hcju/stacksqa huggingface dataset:
Parts (1)(b) and (2)(b) follow formally from the fact that the restriction functor $\pi_{X, *} = i_X^{-1}$ is a right adjoint of the exact functor $\pi_X^{-1}$, see Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives} and Derived Categories, Lemma \ref{derived-lemma-adjoint-preserve-K-injectives}. \medskip\noindent Parts (1)(a) and (2)(a) can be seen in two ways. First proof: We can use that $i_f^{-1}$ is a right adjoint of the exact functor $i_{f, !}$. This functor is constructed in Topologies, Lemma \ref{topologies-lemma-put-in-T-etale} for sheaves of sets and for abelian sheaves in Modules on Sites, Lemma \ref{sites-modules-lemma-g-shriek-adjoint}. It is shown in Modules on Sites, Lemma \ref{sites-modules-lemma-exactness-lower-shriek} that it is exact. Second proof. We can use that $i_f = i_Y \circ f_{big}$ as is shown in Topologies, Lemma \ref{topologies-lemma-morphism-big-small-etale}. Since $f_{big}$ is a localization, we see that pullback by it preserves injectives and K-injectives, see Cohomology on Sites, Lemmas \ref{sites-cohomology-lemma-cohomology-of-open} and \ref{sites-cohomology-lemma-restrict-K-injective-to-open}. Then we apply the already proved parts (1)(b) and (2)(b) to the functor $i_Y^{-1}$ to conclude. \medskip\noindent To see a counter example for the case of modules we refer to \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-compare-injectives}.
Part (1) follows from Lemma \ref{lemma-compare-injectives} and (\ref{equation-compare-big-small}) on choosing a K-injective complex of abelian sheaves representing $K$. \medskip\noindent Part (3) follows from Lemma \ref{lemma-compare-injectives} and Topologies, Lemma \ref{topologies-lemma-morphism-big-small-cartesian-diagram-etale} on choosing a K-injective complex of abelian sheaves representing $K$. \medskip\noindent Part (5) is Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-localize-cartesian-square}. \medskip\noindent Part (6) is  Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-localize-cartesian-square-modules}. \medskip\noindent Part (2) can be proved as follows. Above we have seen that $\pi_X \circ f_{big} = f_{small} \circ \pi_Y$ as morphisms of ringed sites. Hence we obtain $R\pi_{X, *} \circ Rf_{big, *} = Rf_{small, *} \circ R\pi_{Y, *}$ by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-derived-pushforward-composition}. Since the restriction functors $\pi_{X, *}$ and $\pi_{Y, *}$ are exact, we conclude. \medskip\noindent Part (4) follows from part (6) and part (2) applied to $f' : Y' \to X'$.
To prove (5) represent $M$ by a K-injective complex of abelian sheaves and apply Lemma \ref{lemma-compare-injectives} and work out the definitions. Part (3) follows from this as $i_f^{-1}\pi_X^{-1} = f_{small}^{-1}$. Part (1) is a special case of (3). \medskip\noindent Part (6) follows from the very general Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-pullback-same-cohomology}. Then part (4) follows because $Lf_{small}^* = i_f^* \circ L\pi_X^*$. Part (2) is a special case of (4).
