hcju/stacksqa huggingface dataset:
We will use the results of Lemma \ref{lemma-tangent-functor} without further mention. Denote $h : \textit{Alg}_A \to \textit{Sets}$ the functor given by $h(C) = \Mor_A(B[N], C)$. Similarly for $h_1$ and $h_2$. The map $B[N] \to B[N_2]$ corresponding to the surjection $N \to N_2$ is surjective. It corresponds to a map $h_2 \to h$ such that $h_2(C) \to h(C)$ is injective for all $A$-algebras $C$. On the other hand, there are two maps $p, q : h \to h_1$, corresponding to the zero map $N_1 \to N$ and the injection $N_1 \to N$. Note that $$ \xymatrix{ h_2 \ar[r] & h \ar@<1ex>[r] \ar@<-1ex>[r] & h_1 } $$ is an equalizer diagram. Denote $\mathcal{O}_h$ the module-valued functor $C \mapsto \bigoplus_{h(C)} C$. Similarly for $\mathcal{O}_{h_1}$ and $\mathcal{O}_{h_2}$. Note that $$ \Hom_\mathcal{P}(\mathcal{O}_h, F) = F(B[N]) $$ where $\mathcal{P}$ is the category of module-valued functors on $\textit{Alg}_A$. We claim there is an equalizer diagram $$ \xymatrix{ \mathcal{O}_{h_2} \ar[r] & \mathcal{O}_h \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{O}_{h_1} } $$ in $\mathcal{P}$. Namely, suppose that $C \in \Ob(\textit{Alg}_A)$ and $\xi = \sum_{i = 1, \ldots, n} c_i \cdot f_i$ where $c_i \in C$ and $f_i : B[N] \to C$ is an element of $\mathcal{O}_h(C)$. If $p(\xi) = q(\xi)$, then we see that $$ \sum c_i \cdot f_i \circ z = \sum c_i \cdot f_i \circ y $$ where $z, y : B[N_1] \to B[N]$ are the maps $z : (b, m_1) \mapsto (b, 0)$ and $y : (b, m_1) \mapsto (b, m_1)$. This means that for every $i$ there exists a $j$ such that $f_j \circ z = f_i \circ y$. Clearly, this implies that $f_i(N_1) = 0$, i.e., $f_i$ factors through a unique map $\overline{f}_i : B[N_2] \to C$. Hence $\xi$ is the image of $\overline{\xi} = \sum c_i \cdot \overline{f}_i$. Since $I$ is injective, it transforms this equalizer diagram into a coequalizer diagram $$ \xymatrix{ I(B[N_1]) \ar@<1ex>[r] \ar@<-1ex>[r] & I(B[N]) \ar[r] & I(B[N_2]) } $$ This diagram is compatible with the direct sum decompositions $I(B[N]) = I(B) \oplus TI(B, N)$ and $I(B[N_i]) = I(B) \oplus TI(B, N_i)$. The zero map $N \to N_1$ induces the zero map $TI(B, N) \to TI(B, N_1)$. Thus we see that the coequalizer property above means we have an exact sequence $TI(B, N_1) \to TI(B, N) \to TI(B, N_2) \to 0$ as desired.
We will use the results of Lemma \ref{lemma-tangent-functor} without further mention. The maps $N_1 \to N_1 \oplus N_2$ and $N_2 \to N_1 \oplus N_2$ give a map $TF(B, N_1) \oplus TF(B, N_2) \to TF(B, N_1 \oplus N_2)$ which is injective since the maps $N_1 \oplus N_2 \to N_1$ and $N_1 \oplus N_2 \to N_2$ induce an inverse. Since $TF$ is right exact we see that $TF(B, N_1) \to TF(B, N_1 \oplus N_2) \to TF(B, N_2) \to 0$ is exact. Hence $TF(B, N_1) \oplus TF(B, N_2) \to TF(B, N_1 \oplus N_2)$ is an isomorphism. This proves (1). \medskip\noindent To see (2) the only thing we need to show is that $x \cdot (b_1 + b_2) = x \cdot b_1 + x \cdot b_2$. (Associativity and additivity are clear.) To see this consider $$ N \xrightarrow{(b_1, b_2)} N \oplus N \xrightarrow{+} N $$ and apply $TF(B, -)$. \medskip\noindent Part (3) follows immediately from the fact that $N \otimes_B F(B) \to TF(B, N)$ is functorial in the pair $(B, N)$. \medskip\noindent Suppose $N$ is a finitely presented $B$-module. Choose a presentation $B^{\oplus m} \to B^{\oplus n} \to N \to 0$. This gives an exact sequence $$ TF(B, B^{\oplus m}) \to TF(B, B^{\oplus n}) \to TF(B, N) \to 0 $$ by right exactness of $TF(B, -)$. By part (1) we can write $TF(B, B^{\oplus m}) = TF(B, B)^{\oplus m}$ and $TF(B, B^{\oplus n}) = TF(B, B)^{\oplus n}$. Next, suppose that $B^{\oplus m} \to B^{\oplus n}$ is given by the matrix $T = (b_{ij})$. Then the induced map $TF(B, B)^{\oplus m} \to TF(B, B)^{\oplus n}$ is given by the matrix with entries $TF(B, b_{ij} \cdot 1_B)$. This combined with right exactness of $\otimes$ proves (4).
