hcju/stacksqa huggingface dataset:
Proof of (1). Because we have the ascending chain condition for subsheaves of $\mathcal{F}$ (Lemma \ref{lemma-constructible-over-noetherian-noetherian}), it suffices to show that for every point $x \in X$ we can find a map $\varphi : \mathcal{F} \to f_*\underline{E}$ where $f : Y \to X$ is finite and $E$ is a finite set such that $\varphi_{\overline{x}} : \mathcal{F}_{\overline{x}} \to (f_*S)_{\overline{x}}$ is injective. (This argument can be avoided by picking a partition of $X$ as in Lemma \ref{lemma-constructible-quasi-compact-quasi-separated} and constructing a $Y_i \to X$ for each irreducible component of each part.) Let $Z \subset X$ be the induced reduced scheme structure (Schemes, Definition \ref{schemes-definition-reduced-induced-scheme}) on $\overline{\{x\}}$. Since $\mathcal{F}$ is constructible, there is a finite separable extension $K/\kappa(x)$ such that $\mathcal{F}|_{\Spec(K)}$ is the constant sheaf with value $E$ for some finite set $E$. Let $Y \to Z$ be the normalization of $Z$ in $\Spec(K)$. By Morphisms, Lemma \ref{morphisms-lemma-normal-normalization} we see that $Y$ is a normal integral scheme. As $K/\kappa(x)$ is a finite extension, it is clear that $K$ is the function field of $Y$. Denote $g : \Spec(K) \to Y$ the inclusion. The map $\mathcal{F}|_{\Spec(K)} \to \underline{E}$ is adjoint to a map $\mathcal{F}|_Y \to g_*\underline{E} = \underline{E}$ (Lemma \ref{lemma-push-constant-sheaf-from-open}). This in turn is adjoint to a map $\varphi : \mathcal{F} \to f_*\underline{E}$. Observe that the stalk of $\varphi$ at a geometric point $\overline{x}$ is injective: we may take a lift $\overline{y} \in Y$ of $\overline{x}$ and the commutative diagram $$ \xymatrix{ \mathcal{F}_{\overline{x}} \ar@{=}[r] \ar[d] & (\mathcal{F}|_Y)_{\overline{y}} \ar@{=}[d] \\ (f_*\underline{E})_{\overline{x}} \ar[r] & \underline{E}_{\overline{y}} } $$ proves the injectivity. We are not yet done, however, as the morphism $f : Y \to Z$ is integral but in general not finite\footnote{If $X$ is a Nagata scheme, for example of finite type over a field, then $Y \to Z$ is finite.}. \medskip\noindent To fix the problem stated in the last sentence of the previous paragraph, we write $Y = \lim_{i \in I} Y_i$ with $Y_i$ irreducible, integral, and finite over $Z$. Namely, apply Properties, Lemma \ref{properties-lemma-integral-algebra-directed-colimit-finite} to $f_*\mathcal{O}_Y$ viewed as a sheaf of $\mathcal{O}_Z$-algebras and apply the functor $\underline{\Spec}_Z$. Then $f_*\underline{E} = \colim f_{i, *}\underline{E}$ by Lemma \ref{lemma-relative-colimit}. By Lemma \ref{lemma-constructible-is-compact} the map $\mathcal{F} \to f_*\underline{E}$ factors through $f_{i, *}\underline{E}$ for some $i$. Since $Y_i \to Z$ is a finite morphism of integral schemes and since the function field extension induced by this morphism is finite separable, we see that the morphism is finite \'etale over a nonempty open of $Z$ (use Algebra, Lemma \ref{algebra-lemma-smooth-at-generic-point}; details omitted). This finishes the proof of (1). \medskip\noindent The proofs of (2) and (3) are identical to the proof of (1).