Proof of (1). Let $\mathcal{F}$ be a sheaf on $Y_\etale$. Let $g : X' \to X$ be an object of $(\textit{Spaces}/X)_\etale$. Consider the fibre product $$ \xymatrix{ Y' \ar[r]_{f'} \ar[d]_{g'} & X' \ar[d]^g \\ Y \ar[r]^f & X } $$ Then we have $$ (f_{big, *}\pi_Y^{-1}\mathcal{F})(X') = (\pi_Y^{-1}\mathcal{F})(Y') = ((g'_{small})^{-1}\mathcal{F})(Y')  = (f'_{small, *}(g'_{small})^{-1}\mathcal{F})(X') $$ the second equality by Lemma \ref{lemma-describe-pullback}. On the other hand $$ (\pi_X^{-1}f_{small, *}\mathcal{F})(X') = (g_{small}^{-1}f_{small, *}\mathcal{F})(X') $$ again by Lemma \ref{lemma-describe-pullback}. Hence by proper base change for sheaves of sets (Lemma \ref{lemma-proper-base-change-f-star}) we conclude the two sets are canonically isomorphic. The isomorphism is compatible with restriction mappings and defines an isomorphism $\pi_X^{-1}f_{small, *}\mathcal{F} = f_{big, *}\pi_Y^{-1}\mathcal{F}$. Thus an isomorphism of functors $\pi_X^{-1} \circ f_{small, *} = f_{big, *} \circ \pi_Y^{-1}$. \medskip\noindent Proof of (2). There is a canonical base change map $\pi_X^{-1}Rf_{small, *}K \to Rf_{big, *}\pi_Y^{-1}K$ for any $K$ in $D(Y_\etale)$, see Cohomology on Sites, Remark \ref{sites-cohomology-remark-base-change}. To prove it is an isomorphism, it suffices to prove the pull back of the base change map by $i_g : \Sh(X'_\etale) \to \Sh((\Sch/X)_\etale)$ is an isomorphism for any object $g : X' \to X$ of $(\Sch/X)_\etale$. Let $T', g', f'$ be as in the previous paragraph. The pullback of the base change map is \begin{align*} g_{small}^{-1}Rf_{small, *}K & = i_g^{-1}\pi_X^{-1}Rf_{small, *}K \\ & \to i_g^{-1}Rf_{big, *}\pi_Y^{-1}K \\ & = Rf'_{small, *}(i_{g'}^{-1}\pi_Y^{-1}K) \\ & = Rf'_{small, *}((g'_{small})^{-1}K) \end{align*} where we have used $\pi_X \circ i_g = g_{small}$, $\pi_Y \circ i_{g'} = g'_{small}$, and Lemma \ref{lemma-compare-higher-direct-image}. This map is an isomorphism by the proper base change theorem (Lemma \ref{lemma-proper-base-change}) provided $K$ is bounded below and the cohomology sheaves of $K$ are torsion. \medskip\noindent Proof of (3). If $f$ is finite, then the functors $f_{small, *}$ and $f_{big, *}$ are exact. This follows from Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-finite-higher-direct-image-zero} for $f_{small}$. Since any base change $f'$ of $f$ is finite too, we conclude from Lemma \ref{lemma-compare-higher-direct-image} part (3) that $f_{big, *}$ is exact too (as the higher derived functors are zero). Thus this case follows from part (1).
We have $a_X^{-1}\mathcal{F} = \epsilon_X^{-1} \pi_X^{-1}\mathcal{F}$. By Lemma \ref{lemma-describe-pullback} the \'etale sheaf $\pi_X^{-1}\mathcal{F}$ is a sheaf for the fppf topology and therefore is equal to $a_X^{-1}\mathcal{F}$ (as pulling back by $\epsilon_X$ is given by fppf sheafification). Recall moreover that $\epsilon_{X, *}$ is the identity on underlying presheaves. Now part (1) is immediate from the explicit description of $\pi_X^{-1}$ in Lemma \ref{lemma-describe-pullback}. \medskip\noindent We will prove part (2) by reducing it to the case of schemes -- see part (1) of \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-V-C-all-n-etale-fppf}. This will ``clearly