hcju/stacksqa huggingface dataset:
For each $i$ we can choose syntomic (resp.\ smooth, resp.\ \'etale) morphisms $g_{ij} : V_{ij} \to Y$ and morphisms $V_{ij} \times_Y X \to U_i$ over $X$, such that $\{V_{ij} \times_Y X \to U_i\}$ are Zariski coverings, see Lemma \ref{lemma-closed-immersion-cover-from-below}. This in particular implies that $\bigcup_{ij} g_{ij}(V_{ij})$ contains the closed subset $f(X)$. Hence the family of syntomic (resp.\ smooth, resp.\ \'etale) maps $g_{ij}$ together with the open immersion $Y \setminus f(X) \to Y$ forms the desired syntomic (resp.\ smooth, resp.\ \'etale) covering of $Y$.
We have to show the following: given a morphism $V \to Y$ and any syntomic (resp.\ smooth, resp.\ \'etale) covering $\{U_i \to X \times_Y V\}$, there exists a smooth (resp.\ smooth, resp.\ \'etale) covering $\{V_j \to V\}$ such that for each $j$, either $X \times_Y V_j$ is empty, or $X \times_Y V_j \to Z \times_Y V$ factors through one of the $U_i$. This follows on applying Lemma \ref{lemma-prepare-closed-immersion-almost-cocontinuous} above to the closed immersion $X \times_Y V \to V$.
Recall that $f_{big}$ is the morphism of topoi associated to the continuous and cocontinuous functor $u : (\Sch/X)_\tau \to (\Sch/Y)_\tau$, $U/X \mapsto U/Y$. Moreover, we have $f_{big}^{-1}\mathcal{O} = \mathcal{O}$. Hence the existence of $f_{big!}$ follows from Modules on Sites, Lemma \ref{sites-modules-lemma-g-shriek-adjoint}, respectively Modules on Sites, Lemma \ref{sites-modules-lemma-lower-shriek-modules}. Note that if $U$ is an object of $(\Sch/X)_\tau$ then the functor $u$ induces an equivalence of categories $$ u' : (\Sch/X)_\tau/U \longrightarrow (\Sch/Y)_\tau/U $$ because both sides of the arrow are equal to $(\Sch/U)_\tau$. Hence the agreement of $f_{big!}$ on underlying abelian sheaves follows from the discussion in Modules on Sites, Remark \ref{sites-modules-remark-when-shriek-equal}. The exactness of $f_{big!}$ follows from Modules on Sites, Lemma \ref{sites-modules-lemma-exactness-lower-shriek} as the functor $u$ above which commutes with fibre products and equalizers.
Observe that if $\mathcal{I}$ is an injective abelian sheaf, then $$ 0 \to \mathcal{I}(X) \to \mathcal{I}(U) \oplus \mathcal{I}(V) \to \mathcal{I}(U \cap V) \to 0 $$ is exact. This is true in the first and middle spots as $\mathcal{I}$ is a sheaf. It is true on the right, because $\mathcal{I}(U) \to \mathcal{I}(U \cap V)$ is surjective by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-restriction-along-monomorphism-surjective}. Another way to prove it would be to show that the cokernel of the map $\mathcal{I}(U) \oplus \mathcal{I}(V) \to \mathcal{I}(U \cap V)$ is the first {\v C}ech cohomology group of $\mathcal{I}$ with respect to the covering $X = U \cup V$ which vanishes by Lemmas \ref{lemma-hom-injective} and \ref{lemma-forget-injectives}. Thus, if $\mathcal{F} \to \mathcal{I}^\bullet$ is an injective resolution, then $$ 0 \to \mathcal{I}^\bullet(X) \to \mathcal{I}^\bullet(U) \oplus \mathcal{I}^\bullet(V) \to \mathcal{I}^\bullet(U \cap V) \to 0 $$ is a short exact sequence of complexes and the associated long exact cohomology sequence is the sequence of the statement of the lemma.
