hcju/stacksqa huggingface dataset:
There is a canonical map from left to right by Cohomology on Sites, Section \ref{sites-cohomology-section-projection-formula}. We will check the equality on stalks. Recall that computing derived tensor products commutes with pullbacks. See Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-pullback-tensor-product}. Thus we have $$ (E \otimes_\mathbf{Z}^\mathbf{L} f^{-1}K)_{\overline{x}} = E_{\overline{x}} \otimes_\mathbf{Z}^\mathbf{L} K_{\overline{y}} $$ where $\overline{y}$ is the image of $\overline{x}$ in $Y$. Since $\mathbf{Z}$ has global dimension $1$ we see that this complex has vanishing cohomology in degree $< - 1 + a + b$ if $H^i(E) = 0$ for $i \geq a$ and $H^j(K) = 0$ for $j \geq b$. Moreover, since $H^i(E)$ is a torsion abelian sheaf for each $i$, the same is true for the cohomology sheaves of the complex $E \otimes_\mathbf{Z}^\mathbf{L} K$. Namely, we have $$ (E \otimes_\mathbf{Z}^\mathbf{L} f^{-1}K) \otimes_{\mathbf{Z}}^\mathbf{L} \mathbf{Q} = (E \otimes_\mathbf{Z}^\mathbf{L} \mathbf{Q}) \otimes_{\mathbf{Q}}^\mathbf{L} (f^{-1}K \otimes_{\mathbf{Z}}^\mathbf{L} \mathbf{Q}) $$ which is zero in the derived category. In this way we see that Lemma \ref{lemma-proper-base-change-stalk} applies to both sides to see that it suffices to show $$ R\Gamma(X_{\overline{y}}, E|_{X_{\overline{y}}} \otimes_\mathbf{Z}^\mathbf{L} (X_{\overline{y}} \to \overline{y})^{-1}K_{\overline{y}}) = R\Gamma(X_{\overline{y}}, E|_{X_{\overline{y}}}) \otimes_\mathbf{Z}^\mathbf{L} K_{\overline{y}} $$ This is shown in Lemma \ref{lemma-pull-out-constant}.
Denote $f_i : X \to X_i$ the projections. Let $\mathcal{F}$ be an abelian torsion sheaf on $X_\etale$. Then we have $\mathcal{F} = \lim f_i^{-1}f_{i, *}\mathcal{F}$ by Lemma \ref{lemma-linus-hamann}. Thus $H^q_\etale(X, \mathcal{F}) = \colim H^q_\etale(X_i, f_{i, *}\mathcal{F})$ by Theorem \ref{theorem-colimit}. The lemma follows.
Let $\mathcal{F}$ be an abelian torsion sheaf on $X_\etale$. Consider the Leray spectral sequence for the morphism $f : X \to \Spec(K)$. We obtain $$ E_2^{p, q} = H^p(\Spec(K), R^qf_*\mathcal{F}) $$ converging to $H^{p + q}_\etale(X, \mathcal{F})$. The stalk of $R^qf_*\mathcal{F}$ at a geometric point $\Spec(\overline{K}) \to \Spec(K)$ is the cohomology of the pullback of $\mathcal{F}$ to $X_{\overline{K}}$. Hence it vanishes in degrees $\geq 2$ by Theorem \ref{theorem-vanishing-affine-curves}.
If $\text{trdeg}_K(L) = \infty$, then this is clear. If not then we can find a sequence of extensions $L= L_r/L_{r - 1}/ \ldots /L_1/L_0 = K$ such that $\text{trdeg}_{L_i}(L_{i + 1}) = 1$ and $r = \text{trdeg}_K(L)$. Hence it suffices to prove the lemma in the case that $r = 1$. In this case we can write $L = \colim A_i$ as a filtered colimit of its finite type $K$-subalgebras. By Lemma \ref{lemma-cd-limit} it suffices to prove that $\text{cd}(A_i) \leq 1 + \text{cd}(K)$. This follows from Lemma \ref{lemma-cd-curve-over-field}.