work'' as every algebraic space is \'etale locally a scheme. The details are given below but we urge the reader to skip the proof. \medskip\noindent For an abelian sheaf $\mathcal{H}$ on $(\textit{Spaces}/X)_{fppf}$ the higher direct image $R^p\epsilon_{X, *}\mathcal{H}$ is the sheaf associated to the presheaf $U \mapsto H^p_{fppf}(U, \mathcal{H})$ on $(\textit{Spaces}/X)_\etale$. See Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-higher-direct-images}. Since every object of $(\textit{Spaces}/X)_\etale$ has a covering by schemes, it suffices to prove that given $U/X$ a scheme and $\xi \in H^p_{fppf}(U, a_X^{-1}\mathcal{F})$ we can find an \'etale covering $\{U_i \to U\}$ such that $\xi$ restricts to zero on $U_i$. We have \begin{align*} H^p_{fppf}(U, a_X^{-1}\mathcal{F}) & = H^p((\textit{Spaces}/U)_{fppf}, (a_X^{-1}\mathcal{F})|_{\textit{Spaces}/U}) \\ & = H^p((\Sch/U)_{fppf}, (a_X^{-1}\mathcal{F})|_{\Sch/U}) \end{align*} where the second identification is Lemma \ref{lemma-compare-cohomology-other-topologies} and the first is a general fact about restriction (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-of-open}). Looking at the first paragraph and the corresponding result in the case of schemes (\'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-describe-pullback-pi-fppf}) we conclude that the sheaf $(a_X^{-1}\mathcal{F})|_{\Sch/U}$ matches the pullback by the ``schemes version of $a_U$''. Therefore we can find an \'etale covering $\{U_i \to U\}$ such that our class dies in $H^p((\Sch/U_i)_{fppf}, (a_X^{-1}\mathcal{F})|_{\Sch/U_i})$ for each $i$, see \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-V-C-all-n-etale-fppf} (the precise statement one should use here is that $V_n$ holds for all $n$ which is the statement of part (2) for the case of schemes). Transporting back (using the same formulas as above but now for $U_i$) we conclude $\xi$ restricts to zero over $U_i$ as desired.
We only prove the second statement; the first is easier and proved in exactly the same manner. There is an immediate reduction to the case where $K$ is given by a single abelian sheaf. Namely, represent $K$ by a bounded below complex $\mathcal{F}^\bullet$. By the case of a sheaf we see that $\mathcal{F}^n = a_{X, *} a_X^{-1} \mathcal{F}^n$ and that the sheaves $R^qa_{X, *}a_X^{-1}\mathcal{F}^n$ are zero for $q > 0$. By Leray's acyclicity lemma (Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity}) applied to $a_X^{-1}\mathcal{F}^\bullet$ and the functor $a_{X, *}$ we conclude. From now on assume $K = \mathcal{F}$. \medskip\noindent By Lemma \ref{lemma-comparison-fppf-etale} we have $a_{X, *}a_X^{-1}\mathcal{F} = \mathcal{F}$. Thus it suffices to show that $R^qa_{X, *}a_X^{-1}\mathcal{F} = 0$ for $q > 0$. For this we can use $a_X = \epsilon_X \circ \pi_X$ and the Leray spectral sequence (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-relative-Leray}). By Lemma \ref{lemma-comparison-fppf-etale} we have $R^i\epsilon_{X, *}(a_X^{-1}\mathcal{F}) = 0$ for $i > 0$. We have $\epsilon_{X, *}a_X^{-1}\mathcal{F} = \pi_X^{-1}\mathcal{F}$ and by Lemma \ref{lemma-cohomological-descent-etale} we have $R^j\pi_{X, *}(\pi_X^{-1}\mathcal{F}) = 0$ for $j > 0$. This concludes the proof.