Let $\mathcal{F} \to \mathcal{I}^\bullet$ be an injective resolution of $\mathcal{F}$ on $X_\etale$. We claim that we get a short exact sequence of complexes $$ 0 \to f_*\mathcal{I}^\bullet \to a_*\mathcal{I}^\bullet|_U \oplus b_*\mathcal{I}^\bullet|_V \to c_*\mathcal{I}^\bullet|_{U \cap V} \to 0. $$ Namely, for any $W$ in $Y_\etale$, and for any $n \geq 0$ the corresponding sequence of groups of sections over $W$ $$ 0 \to \mathcal{I}^n(W \times_Y X) \to \mathcal{I}^n(W \times_Y U) \oplus \mathcal{I}^n(W \times_Y V) \to \mathcal{I}^n(W \times_Y (U \cap V)) \to 0 $$ was shown to be short exact in the proof of Lemma \ref{lemma-mayer-vietoris}. The lemma follows by taking cohomology sheaves and using the fact that $\mathcal{I}^\bullet|_U$ is an injective resolution of $\mathcal{F}|_U$ and similarly for $\mathcal{I}^\bullet|_V$, $\mathcal{I}^\bullet|_{U \cap V}$.
Let us first prove this when $X$ and $X_i$ are quasi-compact and quasi-separated for all $i$ (as this is true in all cases of interest). In this case any object of $X_{affine, \etale}$, resp.\ $(X_i)_{affine, \etale}$ is of finite presentation over $X$. Moreover, the category of schemes of finite presentation over $X$ is the colimit of the categories of schemes of finite presentation over $X_i$, see Limits, Lemma \ref{limits-lemma-descend-finite-presentation}. The same holds for the subcategories of affine objects \'etale over $X$ by Limits, Lemmas \ref{limits-lemma-limit-affine} and \ref{limits-lemma-descend-etale}. Finally, if $\{U^j \to U\}$ is a covering of $X_{affine, \etale}$ and if $U_i^j \to U_i$ is morphism of affine schemes \'etale over $X_i$ whose base change to $X$ is $U^j \to U$, then we see that the base change of $\{U^j_i \to U_i\}$ to some $X_{i'}$ is a covering for $i'$ large enough, see Limits, Lemma \ref{limits-lemma-descend-surjective}. \medskip\noindent In the general case, let $U$ be an object of $X_{affine, \etale}$. Then $U \to X$ is \'etale and separated (as $U$ is separated) but in general not quasi-compact. Still, $U \to X$ is locally of finite presentation and hence by Limits, Lemma \ref{limits-lemma-descend-finite-presentation-variant} there exists an $i$, a quasi-compact and quasi-separated scheme $U_i$, and a morphism $U_i \to X_i$ which is locally of finite presentation whose base change to $X$ is $U \to X$. Then $U = \lim_{i' \geq i} U_{i'}$ where $U_{i'} = U_i \times_{X_i} X_{i'}$. After increasing $i$ we may assume $U_i$ is affine, see Limits, Lemma \ref{limits-lemma-limit-affine}. To check that $U_i \to X_i$ is \'etale for $i$ sufficiently large, choose a finite affine open covering $U_i = U_{i, 1} \cup \ldots \cup U_{i, m}$ such that $U_{i, j} \to U_i \to X_i$ maps into an affine open $W_{i, j} \subset X_i$. Then we can apply Limits, Lemma \ref{limits-lemma-descend-etale} to see that $U_{i, j} \to W_{i, j}$ is \'etale after possibly increasing $i$. In this way we see that the functor $\colim (X_i)_{affine, \etale} \to X_{affine, \etale}$ is essentially surjective. Fully faithfulness follows directly from the already used Limits, Lemma \ref{limits-lemma-descend-finite-presentation-variant}. The statement on coverings is proved in exactly the same manner as done in the first paragraph of the proof.