We will reduce this lemma to the Noetherian case by absolute Noetherian approximation. Namely, by Limits, Proposition \ref{limits-proposition-approximate} we can write $X = \lim_{t \in T} X_t$ with each $X_t$ of finite type over $\Spec(\mathbf{Z})$ and with affine transition morphisms. By Lemma \ref{lemma-category-is-colimit} the category of constructible sheaves (of sets, abelian groups, or $\Lambda$-modules) on $X_\etale$ is the colimit of the corresponding categories for $X_t$. Thus our constructible sheaf $\mathcal{F}$ is the pullback of a similar constructible sheaf $\mathcal{F}_t$ over $X_t$ for some $t$. Then we apply the Noetherian case (Lemma \ref{lemma-constructible-maps-into-constant}) to find an injection $$ \mathcal{F}_t \longrightarrow \prod\nolimits_{i = 1, \ldots, n} f_{i, *}\underline{E_i} \quad\text{or}\quad \mathcal{F}_t \longrightarrow \bigoplus\nolimits_{i = 1, \ldots, n} f_{i, *}\underline{M_i} $$ over $X_t$ for some finite morphisms $f_i : Y_i \to X_t$. Since $X_t$ is Noetherian the morphisms $f_i$ are of finite presentation. Since pullback is exact and since formation of $f_{i, *}$ commutes with base change (Lemma \ref{lemma-finite-pushforward-commutes-with-base-change}), 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}. By Proposition \ref{proposition-constructible-over-noetherian} the image $\mathcal{F}'_i \subset \mathcal{F}$ of the map $\mathcal{F}_i \to \mathcal{F}$ is constructible. Thus $\mathcal{F} = \colim \mathcal{F}'_i$ and the support of $\mathcal{F}'_i$ is contained in $E$. Since the support of $\mathcal{F}'_i$ is constructible (by our definition of constructible sheaves), we see that its closure is also contained in $E$, see for example Topology, Lemma \ref{topology-lemma-constructible-stable-specialization-closed}. \medskip\noindent The proof in case (2) is exactly the same and we omit it.
Represent $K$ by a complex $\mathcal{K}^\bullet$ of sheaves of $\Lambda$-modules. Consider the surjection $$ \Ker(\mathcal{K}^b \to \mathcal{K}^{b + 1}) \longrightarrow H^b(K) $$ By Modules on Sites, Lemma \ref{sites-modules-lemma-module-quotient-direct-sum} we may choose a surjection $\bigoplus_{i \in I} j_{U_i!} \underline{\Lambda} \to \Ker(\mathcal{K}^b \to \mathcal{K}^{b + 1})$ with $U_i$ affine. For $I' \subset I$ finite, denote $\mathcal{H}_{I'} \subset H^b(K)$ the image of $\bigoplus_{i \in I'} j_{U_i!} \underline{\Lambda}$. By Lemma \ref{lemma-colimit-constructible} we see that $\mathcal{H}_{I'} = H^b(K)$ for some $I' \subset I$ finite. The lemma follows taking $\mathcal{F} = \bigoplus_{i \in I'} j_{U_i!} \underline{\Lambda}$.
It is clear that (2) implies (3) and that (3) implies (1). Assume $K$ is in $D_c^-(X_\etale, \Lambda)$. Say $H^i(K) = 0$ for $i > b$. By induction on $a$ we will construct a complex $\mathcal{F}^a \to \ldots \to \mathcal{F}^b$ such that each $\mathcal{F}^i$ is a finite direct sum of $j_{U!}\underline{\Lambda}$ with $U \in \Ob(X_\etale)$ affine and a map $\mathcal{F}^\bullet \to K$ which induces an isomorphism $H^i(\mathcal{F}^\bullet) \to H^i(K)$ for $i > a$ and a surjection $H^a(\mathcal{F}^\bullet) \to H^a(K)$. For $a = b$ this can be done by Lemma \ref{lemma-one-constructible}. Given such a datum choose a distinguished triangle $$ \mathcal{F}^\bullet \to K \to L \to \mathcal{F}^\bullet[1] $$ Then we see that $H^i(L) = 0$ for $i \geq a$. Choose $\mathcal{F}^{a - 1}[-a +1] \to L$ as in Lemma \ref{lemma-one-constructible}. The composition $\mathcal{F}^{a - 1}[-a +1] \to L \to \mathcal{F}^\bullet$ corresponds to a map $\mathcal{F}^{a - 1} \to \mathcal{F}^a$ such that the composition with $\mathcal{F}^a \to \mathcal{F}^{a + 1}$ is zero. By TR4 we obtain a map $$ (\mathcal{F}^{a - 1} \to \ldots \to \mathcal{F}^b) \to K $$ in $D(X_\etale, \Lambda)$. This finishes the induction step and the proof of the lemma.