We may assume $X$ is affine. By Noether normalization, after possibly shrinking $X$ again, we can choose a finite morphism $\pi : X \to \mathbf{A}^d_K$, see Algebra, Lemma \ref{algebra-lemma-Noether-normalization-at-point}. Since $\kappa(x)$ is a finite extension of the residue field of $\pi(x)$, this residue field has transcendence degree $a$ over $K$ as well. Thus we can find a finite morphism $\pi' : \mathbf{A}^d_K \to \mathbf{A}^d_K$ such that $\pi'(\pi(x))$ corresponds to the generic point of the linear subspace $\mathbf{A}^a_K \subset \mathbf{A}^d_K$ given by setting the last $d - a$ coordinates equal to zero. Hence the composition $$ X \xrightarrow{\pi' \circ \pi} \mathbf{A}^d_K \xrightarrow{p} \mathbf{A}^a_K $$ of $\pi' \circ \pi$ and the projection $p$ onto the first $a$ coordinates maps $x$ to the generic point $\eta \in \mathbf{A}^a_K$. The induced map $$ K(t_1, \ldots, t_a)^{sep} = \mathcal{O}_{\mathbf{A}^a_k, \eta}^{sh} \longrightarrow \mathcal{O}_{X, x}^{sh} $$ on \'etale local rings satisfies (1) since it is clear that the residue field of $\mathcal{O}_{X, x}^{sh}$ is an algebraic extension of the separably closed field $K(t_1, \ldots, t_a)^{sep}$. On the other hand, if $X = \Spec(B)$, then $\mathcal{O}_{X, x}^{sh} = \colim B_j$ is a filtered colimit of \'etale $B$-algebras $B_j$. Observe that $B_j$ is quasi-finite over $K[t_1, \ldots, t_d]$ as $B$ is finite over $K[t_1, \ldots, t_d]$. We may similarly write $K(t_1, \ldots, t_a)^{sep} = \colim A_i$ as a filtered colimit of \'etale $K[t_1, \ldots, t_a]$-algebras. For every $i$ we can find an $j$ such that $A_i \to K(t_1, \ldots, t_a)^{sep} \to \mathcal{O}_{X, x}^{sh}$ factors through a map $\psi_{i, j} : A_i \to B_j$. Then $B_j$ is quasi-finite over $A_i[t_{a + 1}, \ldots, t_d]$. Hence $$ B_{i, j} = B_j \otimes_{\psi_{i, j}, A_i} K(t_1, \ldots, t_a)^{sep} $$ has dimension $\leq d - a$ as it is quasi-finite over $K(t_1, \ldots, t_a)^{sep}[t_{a + 1}, \ldots, t_d]$. The proof of (2) is now finished as $\mathcal{O}_{X, x}^{sh}$ is a filtered colimit\footnote{Let $R$ be a ring. Let $A = \colim_{i \in I} A_i$ be a filtered colimit of finitely presented $R$-algebras. Let $B = \colim_{j \in J} B_j$ be a filtered colimit of $R$-algebras. Let $A \to B$ be an $R$-algebra map. Assume that for all $i \in I$ there is a $j \in J$ and an $R$-algebra map $\psi_{i, j} : A_i \to B_j$. Say $(i', j', \psi_{i', j'}) \geq (i, j, \psi_{i, j})$ if $i' \geq i$, $j' \geq j$, and $\psi_{i, j}$ and $\psi_{i', j'}$ are compatible. Then the collection of triples forms a directed set and $B = \colim B_j \otimes_{\psi_{i, j} A_i} A$.} of the algebras $B_{i, j}$. Some details omitted.
We can write $\mathcal{F} = \colim \mathcal{F}_i$ with $\mathcal{F}_i$ a torsion abelian sheaf supported on a closed subscheme $Z_i$ contained in $E_a$, see Lemma \ref{lemma-support-in-subset}. Then Proposition \ref{proposition-cd-affine} gives the desired vanishing for $\mathcal{F}_i$. Details omitted; hints: first use Proposition \ref{proposition-closed-immersion-pushforward} to write $\mathcal{F}_i$ as the pushforward of a sheaf on $Z_i$, use the vanishing for this sheaf on $Z_i$, and use the Leray spectral sequence for $Z_i \to Z$ to get the vanishing for $\mathcal{F}_i$. Finally, we conclude by Theorem \ref{theorem-colimit}.