This is a special case of Theorem \ref{theorem-colimit}. We also sketch a direct proof. We prove it for all $X$ at the same time, by induction on $p$. \begin{enumerate} \item For any quasi-compact and quasi-separated scheme $X$ and any \'etale covering $\mathcal{U}$ of $X$, show that there exists a refinement $\mathcal{V} = \{V_j \to X\}_{j\in J}$ with $J$ finite and each $V_j$ quasi-compact and quasi-separated such that all $V_{j_0} \times_X \ldots \times_X V_{j_p}$ are also quasi-compact and quasi-separated. \item Using the previous step and the definition of colimits in the category of sheaves, show that the theorem holds for $p = 0$ and all $X$. \item Using the locality of cohomology (Lemma \ref{lemma-locality-cohomology}), the {\v C}ech-to-cohomology spectral sequence (Theorem \ref{theorem-cech-ss}) and the fact that the induction hypothesis applies to all $V_{j_0} \times_X \ldots \times_X V_{j_p}$ in the above situation, prove the induction step $p \to p + 1$. \end{enumerate}
This is a special case of Theorem \ref{theorem-colimit}. We also outline a direct proof as follows. \begin{enumerate} \item Given $V \to Y$ \'etale with $V$ quasi-compact and quasi-separated, there exist $i\in I$ and $V_i \to Y_i$ such that $V = V_i \times_{Y_i} Y$. If all the schemes considered were affine, this would correspond to the following algebra statement: if $B = \colim B_i$ and $B \to C$ is \'etale, then there exist $i \in I$ and $B_i \to C_i$ \'etale such that $C \cong B \otimes_{B_i} C_i$. This is proved in Algebra, Lemma \ref{algebra-lemma-etale}. \item In the situation of (1) show that $\mathcal{G}(V) = \colim_{i' \geq i} \mathcal{G}_{i'}(V_{i'})$ where $V_{i'}$ is the base change of $V_i$ to $Y_{i'}$. \item By (1), we see that for every \'etale covering $\mathcal{V} = \{V_j \to Y\}_{j\in J}$ with $J$ finite and the $V_j$s quasi-compact and quasi-separated, there exists $i \in I$ and an \'etale covering $\mathcal{V}_i = \{V_{ij} \to Y_i\}_{j \in J}$ such that $\mathcal{V} \cong \mathcal{V}_i \times_{Y_i} Y$. \item Show that (2) and (3) imply $$ \check H^*(\mathcal{V}, \mathcal{G})= \colim_{i\in I} \check H^*(\mathcal{V}_i, \mathcal{G}_i). $$ \item Cleverly use the {\v C}ech-to-cohomology spectral sequence (Theorem \ref{theorem-cech-ss}). \end{enumerate}
Recall (Lemma \ref{lemma-higher-direct-images}) that $R^p g_{i, *} \mathcal{F}_i$ is the sheaf associated to the presheaf $U \mapsto H^p_\etale(U \times_S X_i, \mathcal{F}_i)$ and similarly for $R^pg_*\mathcal{F}$. Moreover, the colimit of a system of sheaves is the sheafification of the colimit on the level of presheaves. Note that every object of $S_\etale$ has a covering by quasi-compact and quasi-separated objects (e.g., affine schemes). Moreover, if $U$ is a quasi-compact and quasi-separated object, then we have $$ \colim H^p_\etale(U \times_S X_i, \mathcal{F}_i) = H^p_\etale(U \times_S X, \mathcal{F}) $$ by Theorem \ref{theorem-colimit}. Thus the lemma follows.
Observe that $\check{\mathcal{C}}^\bullet(\mathcal{V}, f_*\mathcal{I}) = \check{\mathcal{C}}^\bullet(\mathcal{V} \times_Y X, \mathcal{I})$ which has vanishing higher cohomology groups by Lemma \ref{lemma-hom-injective}. This proves (1). The second statement follows as a sheaf which has vanishing higher {\v C}ech cohomology groups for any covering has vanishing higher cohomology groups. This a wonderful exercise in using the {\v C}ech-to-cohomology spectral sequence, but see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cech-vanish-collection} for details and a more precise and general statement. Part (3) is a consequence of (2) and the description of $R^pg_*$ in Lemma \ref{lemma-higher-direct-images}.
