hcju/stacksqa huggingface dataset:
Write $X = \bar X - \{ x_1, \ldots, x_r\}$. Then $\Pic(X) = \Pic(\bar X)/ R$, where $R$ is the subgroup generated by $\mathcal{O}_{\bar X}(x_i)$, $1 \leq i \leq r$. Since $r \geq 1$, we see that $\Pic^0(\bar X) \to \Pic(X)$ is surjective, hence $\Pic(X)$ is divisible. Applying the Kummer sequence, we get (1) and (3). For (2), recall that \begin{align*} H_\etale^1(X, \mu_n) & = \{(\mathcal L, \alpha) | \mathcal L \in \Pic(X), \alpha : \mathcal{L}^{\otimes n} \to \mathcal{O}_X\}/\cong \\ & = \{(\bar{\mathcal L},\ D,\ \bar \alpha)\}/\tilde{R} \end{align*} where $\bar{\mathcal L} \in \Pic^0(\bar X)$, $D$ is a divisor on $\bar X$ supported on $\left\{x_1, \ldots, x_r\right\}$ and $ \bar{\alpha}: \bar{\mathcal L}^{\otimes n} \cong \mathcal{O}_{\bar{X}}(D)$ is an isomorphism. Note that $D$ must have degree 0. Further $\tilde{R}$ is the subgroup of triples of the form $(\mathcal{O}_{\bar X}(D'), n D', 1^{\otimes n})$ where $D'$ is supported on $\left\{x_1, \ldots, x_r\right\}$ and has degree 0. Thus, we get an exact sequence $$ 0 \longrightarrow H_\etale^1(\bar X, \mu_n) \longrightarrow H_\etale^1(X, \mu_n) \longrightarrow \bigoplus_{i = 1}^r \mathbf{Z}/n\mathbf{Z} \xrightarrow{\ \sum\ } \mathbf{Z}/n\mathbf{Z} \longrightarrow 0 $$ where the middle map sends the class of a triple $(\bar{ \mathcal L}, D, \bar \alpha)$ with $D = \sum_{i = 1}^r a_i (x_i)$ to the $r$-tuple $(a_i)_{i = 1}^r$. It now suffices to use Lemma \ref{lemma-cohomology-smooth-projective-curve} to count ranks.
We encourage the reader to prove the first statement by working through the definitions, but here we just use that it is a special case of the very general Modules on Sites, Lemma \ref{sites-modules-lemma-restrict-back}. For the second statement, observe that if $\mathcal{G}$ is an abelian sheaf on $X_\etale$ whose restriction to $U$ is $\mathcal{F}$, then we obtain by adjointness a map $j_!\mathcal{F} \to \mathcal{G}$. This map is then an isomorphism at stalks of geometric points of $U$ by Proposition \ref{proposition-describe-jshriek}. Thus if $\mathcal{G}$ has vanishing stalks at geometric points of $X \setminus U$, then $j_!\mathcal{F} \to \mathcal{G}$ is an isomorphism by Theorem \ref{theorem-exactness-stalks}.
This is true because $j'_! f'^{-1}$ is left adjoint to $f'_* (j')^{-1}$ and $f^{-1} j_!$ is left adjoint to $j^{-1}f_*$. Further $f'_* (j')^{-1} = j^{-1}f_*$ because $f_*$ commutes with \'etale localization (by construction). In fact, the lemma holds very generally in the setting of a morphism of sites, see Modules on Sites, Lemma \ref{sites-modules-lemma-localize-morphism-ringed-sites}.
\medskip\noindent It suffices to check $j_!\mathcal{F} \to j_*\mathcal{F}$ is an isomorphism \'etale locally on $X$. Thus we may assume $U \to X$ is a finite disjoint union of isomorphisms, see \'Etale Morphisms, Lemma \ref{etale-lemma-finite-etale-etale-local}. We omit the proof in this case.