The question is local on $Y$ hence we can assume $Y$ is affine. Then $X$ is affine too and we can choose a diagram $$ \xymatrix{ X \ar[d]_f \ar[r]_i & \mathbf{A}^{n + m}_K \ar[d]^{\text{pr}} \\ Y \ar[r]^j & \mathbf{A}^n_K } $$ where the horizontal arrows are closed immersions and the vertical arrow on the right is the projection (details omitted). Then $j_*R^qf_*\mathcal{F} = R^q\text{pr}_*i_*\mathcal{F}$ by the vanishing of the higher direct images of $i$ and $j$, see Proposition \ref{proposition-finite-higher-direct-image-zero}. Moreover, the description of the stalks of $j_*$ in the proposition shows that it suffices to prove the vanishing for $j_*R^qf_*\mathcal{F}$. Thus we may assume $f$ is the projection morphism $\text{pr} : \mathbf{A}^{n + m}_K \to \mathbf{A}^n_K$ and an abelian torsion sheaf $\mathcal{F}$ on $\mathbf{A}^{n + m}_K$ satisfying the assumption in the statement of the lemma. \medskip\noindent Let $y$ be a point in $\mathbf{A}^n_K$. By Theorem \ref{theorem-higher-direct-images} we have $$ (R^q\text{pr}_*\mathcal{F})_{\overline{y}} = H^q(\mathbf{A}^{n + m}_K \times_{A^n_K} \Spec(\mathcal{O}_{Y, y}^{sh}), \mathcal{F}) = H^q(\mathbf{A}^m_{\mathcal{O}_{Y, y}^{sh}}, \mathcal{F}) $$ Say $b = \text{trdeg}_K(\kappa(y))$. From Lemma \ref{lemma-strictly-henselian} we get an embedding $$ L = K(t_1, \ldots, t_b)^{sep} \longrightarrow \mathcal{O}_{Y, y}^{sh} $$ Write $\mathcal{O}_{Y, y}^{sh} = \colim B_i$ as the filtered colimit of finite type $L$-subalgebras $B_i \subset \mathcal{O}_{Y, y}^{sh}$ containing the ring $K[T_1, \ldots, T_n]$ of regular functions on $\mathbf{A}^n_K$. Then we get $$ \mathbf{A}^m_{\mathcal{O}_{Y, y}^{sh}} = \lim \mathbf{A}^m_{B_i} $$ If $z \in \mathbf{A}^m_{B_i}$ is a point in the support of $\mathcal{F}$, then the image $x$ of $z$ in $\mathbf{A}^{m + n}_K$ satisfies $\text{trdeg}_K(\kappa(x)) \leq a$ by our assumption on $\mathcal{F}$ in the lemma. Since $\mathcal{O}_{Y, y}^{sh}$ is a filtered colimit of \'etale algebras over $K[T_1, \ldots, T_n]$ and since $B_i \subset \mathcal{O}_{Y, y}^{sh}$ we see that $\kappa(z)/\kappa(x)$ is algebraic (some details omitted). Then $\text{trdeg}_K(\kappa(z)) \leq a$ and hence $\text{trdeg}_L(\kappa(z)) \leq a - b$. By Lemma \ref{lemma-interlude-II} we see that $$ H^q(\mathbf{A}^m_{B_i}, \mathcal{F}) = 0\text{ for }q > a - b $$ Thus by Theorem \ref{theorem-colimit} we get $(Rf_*\mathcal{F})_{\overline{y}} = 0$ for $q > a - b$ as desired.
Proof of (1). We may assume $Y$ is affine. We will use the induction principle of  Cohomology of Schemes, Lemma \ref{coherent-lemma-induction-principle} to prove this. If $X$ is affine too, then the result holds by Lemma \ref{lemma-interlude-I}. Thus it suffices to show that if $X = U \cup V$ and the result is true for $U \to Y$, $V \to Y$, and $U \cap V \to Y$, then it is true for $f$. This follows from the relative Mayer-Vietoris sequence, see Lemma \ref{lemma-relative-mayer-vietoris}. \medskip\noindent Proof of (2). We will use the induction principle of Cohomology of Schemes, Lemma \ref{coherent-lemma-induction-principle} to prove this. If $X$ is affine, then the result holds by Proposition \ref{proposition-cd-affine}. Thus it suffices to show that if $X = U \cup V$ and the result is true for $U$, $V$, and $U \cap V $, then it is true for $X$. This follows from the Mayer-Vietoris sequence, see Lemma \ref{lemma-mayer-vietoris}.