If we show that $\Gamma(S, \mathcal{F}) = \mathcal{F}_{\overline{s}}$ then $\Gamma(S, -)$ is exact as the stalk functor is exact. Let $(U, \overline{u})$ be an \'etale neighbourhood of $\overline{s}$. Pick an affine open neighborhood $\Spec(A)$ of $\overline{u}$ in $U$. Then $R \to A$ is \'etale and $\kappa(\overline{s}) = \kappa(\overline{u})$. By Theorem \ref{theorem-henselian} we see that $A \cong R \times A'$ as an $R$-algebra compatible with maps to $\kappa(\overline{s}) = \kappa(\overline{u})$. Hence we get a section $$ \xymatrix{ \Spec(A) \ar[r] & U \ar[d]\\ & S \ar[ul] } $$ It follows that in the system of \'etale neighbourhoods of $\overline{s}$ the identity map $(S, \overline{s}) \to (S, \overline{s})$ is cofinal. Hence $\Gamma(S, \mathcal{F}) = \mathcal{F}_{\overline{s}}$. The final statement of the lemma follows as the higher derived functors of an exact functor are zero, see Derived Categories, Lemma \ref{derived-lemma-right-derived-exact-functor}.
In great generality there is a pullback map $g^{-1}f_*\mathcal{F} \to f'_*(g')^{-1}\mathcal{F}$, see Sites, Section \ref{sites-section-pullback}. It suffices to check on stalks (Theorem \ref{theorem-exactness-stalks}). Let $\overline{y}' : \Spec(k) \to Y'$ be a geometric point. We have \begin{align*} (f'_*(g')^{-1}\mathcal{F})_{\overline{y}'} & = \prod\nolimits_{\overline{x}' : \Spec(k) \to X',\ f' \circ \overline{x}' = \overline{y}'} ((g')^{-1}\mathcal{F})_{\overline{x}'} \\ & = \prod\nolimits_{\overline{x}' : \Spec(k) \to X',\ f' \circ \overline{x}' = \overline{y}'} \mathcal{F}_{g' \circ \overline{x}'} \\ & = \prod\nolimits_{\overline{x} : \Spec(k) \to X,\ f \circ \overline{x} = g \circ \overline{y}'} \mathcal{F}_{\overline{x}} \\ & = (f_*\mathcal{F})_{g \circ \overline{y}'} \\ & = (g^{-1}f_*\mathcal{F})_{\overline{y}'} \end{align*} The first equality by Proposition \ref{proposition-finite-higher-direct-image-zero}. The second equality by Lemma \ref{lemma-stalk-pullback}. The third equality holds because the diagram is a cartesian square and hence the map $$ \{\overline{x}' : \Spec(k) \to X',\ f' \circ \overline{x}' = \overline{y}'\} \longrightarrow \{\overline{x} : \Spec(k) \to X,\ f \circ \overline{x} = g \circ \overline{y}'\} $$ sending $\overline{x}'$ to $g' \circ \overline{x}'$ is a bijection. The fourth equality by Proposition \ref{proposition-finite-higher-direct-image-zero}. The fifth equality by Lemma \ref{lemma-stalk-pullback}.