We obtain the maps by the adjointness properties of the functors involved. For a geometric point $\overline{x}$ in $X$ we have either $\overline{x} \in U$ in which case the map on the left hand side is an isomorphism on stalks and the stalk of $i_*i^{-1}\mathcal{F}$ is zero or $\overline{x} \in Z$ in which case the map on the right hand side is an isomorphism on stalks and the stalk of $j_!j^{-1}\mathcal{F}$ is zero. Here we have used the description of stalks of Lemma \ref{lemma-stalk-pushforward-closed-immersion} and Proposition \ref{proposition-describe-jshriek}.
It is clear that (1) implies (2). \medskip\noindent Assume (2). For every $x \in X$ we can find an $i$ and an affine open neighbourhood $V_x \subset U_i$ of $x$. Hence we can find a finite affine open covering $X = \bigcup V_j$ such that for each $j$ there exists a finite decomposition $V_j = \coprod V_{j, k}$ by locally closed constructible subsets such that $\mathcal{F}|_{V_{j, k}}$ is finite locally constant. By Topology, Lemma \ref{topology-lemma-quasi-compact-open-immersion-constructible-image} each $V_{j, k}$ is constructible as a subset of $X$. By Topology, Lemma \ref{topology-lemma-constructible-partition-refined-by-stratification} we can find a finite stratification $X = \coprod X_l$ with constructible locally closed strata such that each $V_{j, k}$ is a union of $X_l$. Thus (3) holds. \medskip\noindent Assume (3) holds. Let $U \subset X$ be an affine open. Then $U \cap X_i$ is a constructible locally closed subset of $U$ (for example by Properties, Lemma \ref{properties-lemma-locally-constructible}) and $U = \coprod U \cap X_i$ is a partition of $U$ as in Definition \ref{definition-constructible}. Thus (1) holds.
First, we can assume the covering is finite as $X$ is quasi-compact in the spectral topology (Topology, Lemma \ref{topology-lemma-constructible-hausdorff-quasi-compact} and Properties, Lemma \ref{properties-lemma-quasi-compact-quasi-separated-spectral}). Observe that each $T_i$ is a quasi-compact and quasi-separated scheme in its own right (because it is constructible in $X$; details omitted). Thus we can find a finite partition $T_i = \coprod T_{i, j}$ into locally closed constructible parts of $T_i$ such that $\mathcal{F}|_{T_{i, j}}$ is finite locally constant (Lemma \ref{lemma-constructible-quasi-compact-quasi-separated}). By Topology, Lemma \ref{topology-lemma-constructible-in-constructible} we see that $T_{i, j}$ is a constructible locally closed subscheme of $X$. Then we can apply Topology, Lemma \ref{topology-lemma-constructible-partition-refined-by-stratification} to $X = \bigcup T_{i, j}$ to find the desired partition of $X$.
The statement means if $X = \bigcup U_i$ is an open covering such that $\mathcal{F}|_{U_i}$ is constructible, then $\mathcal{F}$ is constructible. If $U \subset X$ is affine open, then $U = \bigcup U \cap U_i$ and $\mathcal{F}|_{U \cap U_i}$ is constructible (it is trivial that the restriction of a constructible sheaf to an open is constructible). It follows from Lemma \ref{lemma-constructible-quasi-compact-quasi-separated} that $\mathcal{F}|_U$ is constructible, i.e., a suitable partition of $U$ exists.
By Lemma \ref{lemma-constructible-local} this reduces to the case where $X$ and $Y$ are affine. By Lemma \ref{lemma-constructible-quasi-compact-quasi-separated} it suffices to find a finite partition of $X$ by constructible locally closed subschemes such that $f^{-1}\mathcal{F}$ is finite locally constant on each of them. To find it we just pull back the partition of $Y$ adapted to $\mathcal{F}$ and use Lemma \ref{lemma-pullback-locally-constant}.