Proof of (1). You can prove this by repeating the proof of Lemma \ref{lemma-compare-higher-direct-image-proper} part (1); we will instead deduce the result from this. As $\epsilon_{Y, *}$ is the identity functor on underlying presheaves, it reflects isomorphisms. Lemma \ref{lemma-comparison-fppf-etale} shows that $\epsilon_{Y, *} \circ a_Y^{-1} = \pi_Y^{-1}$ and similarly for $X$. To show that the canonical map $a_Y^{-1}f_{small, *}\mathcal{F} \to f_{big, fppf, *}a_X^{-1}\mathcal{F}$ is an isomorphism, it suffices to show that \begin{align*} \pi_Y^{-1}f_{small, *}\mathcal{F} & = \epsilon_{Y, *}a_Y^{-1}f_{small, *}\mathcal{F} \\ & \to  \epsilon_{Y, *}f_{big, fppf, *}a_X^{-1}\mathcal{F} \\ & = f_{big, \etale, *} \epsilon_{X, *}a_X^{-1}\mathcal{F} \\ & = f_{big, \etale, *}\pi_X^{-1}\mathcal{F} \end{align*} is an isomorphism. This is part (1) of Lemma \ref{lemma-compare-higher-direct-image-proper}. \medskip\noindent To see (2) we use that \begin{align*} R\epsilon_{Y, *}Rf_{big, fppf, *}a_X^{-1}K & = Rf_{big, \etale, *}R\epsilon_{X, *}a_X^{-1}K \\ & = Rf_{big, \etale, *}\pi_X^{-1}K \\ & = \pi_Y^{-1}Rf_{small, *}K \\ & = R\epsilon_{Y, *} a_Y^{-1}Rf_{small, *}K \end{align*} The first equality by the commutative diagram in Lemma \ref{lemma-push-pull-fppf-etale} and Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-derived-pushforward-composition}. Then second equality is Lemma \ref{lemma-cohomological-descent-etale-fppf}. The third is Lemma \ref{lemma-compare-higher-direct-image-proper} part (2). The fourth is Lemma \ref{lemma-cohomological-descent-etale-fppf} again. Thus the base change map $a_Y^{-1}(Rf_{small, *}K) \to Rf_{big, fppf, *}(a_X^{-1}K)$ induces an isomorphism $$ R\epsilon_{Y, *}a_Y^{-1}Rf_{small, *}K \to R\epsilon_{Y, *}Rf_{big, fppf, *}a_X^{-1}K $$ The proof is finished by the following remark: a map $\alpha : a_Y^{-1}L \to M$ with $L$ in $D^+(Y_\etale)$ and $M$ in $D^+((\textit{Spaces}/Y)_{fppf})$ such that $R\epsilon_{Y, *}\alpha$ is an isomorphism, is an isomorphism. Namely,  we show by induction on $i$ that $H^i(\alpha)$ is an isomorphism. This is true for all sufficiently small $i$. If it holds for $i \leq i_0$, then we see that $R^j\epsilon_{Y, *}H^i(M) = 0$ for $j > 0$ and $i \leq i_0$ by Lemma \ref{lemma-comparison-fppf-etale} because $H^i(M) = a_Y^{-1}H^i(L)$ in this range. Hence $\epsilon_{Y, *}H^{i_0 + 1}(M) = H^{i_0 + 1}(R\epsilon_{Y, *}M)$ by a spectral sequence argument. Thus $\epsilon_{Y, *}H^{i_0 + 1}(M) = \pi_Y^{-1}H^{i_0 + 1}(L) = \epsilon_{Y, *}a_Y^{-1}H^{i_0 + 1}(L)$. This implies $H^{i_0 + 1}(\alpha)$ is an isomorphism (because $\epsilon_{Y, *}$ reflects isomorphisms as it is the identity on underlying presheaves) as desired.
Let $V \to Y$ be a surjective \'etale morphism where $V$ is a scheme. It suffices to prove the base change map is an isomorphism after restricting to $V$. Hence we may assume that $Y$ is a scheme. As the morphism is finite, hence representable, we conclude that we may assume both $X$ and $Y$ are schemes. In this case the result follows from the case of schemes (\'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-V-C-all-n-etale-fppf} part (2)) using the comparison of topoi discussed in Section \ref{section-api} and in particular given in Lemma \ref{lemma-compare-cohomology-other-topologies}. Some details omitted.