Since $L$ is linearly adequate there exists an exact sequence $$ 0 \to L \to \underline{A^{\oplus m}} \to \underline{A^{\oplus n}} \to \underline{P} \to 0 $$ Here $P = \Coker(A^{\oplus m} \to A^{\oplus n})$ is the cokernel of the map of finite free $A$-modules which is given by the definition of linearly adequate functors. By Lemma \ref{lemma-ext-group-zero-key} we have the vanishing of $\Ext^i_\mathcal{P}(\underline{P}, \underline{M})$ and $\Ext^i_\mathcal{P}(\underline{A}, \underline{M})$ for $i > 0$. Let $K = \Ker(\underline{A^{\oplus n}} \to \underline{P})$. By the long exact sequence of Ext groups associated to the exact sequence $0 \to K \to \underline{A^{\oplus n}} \to \underline{P} \to 0$ we conclude that $\Ext^i_\mathcal{P}(K, \underline{M}) = 0$ for $i > 0$. Repeating with the sequence $0 \to L \to \underline{A^{\oplus m}} \to K \to 0$ we win.
Choose an exact sequence $0 \to F \to \underline{M^0} \to \underline{M^1}$. Set $M^2 = \Coker(M^0 \to M^1)$ so that $0 \to F \to \underline{M^0} \to \underline{M^1} \to \underline{M^2} \to 0$ is a resolution. By Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor} we obtain a spectral sequence $$ R^pQ(\underline{M^q}) \Rightarrow R^{p + q}Q(F) $$ Since $Q(\underline{M^q}) = \underline{M^q}$ it suffices to prove $R^pQ(\underline{M}) = 0$, $p > 0$ for any $A$-module $M$. \medskip\noindent Choose an injective resolution $\underline{M} \to I^\bullet$ in the category $\mathcal{P}$. Suppose that $R^iQ(\underline{M})$ is nonzero. Then $\Ker(Q(I^i) \to Q(I^{i + 1}))$ is strictly bigger than the image of $Q(I^{i - 1}) \to Q(I^i)$. Hence by Lemma \ref{lemma-adequate-surjection-from-linear} there exists a linearly adequate functor $L$ and a map $\varphi : L \to Q(I^i)$ mapping into the kernel of $Q(I^i) \to Q(I^{i + 1})$ which does not factor through the image of $Q(I^{i - 1}) \to Q(I^i)$. Because $Q$ is a left adjoint to the inclusion functor the map $\varphi$ corresponds to a map $\varphi' : L \to I^i$ with the same properties. Thus $\varphi'$ gives a nonzero element of $\Ext^i_\mathcal{P}(L, \underline{M})$ contradicting Lemma \ref{lemma-ext-group-zero}.
Let $\{\Spec(A_i) \to S\}_{i \in I}$ be a $\tau$-covering such that $F_{\mathcal{F}, A_i}$ is adequate for all $i \in I$. We can find a standard affine $\tau$-covering $\{\Spec(A'_j) \to \Spec(A)\}_{j = 1, \ldots, m}$ such that $\Spec(A'_j) \to \Spec(A) \to S$ factors through $\Spec(A_{i(j)})$ for some $i(j) \in I$. Then we see that $F_{\mathcal{F}, A'_j}$ is the restriction of $F_{\mathcal{F}, A_{i(j)}}$ to the category of $A'_j$-algebras. Hence $F_{\mathcal{F}, A'_j}$ is adequate by Lemma \ref{lemma-base-change-adequate}. By Lemma \ref{lemma-adequate-product} the sequence $F_{\mathcal{F}, A'_j}$ corresponds to an adequate ``product'' functor $F'$ over $A' = A'_1 \times \ldots \times A'_m$. As $\mathcal{F}$ is a sheaf (for the Zariski topology) this product functor $F'$ is equal to $F_{\mathcal{F}, A'}$, i.e., is the restriction of $F$ to $A'$-algebras. Finally,  $\{\Spec(A') \to \Spec(A)\}$ is a $\tau$-covering. It follows from Lemma \ref{lemma-adequate-descent} that $F_{\mathcal{F}, A}$ is adequate.