We prove (3). We will use the criterion of Homology, Lemma \ref{homology-lemma-characterize-weak-serre-subcategory}. Suppose that $\varphi : \mathcal{F} \to \mathcal{G}$ is a map of constructible sheaves of $\Lambda$-modules. We have to show that $\mathcal{K} = \Ker(\varphi)$ and $\mathcal{Q} = \Coker(\varphi)$ are constructible.  Similarly, suppose that $0 \to \mathcal{F} \to \mathcal{E} \to \mathcal{G} \to 0$ is a short exact sequence of sheaves of $\Lambda$-modules with $\mathcal{F}$, $\mathcal{G}$ constructible. We have to show that $\mathcal{E}$ is constructible. In both cases we can replace $X$ with the members of an affine open covering. Hence we may assume $X$ is affine. Then we may further replace $X$ by the members of a finite partition of $X$ by constructible locally closed subschemes on which $\mathcal{F}$ and $\mathcal{G}$ are of finite type and locally constant. Thus we may apply Lemma \ref{lemma-kernel-finite-locally-constant} to conclude. \medskip\noindent The proofs of (1) and (2) are very similar and are omitted.
Proof of (1). Let $X = \coprod X_i$ be a partition of $X$ by locally closed constructible subschemes such that both $\mathcal{F}$ and $\mathcal{G}$ are finite locally constant over the parts (use Lemma \ref{lemma-constructible-quasi-compact-quasi-separated} for both $\mathcal{F}$ and $\mathcal{G}$ and choose a common refinement). Then apply Lemma \ref{lemma-morphism-locally-constant} to the restriction of the map to each part. \medskip\noindent The proof of (2) and (3) is omitted.
If $U \to X$ is separated, then this is More on Morphisms, Lemma \ref{more-morphisms-lemma-stratify-flat-fp-qf}. In general, we may assume $X$ is affine. Choose a finite affine open covering $U = \bigcup U_j$. Apply the previous case to all the morphisms $U_j \to X$ and $U_j \cap U_{j'} \to X$ and choose a common refinement $X = \coprod X_i$ of the resulting partitions. After refining the partition further we may assume $X_i$ affine as well. Fix $i$ and set $V = U \times_X X_i$. The morphisms $V_j = U_j \times_X X_i \to X_i$ and $V_{jj'} = (U_j \cap U_{j'}) \times_X X_i \to X_i$ are finite \'etale. Hence $V_j$ and $V_{jj'}$ are affine schemes and $V_{jj'} \subset V_j$ is closed as well as open (since $V_{jj'} \to X_i$ is proper, so Morphisms, Lemma \ref{morphisms-lemma-image-proper-scheme-closed} applies). Then $V = \bigcup V_j$ is separated because $\mathcal{O}(V_j) \to \mathcal{O}(V_{jj'})$ is surjective, see Schemes, Lemma \ref{schemes-lemma-characterize-separated}. Thus the previous case applies to $V \to X_i$ and we can further refine the partition if needed (it actually isn't but we don't need this).
By Lemma \ref{lemma-etale-stratified-finite} there is a partition $\coprod_i X_i$ such that $\pi_i : j^{-1}(X_i) \to X_i$ is finite \'etale. The restriction of $h_U$ to $X_i$ is $h_{j^{-1}(X_i)}$ which is finite locally constant by Lemma \ref{lemma-characterize-finite-locally-constant}. For cases (2) and (3) we note that $$ j_!(\underline{M})|_{X_i} = \pi_{i!}(\underline{M}) = \pi_{i*}(\underline{M}) $$ by Lemmas \ref{lemma-shriek-base-change} and \ref{lemma-shriek-equals-star-finite-etale}. Thus it suffices to show the lemma for $\pi : Y \to X$ finite \'etale. This is Lemma \ref{lemma-pushforward-locally-constant}.