It is clear that a finite complex of constructible flat sheaves of $\Lambda$-modules has finite tor dimension. It is also clear that it is an object of $D_c(X_\etale, \Lambda)$. Thus we see that (2) implies (1). \medskip\noindent Assume (1). Choose $a, b \in \mathbf{Z}$ such that $H^i(K \otimes_\Lambda^\mathbf{L} \mathcal{G}) = 0$ if $i \not \in [a, b]$ for all sheaves of $\Lambda$-modules $\mathcal{G}$. We will prove the final assertion holds by induction on $b - a$. If $a = b$, then $K = H^a(K)[-a]$ is a flat constructible sheaf and the result holds. Next, assume $b > a$. Represent $K$ by a complex $\mathcal{K}^\bullet$ of sheaves of $\Lambda$-modules. Consider the surjection $$ \Ker(\mathcal{K}^b \to \mathcal{K}^{b + 1}) \longrightarrow H^b(K) $$ By Lemma \ref{lemma-category-constructible-modules} we can find finitely many affine schemes $U_i$ \'etale over $X$ and a surjection $\bigoplus j_{U_i!}\underline{\Lambda}_{U_i} \to H^b(K)$. After replacing $U_i$ by standard \'etale coverings $\{U_{ij} \to U_i\}$ we may assume this surjection lifts to a map $\mathcal{F} = \bigoplus j_{U_i!}\underline{\Lambda}_{U_i} \to \Ker(\mathcal{K}^b \to \mathcal{K}^{b + 1})$. This map determines a distinguished triangle $$ \mathcal{F}[-b] \to K \to L \to \mathcal{F}[-b + 1] $$ in $D(X_\etale, \Lambda)$. Since $D_{ctf}(X_\etale, \Lambda)$ is a triangulated subcategory we see that $L$ is in it too. In fact $L$ has tor amplitude in $[a, b - 1]$ as $\mathcal{F}$ surjects onto $H^b(K)$ (details omitted). By induction hypothesis we can find a finite complex $\mathcal{F}^a \to \ldots \to \mathcal{F}^{b - 1}$ of flat constructible sheaves of $\Lambda$-modules representing $L$. The map $L \to \mathcal{F}[-b + 1]$ corresponds to a map $\mathcal{F}^b \to \mathcal{F}$ annihilating the image of $\mathcal{F}^{b - 1} \to \mathcal{F}^b$. Then it follows from axiom TR3 that $K$ is represented by the complex $$ \mathcal{F}^a \to \ldots \to \mathcal{F}^{b - 1} \to \mathcal{F}^b $$ which finishes the proof.
Choose an \'etale covering $\{U_i \to X\}$ such that $K|_{U_i}$ is constant, say $K|_{U_i} \cong \underline{M_i^\bullet}_{U_i}$ for some finite complex of finite $\Lambda$-modules $M_i^\bullet$. See Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-locally-constant}. Observe that $U_i \times_X U_j$ is empty if $M_i^\bullet$ is not isomorphic to $M_j^\bullet$ in $D(\Lambda)$. For each complex of $\Lambda$-modules $M^\bullet$ let $I_{M^\bullet} = \{i \in I \mid M_i^\bullet \cong M^\bullet\text{ in }D(\Lambda)\}$. As \'etale morphisms are open we see that $U_{M^\bullet} = \bigcup_{i \in I_{M^\bullet}} \Im(U_i \to X)$ is an open subset of $X$. Then $X = \coprod U_{M^\bullet}$ is a disjoint open covering of $X$. As $X$ is connected only one $U_{M^\bullet}$ is nonempty. As $K$ is in $D_{ctf}(X_\etale, \Lambda)$ we see that $M^\bullet$ is a perfect complex of $\Lambda$-modules, see More on Algebra, Lemma \ref{more-algebra-lemma-perfect}. Hence we may assume $M^\bullet$ is a finite complex of finite projective $\Lambda$-modules.
To prove (1) we write $\mathcal{F} = \bigcup \mathcal{F}[n]$ where $\mathcal{F}[d]$ is the $d$-torsion subsheaf. By Lemma \ref{lemma-colimit} we have $H^n_\etale(X, \mathcal{F}) = \colim H^n_\etale(X, \mathcal{F}[d])$. This proves (1) as $H^n_\etale(X, \mathcal{F}[d])$ is annihilated by $d$. \medskip\noindent To prove (2) we can use the spectral sequence $E_2^{p, q} = H^p_\etale(X, H^q(K))$ converging to $H^n_\etale(X, K)$ (Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}) and the result for sheaves.