The functor $a_X^{-1}$ is fully faithful (as $a_{X, *}a_X^{-1} = \text{id}$ by Lemma \ref{lemma-comparison-fppf-etale}). Hence the forgetful functor $(\mathcal{G}, \mathcal{H}, \alpha) \mapsto \mathcal{H}$ identifies the category of triples with a full subcategory of $\Sh((\Sch/Y)_{fppf})$. Moreover, the functor $a_Y^{-1}$ is fully faithful, hence the functor in the lemma is fully faithful as well. \medskip\noindent Suppose that we have an \'etale covering $\{Y_i \to Y\}$. Let $f_i : X_i \to Y_i$ be the base change of $f$. Denote $f_{ij} = f_i \times f_j : X_i \times_X X_j  \to Y_i \times_Y Y_j$. Claim: if the lemma is true for $f_i$ and $f_{ij}$ for all $i, j$, then the lemma is true for $f$. To see this, note that the given \'etale covering determines an \'etale covering of the final object in each of the four sites $Y_\etale, X_\etale, (\Sch/Y)_{fppf}, (\Sch/X)_{fppf}$. Thus the category of sheaves is equivalent to the category of glueing data for this covering (Sites, Lemma \ref{sites-lemma-mapping-property-glue}) in each of the four cases. A huge commutative diagram of categories then finishes the proof of the claim. We omit the details. The claim shows that we may work \'etale locally on $Y$. In particular, we may assume $Y$ is a scheme. \medskip\noindent Assume $Y$ is a scheme. Choose a scheme $X'$ and a surjective \'etale morphism $s : X' \to X$. Set $f' = f \circ s : X' \to Y$ and observe that $f'$ is surjective, locally of finite presentation, and flat. Claim: if the lemma is true for $f'$, then it is true for $f$. Namely, given a triple $(\mathcal{G}, \mathcal{H}, \alpha)$ for $f$, we can pullback by $s$ to get a triple $(s_{small}^{-1}\mathcal{G}, \mathcal{H}, s_{big, fppf}^{-1}\alpha)$ for $f'$. A solution for this triple gives a sheaf $\mathcal{F}$ on $Y_\etale$ with $a_Y^{-1}\mathcal{F} = \mathcal{H}$. By the first paragraph of the proof this means the triple is in the essential image. This reduces us to the case where both $X$ and $Y$ are schemes. This case follows from \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-descent-sheaf-fppf-etale} via the discussion in Section \ref{section-api} and in particular Lemma \ref{lemma-compare-cohomology-other-topologies}.
Part (1) is a consequence of fppf descent of quasi-coherent modules. Namely, suppose that $\{f_i : U_i \to U\}$ is an fpqc covering in $(\textit{Spaces}/X)_\etale$. Denote $g : U \to X$ the structure morphism. Suppose that we have a family of sections $s_i \in \Gamma(U_i , f_i^*g^*\mathcal{F})$ such that $s_i|_{U_i \times_U U_j} = s_j|_{U_i \times_U U_j}$. We have to find the correspond section $s \in \Gamma(U, g^*\mathcal{F})$. We can reinterpret the $s_i$ as a family of maps $\varphi_i : f_i^*\mathcal{O}_U = \mathcal{O}_{U_i} \to f_i^*g^*\mathcal{F}$ compatible with the canonical descent data associated to the quasi-coherent sheaves $\mathcal{O}_U$ and $g^*\mathcal{F}$ on $U$. Hence by Descent on Spaces, Proposition \ref{spaces-descent-proposition-fpqc-descent-quasi-coherent} we see that we may (uniquely) descend these to a map $\mathcal{O}_U \to g^*\mathcal{F}$ which gives us our section $s$. \medskip\noindent We will deduce (2) -- (7) from the corresponding statement for schemes. Choose an \'etale covering $\{X_i \to X\}_{i \in I}$ where each $X_i$ is a scheme. Observe that $X_i \times_X X_j$ is a scheme too. This covering induces a covering of the final object in each of the three sites $(\textit{Spaces}/X)_{fppf}$, $(\textit{Spaces}/X)_\etale$, and $X_\etale$. Hence we see that the category of sheaves on these sites are equivalent to descent data for these coverings, see Sites, Lemma \ref{sites-lemma-mapping-property-glue}. Parts (2), (3) are local (because we have the glueing statement). Being quasi-coherent is a local property, hence parts (4), (5) are local. Clearly (6) and (7) are local. It follows that it suffices to prove parts (2) -- (7) of the lemma when $X$ is a scheme. \medskip\noindent Assume $X$ is a scheme. The embeddings $(\Sch/X)_\etale \subset (\textit{Spaces}/X)_\etale$ and $(\Sch/X)_{fppf} \subset (\textit{Spaces}/X)_{fppf}$ determine equivalences of ringed topoi by Lemma \ref{lemma-compare-cohomology-other-topologies}. We conclude that (2) -- (7) follows from the case of schemes. \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-review-quasi-coherent}. To transport the property of being quasi-coherent via this equivalence use that being quasi-coherent is an intrinsic property of modules as explained in Modules on Sites, Section \ref{sites-modules-section-local}. Some minor details omitted.