Given an adequate module $\mathcal{F}$ the functor $F_{\mathcal{F}, A}$ is adequate by Lemma \ref{lemma-adequate-local}. Given an adequate functor $F$ we choose an exact sequence $0 \to F \to \underline{M} \to \underline{N}$ and we consider the $\mathcal{O}$-module $\mathcal{F} = \Ker(M^a \to N^a)$ where $M^a$ denotes the quasi-coherent $\mathcal{O}$-module on $(\Sch/S)_\tau$ associated to the quasi-coherent sheaf $\widetilde{M}$ on $S$. Note that $F = F_{\mathcal{F}, A}$, in particular the module $\mathcal{F}$ is adequate by definition. We omit the proof that the constructions define mutually inverse equivalences of categories.
The pullback map $f^* : \textit{Mod}((\Sch/S)_\tau, \mathcal{O}) \to \textit{Mod}((\Sch/T)_\tau, \mathcal{O})$ is given by restriction, i.e., $f^*\mathcal{F}(V) = \mathcal{F}(V)$ for any scheme $V$ over $T$. Hence this lemma follows immediately from Lemma \ref{lemma-adequate-local} and the definition.
Let $U = \Spec(A)$ be an affine scheme over $S$. Set $F = F_{\mathcal{F}, A}$. By definition, the functor $F$ is adequate if and only if there exists a map of $A$-modules $M \to N$ such that $F = \Ker(\underline{M} \to \underline{N})$. Combining with Lemmas \ref{lemma-adequate-local} and \ref{lemma-adequate-affine} we see that (1) and (5) are equivalent. \medskip\noindent It is clear that (5) implies (2) and (2) implies (3). If (3) holds then we can refine the covering $\{S_i \to S\}$ such that each $S_i = \Spec(A_i)$ is affine. Then we see, by the preliminary remarks of the proof, that $F_{\mathcal{F}, A_i}$ is adequate. Thus $\mathcal{F}$ is adequate by definition. Hence (3) implies (1). \medskip\noindent Finally, (4) is equivalent to (1) using Lemma \ref{lemma-pullback-adequate} for one direction and that a composition of $\tau$-coverings is a $\tau$-covering for the other.
With $A \to B$ as in the lemma let $F = F_{\mathcal{F}, A}$. This functor is adequate by Lemma \ref{lemma-adequate-local}. By Lemma \ref{lemma-adequate-flat} since $A \to B$, $A \to B \otimes_A B$, etc are flat we see that $F(B) = F(A) \otimes_A B$, $F(B \otimes_A B) = F(A) \otimes_A B \otimes_A B$, etc. Exactness follows from Descent, Lemma \ref{descent-lemma-ff-exact}. \medskip\noindent Thus $\mathcal{F}$ satisfies the sheaf condition for $\tau$-coverings (in particular Zariski coverings) and any faithfully flat covering of an affine by an affine. Arguing as in the proofs of Descent, Lemma \ref{descent-lemma-standard-fpqc-covering} and Descent, Proposition \ref{descent-proposition-fpqc-descent-quasi-coherent} we conclude that $\mathcal{F}$ satisfies the sheaf condition for all fpqc coverings (made out of objects of $(\Sch/S)_\tau$). Details omitted.