Proof of (1). Recall that $R^nf_*\mathcal{F}$ is the sheaf associated to the presheaf $V \mapsto H^n_\etale(X \times_Y V, \mathcal{F})$ on $Y_\etale$. See Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-higher-direct-images}. If we choose $V$ affine, then $X \times_Y V$ is quasi-compact and quasi-separated because $f$ is, hence we can apply Lemma \ref{lemma-torsion-cohomology} to see that $H^n_\etale(X \times_Y V, \mathcal{F})$ is torsion. \medskip\noindent Proof of (2). Recall that $R^nf_*K$ is the sheaf associated to the presheaf $V \mapsto H^n_\etale(X \times_Y V, K)$ on $Y_\etale$. See Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-unbounded-describe-higher-direct-images}. If we choose $V$ affine, then $X \times_Y V$ is quasi-compact and quasi-separated because $f$ is, hence we can apply Lemma \ref{lemma-torsion-cohomology} to see that $H^n_\etale(X \times_Y V, K)$ is torsion.
Recall that $i_{small}^{-1}$ and $i_{small, *}$ is an adjoint pair of exact functors such that $i_{small}^{-1}i_{small, *}$ is isomorphic to the identify functor on abelian sheaves. See Proposition \ref{proposition-closed-immersion-pushforward} and Lemma \ref{lemma-stalk-pullback}. Thus $i_{small, *} : D(Z_\etale) \to D_Z(X_\etale)$ is fully faithful and $i_{small}^{-1}$ determines a left inverse. On the other hand, suppose that $K$ is an object of $D_Z(X_\etale)$ and consider the adjunction map $K \to i_{small, *}i_{small}^{-1}K$. Using exactness of $i_{small, *}$ and $i_{small}^{-1}$ this induces the adjunction maps $H^n(K) \to i_{small, *}i_{small}^{-1}H^n(K)$ on cohomology sheaves. Since these cohomology sheaves are supported on $Z$ we see these adjunction maps are isomorphisms and we conclude that $D(Z_\etale) \to D_Z(X_\etale)$ is an equivalence. \medskip\noindent To finish the proof we have to show that $R\mathcal{H}_Z(K) = i_{small}^{-1}K$ if $K$ is an object of $D_Z(X_\etale)$. To do this we can use that $K = i_{small, *}i_{small}^{-1}K$ as we've just proved this is the case. Then we can choose a K-injective representative $\mathcal{I}^\bullet$ for $i_{small}^{-1}K$. Since $i_{small, *}$ is the right adjoint to the exact functor $i_{small}^{-1}$, the complex $i_{small, *}\mathcal{I}^\bullet$ is K-injective (Derived Categories, Lemma \ref{derived-lemma-adjoint-preserve-K-injectives}). We see that $R\mathcal{H}_Z(K)$ is computed by $\mathcal{H}_Z(i_{small, *}\mathcal{I}^\bullet) = \mathcal{I}^\bullet$ as desired.
Let $j : U \to X$ be the inclusion of the complement of $Z$. The statement (1) on cohomology groups follows from the long exact sequences for cohomology with supports and the agreements $H^q(X_\etale, \mathcal{F}^a) = H^q(X, \mathcal{F})$ and $H^q(U_\etale, \mathcal{F}^a) = H^q(U, \mathcal{F})$, see Theorem \ref{theorem-zariski-fpqc-quasi-coherent}. If $j : U \to X$ is a quasi-compact morphism, i.e., if $U \subset X$ is retrocompact, then $R^qj_*$ transforms quasi-coherent sheaves into quasi-coherent sheaves (Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherence-higher-direct-images}) and commutes with taking associated sheaf on \'etale sites (Descent, Lemma \ref{descent-lemma-higher-direct-images-small-etale}). We conclude by applying Lemma \ref{lemma-cohomology-with-support-triangle}.
Let $\epsilon : S_\etale \to S_{Zar}$ be the morphism of sites given by the inclusion functor. The Zariski sheaf $R^p\epsilon_*\mathcal{F}$ is the sheaf associated to the presheaf $U \mapsto H^p_\etale(U, \mathcal{F})$. Thus the stalk at $x \in X$ is $\colim H^p_\etale(U, \mathcal{F}) = H^p_\etale(\Spec(\mathcal{O}_{X, x}), \mathcal{G}_x)$ where $\mathcal{G}_x$ denotes the pullback of $\mathcal{F}$ to $\Spec(\mathcal{O}_{X, x})$, see Lemma \ref{lemma-directed-colimit-cohomology}. Thus the higher direct images of $R^p\epsilon_*\mathcal{F}$ are zero by Lemma \ref{lemma-vanishing-etale-cohomology-strictly-henselian} and we conclude by the Leray spectral sequence.