Let $\mathcal{B}$ be the collection of quasi-compact and quasi-separated objects of $X_\etale$. By Modules on Sites, Lemma \ref{sites-modules-lemma-module-filtered-colimit-constructibles} any sheaf of sets is a filtered colimit of sheaves of the form $$ \text{Coequalizer}\left( \xymatrix{ \coprod\nolimits_{j = 1, \ldots, m} h_{V_j} \ar@<1ex>[r] \ar@<-1ex>[r] & \coprod\nolimits_{i = 1, \ldots, n} h_{U_i} } \right) $$ with $V_j$ and $U_i$ quasi-compact and quasi-separated objects of $X_\etale$. By Lemmas \ref{lemma-jshriek-constructible} and \ref{lemma-constructible-abelian} these coequalizers are constructible. This proves (1). \medskip\noindent Let $\Lambda$ be a Noetherian ring. By Modules on Sites, Lemma \ref{sites-modules-lemma-module-filtered-colimit-constructibles} $\Lambda$-modules $\mathcal{F}$ is a filtered colimit of modules of the form $$ \Coker\left( \bigoplus\nolimits_{j = 1, \ldots, m} j_{V_j!}\underline{\Lambda}_{V_j} \longrightarrow \bigoplus\nolimits_{i = 1, \ldots, n} j_{U_i!}\underline{\Lambda}_{U_i} \right) $$ with $V_j$ and $U_i$ quasi-compact and quasi-separated objects of $X_\etale$. By Lemmas \ref{lemma-jshriek-constructible} and \ref{lemma-constructible-abelian} these cokernels are constructible. This proves (3). \medskip\noindent Proof of (2). First write $\mathcal{F} = \bigcup \mathcal{F}[n]$ where $\mathcal{F}[n]$ is the $n$-torsion subsheaf. Then we can view $\mathcal{F}[n]$ as a sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules and apply (3).
One implication follows from Lemma \ref{lemma-pullback-constructible}. For the converse, assume $f^{-1}\mathcal{F}$ is constructible. Write $\mathcal{F} = \colim \mathcal{F}_i$ as a filtered colimit of constructible sheaves (of sets, abelian groups, or modules) using Lemma \ref{lemma-torsion-colimit-constructible}. Since $f^{-1}$ is a left adjoint it commutes with colimits (Categories, Lemma \ref{categories-lemma-adjoint-exact}) and we see that $f^{-1}\mathcal{F} = \colim f^{-1}\mathcal{F}_i$. By Lemma \ref{lemma-colimit-constructible} we see that $f^{-1}\mathcal{F}_i \to f^{-1}\mathcal{F}$ is surjective for all $i$ large enough. Since $f$ is surjective we conclude (by looking at stalks using Lemma \ref{lemma-stalk-pullback} and Theorem \ref{theorem-exactness-stalks}) that $\mathcal{F}_i \to \mathcal{F}$ is surjective for all $i$ large enough. Thus $\mathcal{F}$ is the quotient of a constructible sheaf $\mathcal{G}$. Applying the argument once more to $\mathcal{G} \times_\mathcal{F} \mathcal{G}$ or the kernel of $\mathcal{G} \to \mathcal{F}$ we conclude using that $f^{-1}$ is exact and that the category of constructible sheaves (of sets, abelian groups, or modules) is preserved under finite (co)limits or (co)kernels inside $\Sh(Y_\etale)$, $\Sh(X_\etale)$, $\textit{Ab}(Y_\etale)$, $\textit{Ab}(X_\etale)$, $\textit{Mod}(Y_\etale, \Lambda)$, and $\textit{Mod}(X_\etale, \Lambda)$, see Lemma \ref{lemma-constructible-abelian}.
By Lemma \ref{lemma-constructible-local} it suffices to check this Zariski locally on $Y$ and by Lemma \ref{lemma-check-constructible} we may replace $Y$ by an \'etale cover (the construction of $f_*$ commutes with \'etale localization). A finite \'etale morphism is \'etale locally isomorphic to a disjoint union of isomorphisms, see \'Etale Morphisms, Lemma \ref{etale-lemma-finite-etale-etale-local}. Thus, in the case of sheaves of sets, the lemma says that if $\mathcal{F}_i$, $i = 1, \ldots, n$ are constructible sheaves of sets, then $\prod_{i = 1, \ldots, n} \mathcal{F}_i$ is too. This is clear. Similarly for sheaves of abelian groups and modules.