By Lemma \ref{lemma-adequate-flat} and Lemma \ref{lemma-adequate-local} we see that for any flat morphism of affines $U \to V$ over $S$ we have $\mathcal{F}(U) = \mathcal{F}(V) \otimes_{\mathcal{O}(V)} \mathcal{O}(U)$. This works in particular if $U \subset V \subset S$ are affine opens of $S$, hence $\mathcal{F}|_{S_{Zar}}$ is quasi-coherent. Thus (1) holds. \medskip\noindent Let $S' \to S$ be an \'etale morphism of schemes. Then for $U \subset S'$ affine open mapping into an affine open $V \subset S$ we see that $\mathcal{F}(U) = \mathcal{F}(V) \otimes_{\mathcal{O}(V)} \mathcal{O}(U)$ because $U \to V$ is \'etale, hence flat. Therefore $\mathcal{F}|_{S'_{Zar}}$ is the pullback of $\mathcal{F}|_{S_{Zar}}$. This proves (2). \medskip\noindent We are going to apply Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cech-vanish-collection} to the site $(\Sch/S)_\tau$ with $\mathcal{B}$ the set of affine schemes over $S$ and $\text{Cov}$ the set of standard affine $\tau$-coverings. Assumption (3) of the lemma is satisfied by Descent, Lemma \ref{descent-lemma-standard-covering-Cech} and Lemma \ref{lemma-adequate-fpqc} for the case of a covering by a single affine. Hence we conclude that $H^p(U, \mathcal{F}) = 0$ for every affine scheme $U$ over $S$. This proves (3). In exactly the same way as in the proof of Descent, Proposition \ref{descent-proposition-same-cohomology-quasi-coherent} this implies the equality of cohomologies (4).
Let $U = \Spec(A)$ be an affine scheme over $S$. Set $F = F_{\mathcal{F}, A}$. The sheafification $\mathcal{F}^\# = (\mathcal{F}^+)^+$, see Sites, Section \ref{sites-section-sheafification}. By construction $$ (\mathcal{F})^+(U) = \colim_\mathcal{U} \check{H}^0(\mathcal{U}, \mathcal{F}) $$ where the colimit is over coverings in the site $(\Sch/S)_\tau$. Since $U$ is affine it suffices to take the limit over standard affine $\tau$-coverings $\mathcal{U} = \{U_i \to U\}_{i \in I} = \{\Spec(A_i) \to \Spec(A)\}_{i \in I}$ of $U$. Since each $A \to A_i$ and $A \to A_i \otimes_A A_j$ is flat we see that $$ \check{H}^0(\mathcal{U}, \mathcal{F}) = \Ker(\prod F(A) \otimes_A A_i \to \prod F(A) \otimes_A A_i \otimes_A A_j) $$ by Lemma \ref{lemma-adequate-flat}. Since $A \to \prod A_i$ is faithfully flat we see that this always is canonically isomorphic to $F(A)$ by Descent, Lemma \ref{descent-lemma-ff-exact}. Thus the presheaf $(\mathcal{F})^+$ has the same value as $\mathcal{F}$ on all affine schemes over $S$. Repeating the argument once more we deduce the same thing for $\mathcal{F}^\# = ((\mathcal{F})^+)^+$. Thus $F_{\mathcal{F}, A} = F_{\mathcal{F}^\#, A}$ and we conclude that $\mathcal{F}^\#$ is adequate.
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of adequate $\mathcal{O}$-modules. To prove (1) and (2) it suffices to show that $\mathcal{K} = \Ker(\varphi)$ and $\mathcal{Q} = \Coker(\varphi)$ computed in $\textit{Mod}((\Sch/S)_\tau, \mathcal{O})$ are adequate. Let $U = \Spec(A)$ be an affine scheme over $S$. Let $F = F_{\mathcal{F}, A}$ and $G = F_{\mathcal{G}, A}$. By Lemmas \ref{lemma-kernel-adequate} and \ref{lemma-cokernel-adequate} the kernel $K$ and cokernel $Q$ of the induced map $F \to G$ are adequate functors. Because the kernel is computed on the level of presheaves, we see that $K = F_{\mathcal{K}, A}$ and we conclude $\mathcal{K}$ is adequate. To prove the result for the cokernel, denote $\mathcal{Q}'$ the presheaf cokernel of $\varphi$. Then $Q = F_{\mathcal{Q}', A}$ and $\mathcal{Q} = (\mathcal{Q}')^\#$. Hence $\mathcal{Q}$ is adequate by Lemma \ref{lemma-sheafification-adequate}. \medskip\noindent Let $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ is a short exact sequence of $\mathcal{O}$-modules and $\mathcal{F}_1$ and $\mathcal{F}_3$ are adequate. Let $U = \Spec(A)$ be an affine scheme over $S$. Let $F_i = F_{\mathcal{F}_i, A}$. The sequence of functors $$ 0 \to F_1 \to F_2 \to F_3 \to 0 $$ is exact, because for $V = \Spec(B)$ affine over $U$ we have $H^1(V, \mathcal{F}_1) = 0$ by Lemma \ref{lemma-same-cohomology-adequate}. Since $F_1$ and $F_3$ are adequate functors by Lemma \ref{lemma-adequate-local} we see that $F_2$ is adequate by Lemma \ref{lemma-extension-adequate}. Thus $\mathcal{F}_2$ is adequate. \medskip\noindent Let $\mathcal{I} \to \textit{Adeq}(\mathcal{O})$, $i \mapsto \mathcal{F}_i$ be a diagram. Denote $\mathcal{F} = \colim_i \mathcal{F}_i$ the colimit computed in $\textit{Mod}((\Sch/S)_\tau, \mathcal{O})$. To prove (4) it suffices to show that $\mathcal{F}$ is adequate. Let $\mathcal{F}' = \colim_i \mathcal{F}_i$ be the colimit computed in presheaves of $\mathcal{O}$-modules. Then $\mathcal{F} = (\mathcal{F}')^\#$. Let $U = \Spec(A)$ be an affine scheme over $S$. Let $F_i = F_{\mathcal{F}_i, A}$. By Lemma \ref{lemma-colimit-adequate} the functor $\colim_i F_i = F_{\mathcal{F}', A}$ is adequate. Lemma \ref{lemma-sheafification-adequate} shows that $\mathcal{F}$ is adequate.
The implications (2) $\Rightarrow$ (3) $\Rightarrow$ (4) $\Rightarrow$ (5) are immediate from the definitions. Assume (5). Suppose that $S = \bigcup U_i$ is an affine open covering such that $\mathcal{F}(U_i) = 0$ for all $i$. Let $V \to S$ be a flat morphism. There exists an affine open covering $V = \bigcup V_j$ such that each $V_j$ maps into some $U_i$. As the morphism $V_j \to S$ is flat, also $V_j \to U_i$ is flat.  Hence the corresponding ring map $A_i = \mathcal{O}(U_i) \to \mathcal{O}(V_j) = B_j$ is flat. Thus by Lemma \ref{lemma-adequate-local} and Lemma \ref{lemma-adequate-flat} we see that $\mathcal{F}(U_i) \otimes_{A_i} B_j \to \mathcal{F}(V_j)$ is an isomorphism. Hence $\mathcal{F}(V_j) = 0$. Since $\mathcal{F}$ is a sheaf for the Zariski topology we conclude that $\mathcal{F}(V) = 0$. In this way we see that (5) implies (2). \medskip\noindent This proves the equivalence of (2), (3), (4), and (5). As (1) is equivalent to (3) (see Remark \ref{remark-compare}) we conclude that all five conditions are equivalent.
The category $\mathcal{C}$ is the kernel of the exact functor $v : \textit{Adeq}(\mathcal{O}) \to \QCoh(\mathcal{O}_S)$, see Lemma \ref{lemma-parasitic-adequate}. Hence it is a Serre subcategory by Homology, Lemma \ref{homology-lemma-kernel-exact-functor}. By Homology, Lemma \ref{homology-lemma-serre-subcategory-is-kernel} we obtain an induced exact functor $\overline{v} : \textit{Adeq}(\mathcal{O}) / \mathcal{C} \to \QCoh(\mathcal{O}_S)$. Because $u$ is a right inverse to $v$ we see right away that $\overline{v}$ is essentially surjective. We see that $\overline{v}$ is faithful by Homology, Lemma \ref{homology-lemma-quotient-by-kernel-exact-functor}. Because $u$ is a right inverse to $v$ we finally conclude that $\overline{v}$ is fully faithful.
We have already seen in Lemma \ref{lemma-direct-image-adequate} that these higher direct images are adequate. Hence it suffices to show that $(R^if_*\mathcal{F})(U_i) = 0$ for any $\tau$-covering $\{U_i \to S\}$ open. And $R^if_*\mathcal{F}$ is parasitic by Descent, Lemma \ref{descent-lemma-direct-image-parasitic}.