Proof of (1). Since $\text{cd}(f) < \infty$ we see that $$ f_* : \textit{Mod}(X_\etale, \mathbf{Z}/n\mathbf{Z}) \longrightarrow \textit{Mod}(Y_\etale, \mathbf{Z}/n\mathbf{Z}) $$ has finite cohomological dimension in the sense of Derived Categories, Lemma \ref{derived-lemma-unbounded-right-derived}. Let $I$ be a set and for $i \in I$ let $E_i$ be an object of $D(X_\etale, \mathbf{Z}/n\mathbf{Z})$. Choose a K-injective complex $\mathcal{I}_i^\bullet$ of $\mathbf{Z}/n\mathbf{Z}$-modules each of whose terms $\mathcal{I}_i^n$ is an injective sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules representing $E_i$. See Injectives, Theorem \ref{injectives-theorem-K-injective-embedding-grothendieck}. Then $\bigoplus E_i$ is represented by the complex $\bigoplus \mathcal{I}_i^\bullet$ (termwise direct sum), see Injectives, Lemma \ref{injectives-lemma-derived-products}. By Lemma \ref{lemma-relative-colimit} we have $$ R^qf_*(\bigoplus \mathcal{I}_i^n) = \bigoplus R^qf_*(\mathcal{I}_i^n) = 0 $$ for $q > 0$ and any $n$. Hence we conclude by  Derived Categories, Lemma \ref{derived-lemma-unbounded-right-derived} that we may compute $Rf_*(\bigoplus E_i)$ by the complex $$ f_*(\bigoplus \mathcal{I}_i^\bullet) = \bigoplus f_*(\mathcal{I}_i^\bullet) $$ (equality again by Lemma \ref{lemma-relative-colimit}) which represents $\bigoplus Rf_*E_i$ by the already used Injectives, Lemma \ref{injectives-lemma-derived-products}. \medskip\noindent Proof of (2). This is identical to the proof of (1) and we omit it.
It is enough to prove this when $Y$ is quasi-compact. By Morphisms, Lemma \ref{morphisms-lemma-morphism-finite-type-bounded-dimension} we see that the dimension of the fibres of $f : X \to Y$ is bounded. Thus Lemma \ref{lemma-cohomological-dimension-proper} implies that $\text{cd}(f) < \infty$. Hence the result by Lemma \ref{lemma-finite-cd-mod-n-direct-sums}.
There is a canonical map from left to right by Cohomology on Sites, Section \ref{sites-cohomology-section-projection-formula}. Let $T(K)$ be the property that the statement of the lemma holds for $K \in D(\mathbf{Z}/n\mathbf{Z})$. We will check conditions (1), (2), and (3) of More on Algebra, Remark \ref{more-algebra-remark-P-resolution} hold for $T$ to conclude. Property (1) holds because both sides of the equality commute with direct sums, see Lemma \ref{lemma-finite-cd-mod-n-direct-sums}. Property (2) holds because we are comparing exact functors between triangulated categories and we can use Derived Categories, Lemma \ref{derived-lemma-third-isomorphism-triangle}. Property (3) says the lemma holds when $K = \mathbf{Z}/n\mathbf{Z}[k]$ for any shift $k \in \mathbf{Z}$ and this is obvious.
There is a canonical map from left to right by Cohomology on Sites, Section \ref{sites-cohomology-section-projection-formula}. We will check the equality on stalks at $\overline{y}$. By the proper base change (in the form of Lemma \ref{lemma-proper-base-change-mod-n} where $Y' = \overline{y}$) this reduces to the case where $Y$ is the spectrum of an algebraically closed field. This is shown in Lemma \ref{lemma-pull-out-constant-mod-n} where we use that $\text{cd}(X) < \infty$ by Lemma \ref{lemma-cohomological-dimension-proper}.
Proof of (1). By Lemma \ref{lemma-projection-formula-proper} we have $$ R\text{pr}_{2, *}( \text{pr}_1^{-1}E \otimes_\mathbf{Z}^\mathbf{L} \text{pr}_2^{-1}K) = R\text{pr}_{2, *}(\text{pr}_1^{-1}E) \otimes_\mathbf{Z}^\mathbf{L} K $$ By proper base change (in the form of Lemma \ref{lemma-proper-base-change}) this is equal to the object $$ \underline{R\Gamma(X, E)} \otimes_\mathbf{Z}^\mathbf{L} K $$ of $D(Y_\etale)$. Taking $R\Gamma(Y, -)$ on this object reproduces the left hand side of the equality in (1) by the Leray spectral sequence for $\text{pr}_2$. Thus we conclude by Lemma \ref{lemma-pull-out-constant}. \medskip\noindent Proof of (2). This is exactly the same as the proof of (1) except that we use Lemmas \ref{lemma-projection-formula-proper-mod-n}, \ref{lemma-proper-base-change-mod-n}, and \ref{lemma-pull-out-constant-mod-n} as well as $\text{cd}(Y) < \infty$ by Lemma \ref{lemma-finite-cd}.