In the proof of Lemma \ref{lemma-torsion-colimit-constructible} we have seen that sheaves of this form are constructible. For the converse, suppose that for every constructible sheaf of sets $\mathcal{F}$ we can find a surjection $\mathcal{F}_0 \to \mathcal{F}$ with $\mathcal{F}_0$ as in the lemma. Then we find our surjection $\mathcal{F}_1 \to \mathcal{F}_0 \times_\mathcal{F} \mathcal{F}_0$ because the latter is constructible by Lemma \ref{lemma-constructible-abelian}. \medskip\noindent By Topology, Lemma \ref{topology-lemma-constructible-partition-refined-by-stratification} we may choose a finite stratification $X = \coprod_{i \in I} X_i$ such that $\mathcal{F}$ is finite locally constant on each stratum. We will prove the result by induction on the cardinality of $I$. Let $i \in I$ be a minimal element in the partial ordering of $I$. Then $X_i \subset X$ is closed. By induction, there exist finitely many quasi-compact and quasi-separated objects $U_\alpha$ of $(X \setminus X_i)_\etale$ and a surjective map $\coprod h_{U_\alpha} \to \mathcal{F}|_{X \setminus X_i}$. These determine a map $$ \coprod h_{U_\alpha} \to \mathcal{F} $$ which is surjective after restricting to $X \setminus X_i$. By Lemma \ref{lemma-characterize-finite-locally-constant} we see that $\mathcal{F}|_{X_i} = h_V$ for some scheme $V$ finite \'etale over $X_i$. Let $\overline{v}$ be a geometric point of $V$ lying over $\overline{x} \in X_i$. We may think of $\overline{v}$ as an element of the stalk $\mathcal{F}_{\overline{x}} = V_{\overline{x}}$. Thus we can find an \'etale neighbourhood $(U, \overline{u})$ of $\overline{x}$ and a section $s \in \mathcal{F}(U)$ whose stalk at $\overline{x}$ gives $\overline{v}$. Thinking of $s$ as a map $s : h_U \to \mathcal{F}$, restricting to $X_i$ we obtain a morphism $s|_{X_i} : U \times_X X_i \to V$ over $X_i$ which maps $\overline{u}$ to $\overline{v}$. Since $V$ is quasi-compact (finite over the closed subscheme $X_i$ of the quasi-compact scheme $X$) a finite number $s^{(1)}, \ldots, s^{(m)}$ of these sections of $\mathcal{F}$ over $U^{(1)}, \ldots, U^{(m)}$ will determine a jointly surjective map $$ \coprod s^{(j)}|_{X_i} : \coprod U^{(j)} \times_X X_i \longrightarrow V $$ Then we obtain the surjection $$ \coprod h_{U_\alpha} \amalg \coprod h_{U^{(j)}} \to \mathcal{F} $$ as desired.
In the proof of Lemma \ref{lemma-torsion-colimit-constructible} we have seen modules of this form are constructible. Since the category of constructible modules is abelian (Lemma \ref{lemma-constructible-abelian}) it suffices to prove that given a constructible module $\mathcal{F}$ there is a surjection $$ \bigoplus\nolimits_{i = 1, \ldots, n} j_{U_i!}\underline{\Lambda}_{U_i} \longrightarrow \mathcal{F} $$ for some affine objects $U_i$ in $X_\etale$. By Modules on Sites, Lemma \ref{sites-modules-lemma-module-filtered-colimit-constructibles} there is a surjection $$ \Psi : \bigoplus\nolimits_{i \in I} j_{U_i!}\underline{\Lambda}_{U_i} \longrightarrow \mathcal{F} $$ with $U_i$ affine and the direct sum over a possibly infinite index set $I$. For every finite subset $I' \subset I$ set $$ T_{I'} = \text{Supp}(\Coker( \bigoplus\nolimits_{i \in I'} j_{U_i!}\underline{\Lambda}_{U_i} \longrightarrow \mathcal{F})) $$ By the very definition of constructible sheaves, the set $T_{I'}$ is a constructible subset of $X$. We want to show that $T_{I'} = \emptyset$ for some $I'$. Since every stalk $\mathcal{F}_{\overline{x}}$ is a finite type $\Lambda$-module and since $\Psi$ is surjective, for every $x \in X$ there is an $I'$ such that $x \not \in T_{I'}$. In other words we have $\emptyset = \bigcap_{I' \subset I\text{ finite}} T_{I'}$. Since $X$ is a spectral space by Properties, Lemma \ref{properties-lemma-quasi-compact-quasi-separated-spectral} the constructible topology on $X$ is quasi-compact by Topology, Lemma \ref{topology-lemma-constructible-hausdorff-quasi-compact}. Thus $T_{I'} = \emptyset$ for some $I' \subset I$ finite as desired.