For any object $f : U \to X$ of $(\textit{Spaces}/X)_\etale$ consider the restriction $\mathcal{H}_\etale|_{U_\etale}$ of $\mathcal{H}_\etale$ to $U_\etale$ via the functor $i_f^* = i_f^{-1}$ discussed in Section \ref{section-compare}. The sheaf $\mathcal{H}_\etale|_{U_\etale}$ is equal to the homology of complex $f^*\mathcal{F}_\bullet$ in degree $1$. This is true because $i_f \circ \pi_X = f$ as morphisms of ringed sites $U_\etale \to X_\etale$. In particular we see that $\mathcal{H}_\etale|_{U_\etale}$ is a quasi-coherent $\mathcal{O}_U$-module. Next, let $g : V \to U$ be a flat morphism in $(\textit{Spaces}/X)_\etale$. Since $$ i_{f \circ g}^* \circ \pi_X^* = (f \circ g)^* = g^* \circ f^* $$ as morphisms of sites $V_\etale \to X_\etale$ and since $g$ is flat hence $g^*$ is exact, we obtain $$ \mathcal{H}_\etale|_{V_\etale} = g^*\left(\mathcal{H}_\etale|_{U_\etale}\right) $$ With these preparations we are ready to prove the lemma. \medskip\noindent Let $\mathcal{U} = \{g_i : U_i \to U\}_{i \in I}$ be an fppf covering with $f : U \to X$ as above. The sheaf propery holds for $\mathcal{H}_\etale$ and the covering $\mathcal{U}$ by (1) of Lemma \ref{lemma-review-quasi-coherent} applied to $\mathcal{H}_\etale|_{U_\etale}$ and the above. Therefore we see that $\mathcal{H}_\etale$ is already an fppf sheaf and this means that $\mathcal{H}_{fppf}$ is equal to $\mathcal{H}_\etale$ as a presheaf. In particular $\mathcal{H}_\etale = \epsilon_{X, *}\mathcal{H}_{fppf}$. \medskip\noindent Finally, to prove the vanishing, we use Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cech-vanish-collection}. We let $\mathcal{B}$ be the affine objects of $(\textit{Spaces}/X)_{fppf}$ and we let $\text{Cov}$ be the set of finite fppf coverings $\mathcal{U} = \{U_i \to U\}_{i = 1, \ldots, n}$ with $U$, $U_i$ affine. We have $$ {\check H}^p(\mathcal{U}, \mathcal{H}_\etale) = {\check H}^p(\mathcal{U}, \left(\mathcal{H}_\etale|_{U_\etale}\right)^a) $$ because the values of $\mathcal{H}_\etale$ on the affine schemes $U_{i_0} \times_U \ldots \times_U U_{i_p}$ flat over $U$ agree with the values of the pullback of the quasi-coherent module $\mathcal{H}_\etale|_{U_\etale}$ by the first paragraph. Hence we obtain vanishing by Descent, Lemma \ref{descent-lemma-standard-covering-Cech-quasi-coherent}. This finishes the proof.