The question is local on $Y$ and hence we may assume $Y$ is affine. Then we can write $X = \lim X_i$ with $f_i : X_i \to Y$ finite (this is easy in the affine case, but see Limits, Lemma \ref{limits-lemma-integral-limit-finite-and-finite-presentation} for a reference). Denote $p_{i'i} : X_{i'} \to X_i$ the transition morphisms and $p_i : X \to X_i$ the projections. Setting $\mathcal{F}_i = p_{i, *}\mathcal{F}$ we obtain from Lemma \ref{lemma-linus-hamann} a system $(\mathcal{F}_i, \varphi_{i'i})$ with $\mathcal{F} = \colim p_i^{-1}\mathcal{F}_i$. We get $f_*\mathcal{F} = \colim f_{i, *}\mathcal{F}_i$ from Lemma \ref{lemma-relative-colimit}. Set $X'_i = Y' \times_Y X_i$ with projections $f'_i$ and $g'_i$. Then $X' = \lim X'_i$ as limits commute with limits. Denote $p'_i : X' \to X'_i$ the projections. We have \begin{align*} g^{-1}f_*\mathcal{F} & = g^{-1} \colim f_{i, *}\mathcal{F}_i \\ & = \colim g^{-1}f_{i, *}\mathcal{F}_i \\ & = \colim f'_{i, *}(g'_i)^{-1}\mathcal{F}_i \\ & = f'_*(\colim (p'_i)^{-1}(g'_i)^{-1}\mathcal{F}_i) \\ & = f'_*(\colim (g')^{-1}p_i^{-1}\mathcal{F}_i) \\ & = f'_*(g')^{-1} \colim p_i^{-1}\mathcal{F}_i \\ & = f'_*(g')^{-1}\mathcal{F} \end{align*} as desired. For the first equality see above. For the second use that pullback commutes with colimits. For the third use the finite case, see Lemma \ref{lemma-finite-pushforward-commutes-with-base-change}. For the fourth use Lemma \ref{lemma-relative-colimit}. For the fifth use that $g'_i \circ p'_i = p_i \circ g'$. For the sixth use that pullback commutes with colimits. For the seventh use $\mathcal{F} = \colim p_i^{-1}\mathcal{F}_i$.
If we prove the first statement of the lemma, then we obtain a spectral sequence with $E_1^{p, q} = H^q_\etale(Y, \mathcal{F})$ converging to $H^{p + q}(Y_\etale, \mathcal{F})$, see Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}. On the other hand, since $R^if_{p, *}f_p^{-1}\mathcal{F} = 0$ for $i > 0$ (Proposition \ref{proposition-finite-higher-direct-image-zero}) we get $$ H^q_\etale(X_p, f_p^{-1}\mathcal{F}) = H^q_\etale(Y, f_{p, *}f_p^{-1} \mathcal{F}) = H^q_\etale(Y, \mathcal{F}_p) $$ by Proposition \ref{proposition-leray} and we get the spectral sequence of the lemma. \medskip\noindent To prove the first statement of the lemma, observe that $X_n$ forms a simplicial scheme over $Y$, see Simplicial, Example \ref{simplicial-example-fibre-products-simplicial-object}. Observe moreover, that for each of the projections $d_j : X_{n + 1} \to X_n$ there is a map $d_j^{-1} f_n^{-1}\mathcal{F} \to f_{n + 1}^{-1}\mathcal{F}$. These maps induce maps $$ \delta_j : \mathcal{F}_n \to \mathcal{F}_{n + 1} $$ for $j = 0, \ldots, n + 1$. We use the alternating sum of these maps to define the differentials $\mathcal{F}_n \to \mathcal{F}_{n + 1}$. Similarly, there is a canonical augmentation $\mathcal{F} \to \mathcal{F}_0$, namely this is just the canonical map $\mathcal{F} \to f_*f^{-1}\mathcal{F}$. To check that this sequence of sheaves is an exact complex it suffices to check on stalks at geometric points (Theorem \ref{theorem-exactness-stalks}). Thus we let $\overline{y} : \Spec(k) \to Y$ be a geometric point. Let $E = \{\overline{x} : \Spec(k) \to X \mid f(\overline{x}) = \overline{y}\}$. Then $E$ is a finite nonempty set and we see that $$ (\mathcal{F}_n)_{\overline{y}} = \bigoplus\nolimits_{e \in E^{n + 1}} \mathcal{F}_{\overline{y}} $$ by Proposition \ref{proposition-finite-higher-direct-image-zero} and Lemma \ref{lemma-stalk-pullback}. Thus we have to see that given an abelian group $M$ the sequence $$ 0 \to M \to \bigoplus\nolimits_{e \in E} M \to \bigoplus\nolimits_{e \in E^2} M \to \ldots $$ is exact. Here the first map is the diagonal map and the map $\bigoplus_{e \in E^{n + 1}} M  \to \bigoplus_{e \in E^{n + 2}} M$ is the alternating sum of the maps induced by the $(n + 2)$ projections $E^{n + 2} \to E^{n + 1}$. This can be shown directly or deduced by applying Simplicial, Lemma \ref{simplicial-lemma-fibre-products-simplicial-object-w-section} to the map $E \to \{*\}$.