Condition (1) implies that the underlying topological space of $S$ is profinite, see Algebra, Lemma \ref{algebra-lemma-ring-with-only-minimal-primes}. Thus the higher cohomology groups of an abelian sheaf on the topological space $S$ (i.e., Zariski cohomology) is trivial, see Cohomology, Lemma \ref{cohomology-lemma-vanishing-for-profinite}. The local rings are strictly henselian by Algebra, Lemma \ref{algebra-lemma-local-dimension-zero-henselian}. Thus \'etale cohomology of $S$ is computed by Zariski cohomology by Lemma \ref{lemma-local-rings-strictly-henselian} and the proof is done.
Let $R$ be a normal domain whose fraction field is separably algebraically closed. Let $R \to A$ be an \'etale ring map. Then $A \otimes_R K$ is as a $K$-algebra a finite product $\prod_{i = 1, \ldots, n} K$ of copies of $K$. Let $e_i$, $i = 1, \ldots, n$ be the corresponding idempotents of $A \otimes_R K$. Since $A$ is normal (Algebra, Lemma \ref{algebra-lemma-normal-goes-up}) the idempotents $e_i$ are in $A$ (Algebra, Lemma \ref{algebra-lemma-normal-ring-integrally-closed}). Hence $A = \prod Ae_i$ and we may assume $A \otimes_R K = K$. Since $A \subset A \otimes_R K = K$ (by flatness of $R \to A$ and since $R \subset K$) we conclude that $A$ is a domain. By the same argument we conclude that $A \otimes_R A \subset (A \otimes_R A) \otimes_R K = K$. It follows that the map $A \otimes_R A \to A$ is injective as well as surjective. Thus $R \to A$ defines an open immersion by Morphisms, Lemma \ref{morphisms-lemma-universally-injective} and \'Etale Morphisms, Theorem \ref{etale-theorem-etale-radicial-open}. \medskip\noindent Let $f : U \to X$ be a separated \'etale morphism. Let $\eta \in X$ be the generic point and let $f^{-1}(\{\eta\}) = \{\xi_i\}_{i \in I}$. The result of the previous paragraph shows the following: For any affine open $U' \subset U$ whose image in $X$ is contained in an affine we have $U' = \coprod_{i \in I} U'_i$ where $U'_i$ is the set of point of $U'$ which are specializations of $\xi_i$. Moreover, the morphism $U'_i \to X$ is an open immersion. It follows that $U_i = \overline{\{\xi_i\}}$ is an open and closed subscheme of $U$ and that $U_i \to X$ is locally on the source an isomorphism. By Morphisms, Lemma \ref{morphisms-lemma-distinct-local-rings} the fact that $U_i \to X$ is separated, implies that $U_i \to X$ is injective and we conclude that $U_i \to X$ is an open immersion, i.e., (1) holds. \medskip\noindent Part (2) follows from part (1) and the description of the strict henselization of $\mathcal{O}_{X, x}$ as the local ring at $\overline{x}$ on the \'etale site of $X$ (Lemma \ref{lemma-describe-etale-local-ring}). It can also be proved directly, see Fundamental Groups, Lemma \ref{pione-lemma-normal-local-domain-separablly-closed-fraction-field}.
Recall that $R^qf_*\underline{M}$ is the sheaf associated to the presheaf $V \mapsto H^q_\etale(V \times_Y X, M)$ on $Y_\etale$, see Lemma \ref{lemma-higher-direct-images}. If $V$ is affine, then $V \times_Y X \to X$ is separated and \'etale. Hence $V \times_Y X = \coprod U_i$ is a disjoint union of open subschemes $U_i$ of $X$, see Lemma \ref{lemma-normal-scheme-with-alg-closed-function-field}. By Lemma \ref{lemma-local-rings-strictly-henselian} we see that $H^q_\etale(U_i, M)$ is equal to $H^q_{Zar}(U_i, M)$. This vanishes by Cohomology, Lemma \ref{cohomology-lemma-irreducible-constant-cohomology-zero}.
