ref{theorem-exactness-stalks}.
Observe that $\Spec(K) \times_S \Spec(\mathcal{O}^{sh}_{S, \overline{s}})$ is the spectrum of a filtered colimit of \'etale algebras over $K$. Since $K$ is separably closed, each \'etale $K$-algebra is a finite product of copies of $K$. Thus we can write $$ \Spec(K) \times_S \Spec(\mathcal{O}^{sh}_{S, \overline{s}}) = \lim_{i \in I} \coprod\nolimits_{a \in A_i} \Spec(K) $$ as a cofiltered limit where each term is a disjoint union of copies of $\Spec(K)$ over a finite set $A_i$. Note that $A_i$ is nonempty as we are given $\Spec(K) \to \Spec(\mathcal{O}^{sh}_{S, \overline{s}})$. It follows that \begin{align*} \Spec(\mathcal{O}^{sh}_{X, \overline{x}}) \times_S \Spec(K) & = \Spec(\mathcal{O}^{sh}_{X, \overline{x}}) \times_{\Spec(\mathcal{O}^{sh}_{S, \overline{s}})} \left( \Spec(\mathcal{O}^{sh}_{S, \overline{s}}) \times_S \Spec(K)\right) \\ & = \colim_{i \in I} \coprod\nolimits_{a \in A_i} \Spec(\mathcal{O}^{sh}_{X, \overline{x}}) \times_{\Spec(\mathcal{O}^{sh}_{S, \overline{s}})} \Spec(K) \end{align*} Since taking cohomology in our setting commutes with limits of schemes (Theorem \ref{theorem-colimit}) we conclude.