We can prove this using formal arguments and the result of Theorem \ref{theorem-equivalence-sheaves-point} as follows. Given a sheaf $\mathcal{F}$ corresponding to the $G$-set $M = \mathcal{F}_{\overline{s}}$ we have \begin{eqnarray*} \Gamma(S, \mathcal{F}) & = & \Mor_{\Sh(S_\etale)}(h_{\Spec(K)},  \mathcal{F}) \\ & = & \Mor_{G\textit{-Sets}}(\{*\}, M) \\ & = & M^G \end{eqnarray*} Here the first identification is explained in Sites, Sections \ref{sites-section-presheaves} and \ref{sites-section-representable-sheaves}, the second results from Theorem \ref{theorem-equivalence-sheaves-point} and the third is clear. We will also give a direct proof\footnote{For the doubting Thomases out there.}. \medskip\noindent Suppose that $t \in \Gamma(S, \mathcal{F})$ is a global section. Then the triple $(S, \overline{s}, t)$ defines an element of $\mathcal{F}_{\overline{s}}$ which is clearly invariant under the action of $G$. Conversely, suppose that $(U, \overline{u}, t)$ defines an element of $\mathcal{F}_{\overline{s}}$ which is invariant. Then we may shrink $U$ and assume $U = \Spec(L)$ for some finite separable field extension of $K$, see Proposition \ref{proposition-etale-morphisms}. In this case the map $\mathcal{F}(U) \to \mathcal{F}_{\overline{s}}$ is injective, because for any morphism of \'etale neighbourhoods $(U', \overline{u}') \to (U, \overline{u})$ the restriction map $\mathcal{F}(U) \to \mathcal{F}(U')$ is injective since $U' \to U$ is a covering of $S_\etale$. After enlarging $L$ a bit we may assume $K \subset L$ is a finite Galois extension. At this point we use that $$ \Spec(L) \times_{\Spec(K)} \Spec(L) = \coprod\nolimits_{\sigma \in \text{Gal}(L/K)} \Spec(L) $$ where the maps $\Spec(L) \to \Spec(L \otimes_K L)$ come from the ring maps $a \otimes b \mapsto a\sigma(b)$. Hence we see that the condition that $(U, \overline{u}, t)$ is invariant under all of $G$ implies that $t \in \mathcal{F}(\Spec(L))$ maps to the same element of $\mathcal{F}(\Spec(L) \times_{\Spec(K)} \Spec(L))$ via restriction by either projection (this uses the injectivity mentioned above; details omitted). Hence the sheaf condition of $\mathcal{F}$ for the \'etale covering $\{\Spec(L) \to \Spec(K)\}$ kicks in and we conclude that $t$ comes from a unique section of $\mathcal{F}$ over $\Spec(K)$.