(If $\mathcal{F}$ is torsion, then the vanishing follows immediately from Lemma \ref{lemma-interlude-II}.) By Lemma \ref{lemma-support-in-subset} we can write $\mathcal{F}$ as a filtered colimit of constructible sheaves $\mathcal{F}_i$ of $\mathbf{Z}$-modules whose supports $Z_i \subset X$ are finite sets of closed points. By Proposition \ref{proposition-closed-immersion-pushforward} such a sheaf is of the form $(Z_i \to X)_*\mathcal{G}_i$ where $\mathcal{G}_i$ is a sheaf on $Z_i$. As $K$ is separably closed, the scheme $Z_i$ is a finite disjoint union of spectra of separably closed fields. Recall that $H^q(Z_i, \mathcal{G}_i) = H^q(X, \mathcal{F}_i)$ by the Leray spectral sequence for $Z_i \to X$ and vanising of higher direct images for this morphism (Proposition \ref{proposition-finite-higher-direct-image-zero}). By Lemmas \ref{lemma-equivalence-abelian-sheaves-point} and \ref{lemma-compare-cohomology-point} we see that $H^q(Z_i, \mathcal{G}_i)$ is zero for $q > 0$ and that $H^0(Z_i, \mathcal{G}_i)$ generates $\mathcal{G}_i$. We conclude the vanishing of $H^q(X, \mathcal{F}_i)$ for $q > 0$ and that $\mathcal{F}_i$ is generated by global sections. By Theorem \ref{theorem-colimit} we see that $H^q(X, \mathcal{F}) = 0$ for $q > 0$. The proof is now done because a filtered colimit of globally generated sheaves of abelian groups is globally generated (details omitted).
The implication from left to right is trivial. Thus we need to prove the reverse implication. \medskip\noindent Assume $Q$ is bounded below; this cases suffices for almost all applications. If $Q$ is not zero, then we can look at the smallest $i$ such that the cohomology sheaf $H^i(Q)$ is nonzero. By Lemma \ref{lemma-supported-in-closed-points} we have $H^i(X, Q) = H^0(X, H^i(Q)) \not = 0$ and we conclude. \medskip\noindent General case. Let $\mathcal{B} \subset \Ob(X_\etale)$ be the quasi-compact objects. By Lemma \ref{lemma-supported-in-closed-points} the assumptions of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-over-U-trivial} are satisfied. We conclude that $H^q(U, Q) = H^0(U, H^q(Q))$ for all $U \in \mathcal{B}$. In particular, this holds for $U = X$. Thus the conclusion by Lemma \ref{lemma-supported-in-closed-points} as $Q$ is zero in $D(X_\etale)$ if and only if $H^q(Q)$ is zero for all $q$.
We will reduce this to Lemma \ref{lemma-punctual-base-change-upgrade} using that our functors commute with direct sums. We suggest the reader skip the proof. Recall that derived tensor product commutes with direct sums. Recall that (derived) pullback commutes with direct sums. Recall that $Rh_*$ and $Rg_*$ commute with direct sums, see Lemmas \ref{lemma-finite-cd} and \ref{lemma-finite-cd-mod-n-direct-sums} (this is where we use our schemes are of finite type over $K$). \medskip\noindent To finish the proof we can argue as follows. First we write $E = \text{hocolim} \tau_{\leq N} E$. Since our functors commute with direct sums, they commute with homotopy colimits. Hence it suffices to prove the lemma for $E$ bounded above. Similarly for $F$ we may assume $F$ is bounded above. Then we can represent $E$ by a bounded above complex $\mathcal{E}^\bullet$ of sheaves of $\mathbf{Z}/n\mathbf{Z}$-modules. Then $$ \mathcal{E}^\bullet = \colim \sigma_{\geq -N}\mathcal{E}^\bullet $$ (stupid truncations). Thus we may assume $\mathcal{E}^\bullet$ is a bounded complex of sheaves of $\mathbf{Z}/n\mathbf{Z}$-modules. For $F$ we choose a bounded above complex of flat(!) sheaves of $\mathbf{Z}/n\mathbf{Z}$-modules. Then we reduce to the case where $F$ is represented by a bounded complex of flat sheaves of $\mathbf{Z}/n\mathbf{Z}$-modules. At this point Lemma \ref{lemma-punctual-base-change-upgrade} kicks in and we conclude.