If $\varphi : \mathcal{F}^\bullet \to \mathcal{G}^\bullet$ is a morphism of complexes of quasi-coherent $\mathcal{O}_U$-modules, then $u\varphi : u\mathcal{F}^\bullet \to u\mathcal{G}^\bullet$ is a quasi-isomorphism if and only if $\varphi$ is a universal quasi-isomorphism. Hence the collection $S$ of universal quasi-isomorphisms is a saturated multiplicative system compatible with the triangulated structure by Derived Categories, Lemma \ref{derived-lemma-triangle-functor-localize}. Hence $S^{-1}K^+(\QCoh(\mathcal{O}_U))$ exists and is a triangulated category, see Derived Categories, Proposition \ref{derived-proposition-construct-localization}. We obtain a canonical functor $can : S^{-1}K^+(\QCoh(\mathcal{O}_U)) \to D^{+}(\textit{Adeq}(\mathcal{O}))$ by Derived Categories, Lemma \ref{derived-lemma-universal-property-localization}. \medskip\noindent Note that, almost by definition, every adequate module on $U$ has an embedding into a quasi-coherent sheaf, see Lemma \ref{lemma-adequate-characterize}. Hence by Derived Categories, Lemma \ref{derived-lemma-subcategory-right-resolution} given $\mathcal{F}^\bullet \in \Ob(K^+(\textit{Adeq}(\mathcal{O})))$ there exists a quasi-isomorphism $\mathcal{F}^\bullet \to u\mathcal{G}^\bullet$ where $\mathcal{G}^\bullet \in \Ob(K^+(\QCoh(\mathcal{O}_U)))$. This proves that $can$ is essentially surjective. \medskip\noindent Similarly, suppose that $\mathcal{F}^\bullet$ and $\mathcal{G}^\bullet$ are bounded below complexes of quasi-coherent $\mathcal{O}_U$-modules. A morphism in $D^+(\textit{Adeq}(\mathcal{O}))$ between these consists of a pair $f : u\mathcal{F}^\bullet \to \mathcal{H}^\bullet$ and $s : u\mathcal{G}^\bullet \to \mathcal{H}^\bullet$ where $s$ is a quasi-isomorphism. Pick a quasi-isomorphism $s' : \mathcal{H}^\bullet \to u\mathcal{E}^\bullet$. Then we see that $s' \circ f : \mathcal{F} \to \mathcal{E}^\bullet$ and the universal quasi-isomorphism $s' \circ s : \mathcal{G}^\bullet \to \mathcal{E}^\bullet$ give a morphism in $S^{-1}K^{+}(\QCoh(\mathcal{O}_U))$ mapping to the given morphism. This proves the "fully" part of full faithfulness. Faithfulness is proved similarly.
By Topologies, Lemma \ref{topologies-lemma-affine-big-site-fppf} (and similarly for the other topologies) we may work with $\mathcal{O}$-modules on $(\textit{Aff}/U)_\tau$. Denote $\mathcal{P}$ the category of module-valued functors on $\textit{Alg}_A$ and $\mathcal{A}$ the category of adequate functors on $\textit{Alg}_A$. Denote $i : \mathcal{A} \to \mathcal{P}$ the inclusion functor. Denote $Q : \mathcal{P} \to \mathcal{A}$ the construction of Lemma \ref{lemma-adjoint}. We have the commutative diagram \begin{equation} \label{equation-categories} \vcenter{ \xymatrix{ \textit{Adeq}(\mathcal{O}) \ar[r]_-k \ar@{=}[d] & \textit{Mod}((\textit{Aff}/U)_\tau, \mathcal{O}) \ar[r]_-j & \textit{PMod}((\textit{Aff}/U)_\tau, \mathcal{O}) \ar@{=}[d] \\ \mathcal{A} \ar[rr]^-i & & \mathcal{P} } } \end{equation} The left vertical equality is Lemma \ref{lemma-adequate-affine} and the right vertical equality was explained in Section \ref{section-quasi-coherent}. Define $A(\mathcal{F}) = Q(j(\mathcal{F}))$. Since $j$ is fully faithful it follows immediately that $A$ is a right adjoint of the inclusion functor $k$. Also, since $k$ is fully faithful too, the final assertion follows formally.
With notation as in the proof of Lemma \ref{lemma-right-adjoint-adequate} choose an injective resolution $k(\mathcal{F}) \to \mathcal{I}^\bullet$ in the category of $\mathcal{O}$-modules on $(\textit{Aff}/U)_\tau$. By Cohomology on Sites, Lemmas \ref{sites-cohomology-lemma-include-modules} and Lemma \ref{lemma-same-cohomology-adequate} the complex $j(\mathcal{I}^\bullet)$ is exact. On the other hand, each $j(\mathcal{I}^n)$ is an injective object of the category of presheaves of modules by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-injective-module-injective-presheaf}. It follows that $R^pA(\mathcal{F}) = R^pQ(j(k(\mathcal{F})))$. Hence the result now follows from Lemma \ref{lemma-RQ-zero}.