Let us denote the forgetful functor $F : \text{Mod}_{R, G} \to \text{Mod}_G$. Then $F$ has a left adjoint $H : \text{Mod}_G \to \text{Mod}_{R, G}$ given by $H(M) = M \otimes_\mathbf{Z} R$. Observe that every object of $\text{Mod}_G$ is a quotient of a direct sum of modules of the form $\mathbf{Z}[G/U]$ where $U \subset G$ is an open subgroup. Here $\mathbf{Z}[G/U]$ denotes the $G$-modules of finite $\mathbf{Z}$-linear combinations of right $U$ congruence classes in $G$ endowed with left $G$-action. Thus every bounded above complex in $\text{Mod}_G$ is quasi-isomorphic to a bounded above complex in $\text{Mod}_G$ whose underlying terms are flat $\mathbf{Z}$-modules (Derived Categories, Lemma \ref{derived-lemma-subcategory-left-resolution}). Thus it is clear that $LH$ exists on $D^-(\text{Mod}_G)$ and is computed by evaluating $H$ on any complex whose terms are flat $\mathbf{Z}$-modules; this follows from Derived Categories, Lemma \ref{derived-lemma-subcategory-left-acyclics} and Proposition \ref{derived-proposition-enough-acyclics}. We conclude from Derived Categories, Lemma \ref{derived-lemma-pre-derived-adjoint-functors} that $$ \text{Ext}^i(\mathbf{Z}, F(M)) = \text{Ext}^i(R, M) $$ for $M$ in $\textit{Mod}_{R, G}$. Observe that $H^0(G, -) = \Hom(\mathbf{Z}, -)$ on $\text{Mod}_G$ where $\mathbf{Z}$ denotes the $G$-module with trivial action. Hence $H^i(G, -) = \text{Ext}^i(\mathbf{Z}, -)$ on $\text{Mod}_G$. Similarly we have $H^i(G, -) = \text{Ext}^i(R, -)$ on $\text{Mod}_{R, G}$. Combining everything we see that the lemma is true.
The module $M^\vee = \Hom_R(M, R)$ endowed with the contragredient action of $G$. Namely $(g \cdot \lambda)(m) = \lambda(g^{-1} \cdot m)$ for $g \in G$, $\lambda \in M^\vee$, $m \in M$. The action of $G$ on $M^\vee \otimes_R N$ is the diagonal one, i.e., given by $g \cdot (\lambda \otimes n) = g \cdot \lambda \otimes g \cdot n$. Note that for a third $R\text{-}G$-module $E$ we have $\Hom(E, M^\vee \otimes_R N) = \Hom(M \otimes_R E, N)$. Namely, this is true on the level of $R$-modules by Algebra, Lemmas \ref{algebra-lemma-hom-from-tensor-product} and \ref{algebra-lemma-evaluation-map-iso-finite-projective} and the definitions of $G$-actions are chosen such that it remains true for $R\text{-}G$-modules. It follows that $M^\vee \otimes_R N$ is an injective $R\text{-}G$-module if $N$ is an injective $R\text{-}G$-module. Hence if $N \to N^\bullet$ is an injective resolution, then $M^\vee \otimes_R N \to M^\vee \otimes_R N^\bullet$ is an injective resolution. Then $$ \Hom(M, N^\bullet) = \Hom(R, M^\vee \otimes_R N^\bullet) = (M^\vee \otimes_R N^\bullet)^G $$ Since the left hand side computes $\text{Ext}^i(M, N)$ and the right hand side computes $H^i(G, M^\vee \otimes_R N)$ the proof is complete.
Let $g_1, \ldots, g_r \in G$ be elements which topologically generate $G$, i.e., this means that the subgroup generated by $g_1, \ldots, g_r$ is dense. By Lemma \ref{lemma-ext-modules-hom} we see that $H^1(G, V)$ is the $k$-vector space of extensions $$ 0 \to V \to E \to k \to 0 $$ of $k\text{-}G$-modules. Choose $e \in E$ mapping to $1 \in k$. Write $$ g_i \cdot e = v_i + e $$ for some $v_i \in V$. This is possible because $g_i \cdot 1 = 1$. We claim that the list of elements $v_1, \ldots, v_r \in V$ determine the isomorphism class of the extension $E$. Once we prove this the lemma follows as this means that our Ext vector space is isomorphic to a subquotient of the $k$-vector space $V^{\oplus r}$; some details omitted. Since $E$ is an object of the category defined in Definition \ref{definition-G-module-continuous} we know there is an open subgroup $U$ such that $u \cdot e = e$ for all $u \in U$. Now pick any $g \in G$. Then $gU$ contains a word $w$ in the elements $g_1, \ldots, g_r$. Say $gu = w$. Since the element $w \cdot e$ is determined by $v_1, \ldots, v_r$, we see that $g \cdot e = (gu) \cdot e = w \cdot e$ is too.