The case of sheaves of sets. By Lemma \ref{lemma-category-constructible-sets} it suffices to prove the lemma for $h_U$ where $U$ is a quasi-compact and quasi-separated object of $X_\etale$. Recall that $\Mor(h_U, \mathcal{G}) = \mathcal{G}(U)$. Hence the result follows from Sites, Lemma \ref{sites-lemma-directed-colimits-sections}. \medskip\noindent In the case of abelian sheaves or sheaves of modules, the result follows in the same way using Lemmas \ref{lemma-category-constructible-abelian} and \ref{lemma-category-constructible-modules}. For the case of abelian sheaves, we add that $\Mor(j_{U!}\underline{\mathbf{Z}/n\mathbf{Z}}, \mathcal{G})$ is equal to the $n$-torsion elements of $\mathcal{G}(U)$.
It suffices to prove this when $X$ and $Y$ are affine by Lemma \ref{lemma-constructible-local}. By Lemmas \ref{lemma-finite-pushforward-commutes-with-base-change} and \ref{lemma-check-constructible} we may base change to any affine scheme surjective over $X$. By Lemma \ref{lemma-decompose-quasi-finite-morphism} this reduces us to the case of a finite \'etale morphism (because a thickening leads to an equivalence of \'etale topoi and even small \'etale sites, see Theorem \ref{theorem-topological-invariance}). The finite \'etale case is Lemma \ref{lemma-pushforward-constructible}.
Proof of (1). Denote $f_i : X \to X_i$ the projection maps. There are 3 parts to the proof corresponding to ``faithful'', ``fully faithful'', and ``essentially surjective''. \medskip\noindent Faithful. Choose $0 \in I$ and let $\mathcal{F}_0$, $\mathcal{G}_0$ be constructible sheaves on $X_0$. Suppose that $a, b : \mathcal{F}_0 \to \mathcal{G}_0$ are maps such that $f_0^{-1}a = f_0^{-1}b$. Let $E \subset X_0$ be the set of points $x \in X_0$ such that $a_{\overline{x}} = b_{\overline{x}}$. By Lemma \ref{lemma-support-constructible} the subset $E \subset X_0$ is constructible. By assumption $X \to X_0$ maps into $E$. By Limits, Lemma \ref{limits-lemma-limit-contained-in-constructible} we find an $i \geq 0$ such that $X_i \to X_0$ maps into $E$. Hence $f_{i0}^{-1}a = f_{i0}^{-1}b$. \medskip\noindent Fully faithful. Choose $0 \in I$ and let $\mathcal{F}_0$, $\mathcal{G}_0$ be constructible sheaves on $X_0$. Suppose that $a : f_0^{-1}\mathcal{F}_0 \to f_0^{-1}\mathcal{G}_0$ is a map. We claim there is an $i$ and a map $a_i : f_{i0}^{-1}\mathcal{F}_0 \to f_{i0}^{-1}\mathcal{G}_0$ which pulls back to $a$ on $X$. By Lemma \ref{lemma-category-constructible-sets} we can replace $\mathcal{F}_0$ by a finite coproduct of sheaves represented by quasi-compact and quasi-separated objects of $(X_0)_\etale$. Thus we have to show: If $U_0 \to X_0$ is such an object of $(X_0)_\etale$, then $$ f_0^{-1}\mathcal{G}(U) = \colim_{i \geq 0} f_{i0}^{-1}\mathcal{G}(U_i) $$ where $U = X \times_{X_0} U_0$ and $U_i = X_i \times_{X_0} U_0$. This is a special case of Theorem \ref{theorem-colimit}. \medskip\noindent Essentially surjective. We have to show every constructible $\mathcal{F}$ on $X$ is isomorphic to $f_i^{-1}\mathcal{F}$ for some constructible $\mathcal{F}_i$ on $X_i$. Applying Lemma \ref{lemma-category-constructible-sets} and using the results of the previous two paragraphs, we see that it suffices to prove this for $h_U$ for some quasi-compact and quasi-separated object $U$ of $X_\etale$. In this case we have to show that $U$ is the base change of a quasi-compact and quasi-separated scheme \'etale over $X_i$ for some $i$. This follows from Limits, Lemmas \ref{limits-lemma-descend-finite-presentation} and \ref{limits-lemma-descend-etale}. \medskip\noindent Proof of (3). The argument is very similar to the argument for sheaves of sets, but using Lemma \ref{lemma-category-constructible-modules} instead of Lemma \ref{lemma-category-constructible-sets}. Details omitted. Part (2) follows from part (3) because every constructible abelian sheaf over a quasi-compact scheme is a constructible sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules for some $n$.
Proof of (1). Let $x \in X$ be a point. Let $\mathcal{O}_{X, \overline{x}}$ be a strict henselization of $\mathcal{O}_{X, x}$.  By More on Algebra, Lemma \ref{more-algebra-lemma-henselization-normal} we see that $\mathcal{O}_{X, \overline{x}}$ is a normal domain. Hence $\Spec(K) \times_X \Spec(\mathcal{O}_{X, \overline{x}})$ is irreducible. It follows that the stalk $(g_*\underline{E}_{\underline{x}}$ is equal to $E$, see Theorem \ref{theorem-higher-direct-images}. \medskip\noindent Proof of (2). Since $g$ factors through $j$ there is a map $j_*\underline{E} \to g_*\underline{E}$. This map is injective because for every scheme $V$ \'etale over $X$ the set $\Spec(K) \times_X V$ is dense in $U \times_X V$. On the other hand, we have a map $\underline{E} \to j_*\underline{E}$ and we conclude.
Proof of (1). We can write $\mathcal{F} = \colim_{i \in I} \mathcal{F}_i$ with $\mathcal{F}_i$ constructible abelian by Lemma \ref{lemma-torsion-colimit-constructible}. Choose $i \in I$. Since $\mathcal{F}|_\eta$ is zero by assumption, we see that there exists an $i'(i) \geq i$ such that $\mathcal{F}_i|_\eta \to \mathcal{F}_{i'(i)}|_\eta$ is zero, see Lemma \ref{lemma-colimit-constructible}. Then $\mathcal{G}_i = \Im(\mathcal{F}_i \to \mathcal{F}_{i'(i)})$ is a constructible abelian sheaf (Lemma \ref{lemma-constructible-abelian}) whose stalk at $\eta$ is zero. Hence the support $E_i$ of $\mathcal{G}_i$ is a constructible subset of $X$ not containing $\eta$. Since $\eta$ is a generic point of an irreducible component of $X$, we see that $\eta \not \in Z_i = \overline{E_i}$ by Topology, Lemma \ref{topology-lemma-generic-point-in-constructible}. Define a new directed set $I'$ by using the set $I$ with ordering defined by the rule $i_1$ is bigger or equal to $i_2$ if and only if $i_1 \geq i'(i_2)$. Then the sheaves $\mathcal{G}_i$ form a system over $I'$ with colimit $\mathcal{F}$ and the proof is complete. \medskip\noindent The proof in case (2) is exactly the same and we omit it.