Let $A : \textit{Mod}(\mathcal{O}) \to \textit{Adeq}(\mathcal{O})$ be the right adjoint to the inclusion functor constructed in Lemma \ref{lemma-right-adjoint-adequate}. Since $A$ is left exact and since $\textit{Mod}(\mathcal{O})$ has enough injectives, $A$ has a right derived functor $RA : D^+_{\textit{Adeq}}(\mathcal{O}) \to D^+(\textit{Adeq}(\mathcal{O}))$. We claim that $RA$ is a quasi-inverse to (\ref{equation-compare-bounded-adequate}). To see this the key fact is that if $\mathcal{F}$ is an adequate module, then the adjunction map $\mathcal{F} \to RA(\mathcal{F})$ is a quasi-isomorphism by Lemma \ref{lemma-RA-zero}. \medskip\noindent Namely, to prove the lemma in full it suffices to show: \begin{enumerate} \item Given $\mathcal{F}^\bullet \in K^+(\textit{Adeq}(\mathcal{O}))$ the canonical map $\mathcal{F}^\bullet \to RA(\mathcal{F}^\bullet)$ is a quasi-isomorphism, and \item given $\mathcal{G}^\bullet \in K^+(\textit{Mod}(\mathcal{O}))$ the canonical map $RA(\mathcal{G}^\bullet) \to \mathcal{G}^\bullet$ is a quasi-isomorphism. \end{enumerate} Both (1) and (2) follow from the key fact via a spectral sequence argument using one of the spectral sequences of Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}. Some details omitted.
First note that a finitely presented $A$-module is pure projective by Algebra, Theorem \ref{algebra-theorem-universally-exact-criteria}. Hence a direct summand of a direct sum of finitely presented $A$-modules is indeed pure projective. Let $M$ be any $A$-module. Write $M = \colim_{i \in I} P_i$ as a filtered colimit of finitely presented $A$-modules. Consider the sequence $$ 0 \to N \to \bigoplus P_i \to M \to 0. $$ For any finitely presented $A$-module $P$ the map $\Hom_A(P, \bigoplus P_i) \to \Hom_A(P, M)$ is surjective, as any map $P \to M$ factors through some $P_i$. Hence by Algebra, Theorem \ref{algebra-theorem-universally-exact-criteria} this sequence is universally exact. This proves (2). If now $M$ is pure projective, then the sequence is split and we see that $M$ is a direct summand of $\bigoplus P_i$.
Part (1) follows immediately from Lemma \ref{lemma-pure-projective}. Before we prove (2) and (3) note that by Lemma \ref{lemma-split-universally-exact-sequence} we can split the universally exact complex $N_\bullet \to N \to 0$ into universally exact short exact sequences $0 \to K_0 \to N_0 \to N \to 0$ and $0 \to K_i \to N_i \to K_{i - 1} \to 0$. \medskip\noindent Proof of (2). Because $P_0$ is pure projective we can find a map $P_0 \to N_0$ lifting the map $P_0 \to M \to N$. We obtain an induced map $P_1 \to F_0 \to N_0$ wich ends up in $K_0$. Since $P_1$ is pure projective we may lift this to a map $P_1 \to N_1$. This in turn induces a map $P_2 \to P_1 \to N_1$ which maps to zero into $N_0$, i.e., into $K_1$. Hence we may lift to get a map $P_2 \to N_2$. Repeat. \medskip\noindent Proof of (3). To show that $\alpha, \beta$ are homotopic it suffices to show the difference $\gamma = \alpha - \beta$ is homotopic to zero. Note that the image of $\gamma_0 : P_0 \to N_0$ is contained in $K_0$. Hence we may lift $\gamma_0$ to a map $h_0 : P_0 \to N_1$. Consider the map $\gamma_1' = \gamma_1 - h_0 \circ d_{P, 1} : P_1 \to N_1$. By our choice of $h_0$ we see that the image of $\gamma_1'$ is contained in $K_1$. Since $P_1$ is pure projective may lift $\gamma_1'$ to a map $h_1 : P_1 \to N_2$. At this point we have $\gamma_1 = h_0 \circ d_{F, 1} + d_{N, 2} \circ h_1$. Repeat.