We have $a_X^{-1}\mathcal{F} = \epsilon_X^{-1} \pi_X^{-1}\mathcal{F}$. By Lemma \ref{lemma-describe-pullback} the \'etale sheaf $\pi_X^{-1}\mathcal{F}$ is a sheaf for the ph topology and therefore is equal to $a_X^{-1}\mathcal{F}$ (as pulling back by $\epsilon_X$ is given by ph sheafification). Recall moreover that $\epsilon_{X, *}$ is the identity on underlying presheaves. Now part (1) is immediate from the explicit description of $\pi_X^{-1}$ in Lemma \ref{lemma-describe-pullback}. \medskip\noindent We will prove part (2) by reducing it to the case of schemes -- see part (1) of \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-V-C-all-n-etale-ph}. This will ``clearly work'' as every algebraic space is \'etale locally a scheme. The details are given below but we urge the reader to skip the proof. \medskip\noindent For an abelian sheaf $\mathcal{H}$ on $(\textit{Spaces}/X)_{ph}$ the higher direct image $R^p\epsilon_{X, *}\mathcal{H}$ is the sheaf associated to the presheaf $U \mapsto H^p_{ph}(U, \mathcal{H})$ on $(\textit{Spaces}/X)_\etale$. See Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-higher-direct-images}. Since every object of $(\textit{Spaces}/X)_\etale$ has a covering by schemes, it suffices to prove that given $U/X$ a scheme and $\xi \in H^p_{ph}(U, a_X^{-1}\mathcal{F})$ we can find an \'etale covering $\{U_i \to U\}$ such that $\xi$ restricts to zero on $U_i$. We have \begin{align*} H^p_{ph}(U, a_X^{-1}\mathcal{F}) & = H^p((\textit{Spaces}/U)_{ph}, (a_X^{-1}\mathcal{F})|_{\textit{Spaces}/U}) \\ & = H^p((\Sch/U)_{ph}, (a_X^{-1}\mathcal{F})|_{\Sch/U}) \end{align*} where the second identification is Lemma \ref{lemma-compare-cohomology-other-topologies} and the first is a general fact about restriction (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-of-open}). Looking at the first paragraph and the corresponding result in the case of schemes (\'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-describe-pullback-pi-ph}) we conclude that the sheaf $(a_X^{-1}\mathcal{F})|_{\Sch/U}$ matches the pullback by the ``schemes version of $a_U$''. Therefore we can find an \'etale covering $\{U_i \to U\}$ such that our class dies in $H^p((\Sch/U_i)_{ph}, (a_X^{-1}\mathcal{F})|_{\Sch/U_i})$ for each $i$, see \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-V-C-all-n-etale-ph} (the precise statement one should use here is that $V_n$ holds for all $n$ which is the statement of part (2) for the case of schemes). Transporting back (using the same formulas as above but now for $U_i$) we conclude $\xi$ restricts to zero over $U_i$ as desired.
We only prove the second statement; the first is easier and proved in exactly the same manner. There is a reduction to the case where $K$ is given by a single torsion abelian sheaf. Namely, represent $K$ by a bounded below complex $\mathcal{F}^\bullet$ of torsion abelian sheaves. This is possible by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-torsion}. By the case of a sheaf we see that $\mathcal{F}^n = a_{X, *} a_X^{-1} \mathcal{F}^n$ and that the sheaves $R^qa_{X, *}a_X^{-1}\mathcal{F}^n$ are zero for $q > 0$. By Leray's acyclicity lemma (Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity}) applied to $a_X^{-1}\mathcal{F}^\bullet$ and the functor $a_{X, *}$ we conclude. From now on assume $K = \mathcal{F}$ where $\mathcal{F}$ is a torsion abelian sheaf. \medskip\noindent By Lemma \ref{lemma-comparison-ph-etale} we have $a_{X, *}a_X^{-1}\mathcal{F} = \mathcal{F}$. Thus it suffices to show that $R^qa_{X, *}a_X^{-1}\mathcal{F} = 0$ for $q > 0$. For this we can use $a_X = \epsilon_X \circ \pi_X$ and the Leray spectral sequence (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-relative-Leray}). By Lemma \ref{lemma-comparison-ph-etale} we have $R^i\epsilon_{X, *}(a_X^{-1}\mathcal{F}) = 0$ for $i > 0$. We have $\epsilon_{X, *}a_X^{-1}\mathcal{F} = \pi_X^{-1}\mathcal{F}$ and by Lemma \ref{lemma-cohomological-descent-etale} we have $R^j\pi_{X, *}(\pi_X^{-1}\mathcal{F}) = 0$ for $j > 0$. This concludes the proof.