By Algebra, Lemma \ref{algebra-lemma-lift-etale} we can lift $V$ to an affine scheme $U$ \'etale over $X$. Apply Lemma \ref{lemma-normal-scheme-with-alg-closed-function-field} to $U \to X$ to get the first statement. \medskip\noindent The final statement is a consequence of the first. Let $V = \coprod_{i = 1, \ldots, n} V_i$ be a finite decomposition into open and closed subschemes with $V_i \to Z$ an open immersion. As $V \to Z$ is finite we see that $V_i \to Z$ is also closed. Let $U_i \subset Z$ be the image. Then we have a decomposition into open and closed subschemes $$ Z = \coprod\nolimits_{(A, B)} \bigcap\nolimits_{i \in A} U_i \cap \bigcap\nolimits_{i \in B} U_i^c $$ where the disjoint union is over $\{1, \ldots, n\} = A \amalg B$ where $A$ has at least one element. Each of the strata is contained in a single $U_i$ and we find our section.
By Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-torsors-h1} an element of $H^1_\etale(Z, \underline{M})$ corresponds to a $\underline{M}$-torsor $\mathcal{F}$ on $Z_\etale$. Such a torsor is clearly a finite locally constant sheaf. Hence $\mathcal{F}$ is representable by a scheme $V$ finite \'etale over $Z$, Lemma \ref{lemma-characterize-finite-locally-constant}. Of course $V \to Z$ is surjective as a torsor is locally trivial. Since $V \to Z$ has a section by Lemma \ref{lemma-closed-of-affine-normal-scheme-with-alg-closed-function-field} we are done.
We have seen that the result is true for $H^1$ in Lemma \ref{lemma-gabber-for-h1-absolutely-algebraically-closed}. We will prove the result for $q \geq 2$ by induction on $q$. Let $\xi \in H^q_\etale(Z, \underline{M})$. \medskip\noindent Let $X = \Spec(R)$. Let $I \subset R$ be the set of elements $f \in R$ such that $\xi|_{Z \cap D(f)} = 0$. All local rings of $Z$ are strictly henselian by Lemma \ref{lemma-normal-scheme-with-alg-closed-function-field} and Algebra, Lemma \ref{algebra-lemma-quotient-strict-henselization}. Hence \'etale cohomology on $Z$ or open subschemes of $Z$ is equal to Zariski cohomology, see Lemma \ref{lemma-local-rings-strictly-henselian}. In particular $\xi$ is Zariski locally trivial. It follows that for every prime $\mathfrak p$ of $R$ there exists an $f \in I$ with $f \not \in \mathfrak p$. Thus if we can show that $I$ is an ideal, then $1 \in I$ and we're done. It is clear that $f \in I$, $r \in R$ implies $rf \in I$. Thus we now assume that $f, g \in I$ and we show that $f + g \in I$. Note that $$ D(f + g) \cap Z = D(f(f + g)) \cap Z \cup D(g(f + g)) \cap Z $$ By Mayer-Vietoris (Cohomology, Lemma \ref{cohomology-lemma-mayer-vietoris} which applies as \'etale cohomology on open subschemes of $Z$ equals Zariski cohomology) we have an exact sequence $$ \xymatrix{ H^{q - 1}_\etale(D(fg(f + g)) \cap Z, \underline{M}) \ar[d] \\ H^q_\etale(D(f + g) \cap Z, \underline{M}) \ar[d] \\ H^q_\etale(D(f(f + g)) \cap Z, \underline{M}) \oplus H^q_\etale(D(g(f + g)) \cap Z, \underline{M}) } $$ and the result follows as the first group is zero by induction.
Write $X = \Spec(A)$. Write $A = \mathbf{Z}[x_i]/J$ for some ideal $J$. Let $R$ be the integral closure of $\mathbf{Z}[x_i]$ in an algebraic closure of the fraction field of $\mathbf{Z}[x_i]$. Let $A' = R/JR$ and set $X' = \Spec(A')$. This gives an example as in (1) by Lemma \ref{lemma-gabber-for-absolutely-algebraically-closed}. \medskip\noindent Proof of (2). Let $X' \to X$ be the integral surjective morphism we found above. Certainly, $\xi$ maps to zero in $H^q_\etale(X' \times_X Z, \underline{M})$. We may write $X'$ as a limit $X' = \lim X'_i$ of schemes finite and of finite presentation over $X$; this is easy to do in our current affine case, but it is a special case of the more general Limits, Lemma \ref{limits-lemma-integral-limit-finite-and-finite-presentation}. By Lemma \ref{lemma-directed-colimit-cohomology} we see that $\xi$ maps to zero in $H^q_\etale(X'_i \times_X Z, \underline{M})$ for some $i$ large enough.