By Lemma \ref{lemma-punctual-base-change-upgrade-unbounded} we have $$ R\text{pr}_{1, *}( \text{pr}_1^{-1}E \otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L} \text{pr}_2^{-1}K) = E \otimes_{\mathbf{Z}/n\mathbf{Z}}^\mathbf{L} \underline{R\Gamma(Y, K)} $$ We conclude by Lemma \ref{lemma-pull-out-constant-mod-n} which we may use because $\text{cd}(X) < \infty$ by Lemma \ref{lemma-finite-cd}.
Except for a snafu having to do with the empty set, this follows from the very general Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-descent-squares} whose hypotheses hold by Schemes, Lemma \ref{schemes-lemma-sheaf-on-affines} and Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-descent-squares-helper}. \medskip\noindent To get around the snafu, denote $X_{affine, almost-chaotic}$ the site where the empty object $\emptyset$ has two coverings, namely, $\{\emptyset \to \emptyset\}$ and the empty covering (see Sites, Example \ref{sites-example-site-topological} for a discussion). Then we have morphisms of sites $$ X_{affine, Zar} \to X_{affine, almost-chaotic} \to X_{affine, chaotic} $$ The argument above works for the first arrow. Then we leave it to the reader to see that an object $K$ of $D^+(X_{affine, chaotic})$ is in the essential image of the (fully faithful) functor $D(X_{affine, almost-chaotic}) \to D(X_{affine, chaotic})$ if and only if $R\Gamma(\emptyset, K)$ is zero in $D(\textit{Ab})$.
Parts (1)(b) and (2)(b) follow formally from the fact that the restriction functor $\pi_{S, *} = i_S^{-1}$ is a right adjoint of the exact functor $\pi_S^{-1}$, 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 $i_f = i_T \circ f_{big}$ 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 $i_T^{-1}$ to conclude. \medskip\noindent Let $S = \Spec(\mathbf{Z})$ and consider the map $2 : \mathcal{O}_S \to \mathcal{O}_S$. This is an injective map of $\mathcal{O}_S$-modules on $S_\etale$. However, the pullback $\pi_S^*(2) : \mathcal{O} \to \mathcal{O}$ is not injective as we see by evaluating on $\Spec(\mathbf{F}_2)$. Now choose an injection $\alpha : \mathcal{O} \to \mathcal{I}$ into an injective $\mathcal{O}$-module $\mathcal{I}$ on $(\Sch/S)_\etale$. Then consider the diagram $$ \xymatrix{ \mathcal{O}_S \ar[d]_2 \ar[rr]_{\alpha|_{S_\etale}} & & \mathcal{I}|_{S_\etale} \\ \mathcal{O}_S \ar@{..>}[rru] } $$ Then the dotted arrow cannot exist in the category of $\mathcal{O}_S$-modules because it would mean (by adjunction) that the injective map $\alpha$ factors through the noninjective map $\pi_S^*(2)$ which cannot be the case. Thus $\mathcal{I}|_{S_\etale}$ is not an injective $\mathcal{O}_S$-module.
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 $\pi_S \circ f_{big} = f_{small} \circ \pi_T$ as morphisms of ringed sites. Hence we obtain $R\pi_{S, *} \circ Rf_{big, *} = Rf_{small, *} \circ R\pi_{T, *}$ by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-derived-pushforward-composition}. Since the restriction functors $\pi_{S, *}$ and $\pi_{T, *}$ are exact, we conclude. \medskip\noindent Part (4) follows from part (6) and part (2) applied to $f' : T' \to S'$.
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 $i_f^{-1}\pi_S^{-1} = f_{small}^{-1}$. 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 $Lf_{small}^* = i_f^* \circ L\pi_S^*$. Part (2) is a special case of (4).