hcju/stacksqa huggingface dataset:
Proof of (1). By Lemma \ref{lemma-resolve} we can choose a good left differential graded $\mathcal{Q}$ and a quasi-isomorphism $\mathcal{Q} \to \mathcal{N}$. Then $\mathcal{P} \otimes_\mathcal{A} \mathcal{Q}$ is acyclic because $\mathcal{Q}$ is good. Let $\mathcal{N}'$ be the cone on the map $\mathcal{Q} \to \mathcal{N}$. Then $\mathcal{P} \otimes_\mathcal{A} \mathcal{N}'$ is acyclic because $\mathcal{P}$ is good and because $\mathcal{N}'$ is acyclic (as the cone on a quasi-isomorphism). We have a distinguished triangle $$ \mathcal{Q} \to \mathcal{N} \to \mathcal{N}' \to \mathcal{Q}[1] $$ in $K(\text{Mod}_{(\mathcal{A}, \text{d})})$ by our construction of the triangulated structure. Since $\mathcal{P} \otimes_\mathcal{A} -$ sends distinguished triangles to distinguished triangles, we obtain a distinguished triangle $$ \mathcal{P} \otimes_\mathcal{A} \mathcal{Q} \to \mathcal{P} \otimes_\mathcal{A} \mathcal{N} \to \mathcal{P} \otimes_\mathcal{A} \mathcal{N}' \to \mathcal{P} \otimes_\mathcal{A} \mathcal{Q}[1] $$ in $K(\textit{Mod}(\mathcal{O}))$. Thus we conclude. \medskip\noindent Proof of (2). Observe that $f^*\mathcal{P}$ is good by our definition of good modules. Recall that $f^*\mathcal{P} = f^{-1}\mathcal{P} \otimes_{f^{-1}\mathcal{A}} \mathcal{A}'$. Then $f^{-1}\mathcal{P}$ is a good acyclic (because $f^{-1}$ is exact) differential graded $f^{-1}\mathcal{A}$-module. Hence we see that $f^*\mathcal{P}$ is acyclic by part (1).
To prove the existence of $G$ we can use the adjoint functor theorem, see Categories, Theorem \ref{categories-theorem-adjoint-functor} (observe that we have switched the roles of $F$ and $G$). The exactness conditions on $F$ are satisfied by Lemma \ref{lemma-dgm-abelian}. The set theoretic condition can be seen as follows: suppose given a graded $\mathcal{A}$-module $\mathcal{N}$. Then for any map $$ \varphi : \mathcal{N} \longrightarrow F(\mathcal{M}) $$ we can consider the smallest differential graded $\mathcal{A}$-submodule $\mathcal{M}' \subset \mathcal{M}$ with $\Im(\varphi) \subset F(\mathcal{M}')$. It is clear that $\mathcal{M}'$ is the image of the map of graded $\mathcal{A}$-modules $$ \mathcal{N} \oplus \mathcal{N}[-1] \otimes_\mathcal{O} \mathcal{A} \longrightarrow \mathcal{M} $$ defined by $$ (n, \sum n_i \otimes a_i) \longmapsto \varphi(n) + \sum \text{d}(\varphi(n_i)) a_i $$ because the image of this map is easily seen to be a differential graded submodule of $\mathcal{M}$. Thus the number of possible isomorphism classes of these $\mathcal{M}'$ is bounded and we conclude.
Let $T$ and $\mathcal{N}_t \to \mathcal{N}'_t$ be as in Lemma \ref{lemma-characterize-injectives}. Denote $F : \text{Mod}_{(\mathcal{A}, \text{d})} \to \text{Mod}_\mathcal{A}$ the forgetful functor. Let $G$ be the left adjoint functor to $F$ as in Lemma \ref{lemma-dg-hull}. Set $$ \mathcal{M}_t = G(\mathcal{N}_t) \to G(\mathcal{N}'_t) = \mathcal{M}'_t $$ This is an injective map of acyclic differential graded $\mathcal{A}$-modules by Lemma \ref{lemma-dg-hull-acyclic}. Since $G$ is the left adjoint to $F$ we see that there exists a dotted arrow in the diagram $$ \xymatrix{ \mathcal{M}_t \ar[r] \ar[d] & \mathcal{I} \\ \mathcal{M}'_t \ar@{..>}[ru] } $$ if and only if there exists a dotted arrow in the diagram $$ \xymatrix{ \mathcal{N}_t \ar[r] \ar[d] & F(\mathcal{I}) \\ \mathcal{N}'_t \ar@{..>}[ru] } $$ Hence the result follows from the choice of our collection of arrows $\mathcal{N}_t \to \mathcal{N}_t'$.
After replacing $\mathcal{M}'$ by the direct sum of $\mathcal{M}'$ and the cone on the identity on $\mathcal{M}$ (which is acyclic) we may assume $b$ is also injective. Then the cokernel $\mathcal{Q}$ of $b$ is acyclic. Thus we see that $$ \Hom_{K(\text{Mod}_{(\mathcal{A}, \text{d})})}(\mathcal{Q}, \mathcal{I}) = \Hom_{K(\text{Mod}_{(\mathcal{A}, \text{d})})}(\mathcal{Q}, \mathcal{I})[1] = 0 $$ as $\mathcal{I}$ is K-injective. As $\mathcal{I}$ is graded injective by Remark \ref{remark-why-graded-injective} we see that $$ \Hom_{K(\text{Mod}_{(\mathcal{A}, \text{d})})}(\mathcal{M}', \mathcal{I}) \longrightarrow \Hom_{K(\text{Mod}_{(\mathcal{A}, \text{d})})}(\mathcal{M}, \mathcal{I}) $$ is bijective and the proof is complete.
It follows immediately from the definitions that there is a commutative diagram $$ \xymatrix{ \text{Mod}_{(\mathcal{A}, \text{d})} \ar[r] \ar[d] & K(\text{Mod}_{(\mathcal{A}, \text{d})}) \ar[d] \\ \text{Comp}(\mathcal{O}) \ar[r] & K(\textit{Mod}(\mathcal{O})) } $$ Since $H^0(\mathcal{M})$ is defined as the zeroth cohomology sheaf of the underlying complex of $\mathcal{O}$-modules of $\mathcal{M}$ the lemma follows from the case of complexes of $\mathcal{O}$-modules which is a special case of Derived Categories, Lemma \ref{derived-lemma-cohomology-homological}.
Of course an object $\mathcal{M}$ of $K(\text{Mod}_{(\mathcal{A}, \text{d})})$ is in $\text{Ac}$ if and only if $H^i(\mathcal{M}) = H^0(\mathcal{M}[i])$ is zero for all $i$. The lemma follows from this, Lemma \ref{lemma-cohomology-homological}, and Derived Categories, Lemma \ref{derived-lemma-homological-functor-kernel}. See also Derived Categories, Definitions \ref{derived-definition-saturated} and \ref{derived-definition-triangulated-subcategory} and Lemma \ref{derived-lemma-triangulated-subcategory}.
Observe that if $f , g : \mathcal{M} \to \mathcal{N}$ are morphisms of $\text{Mod}_{(\mathcal{A}, \text{d})}$ which are homotopic, then $f$ is a quasi-isomorphism if and only if $g$ is a quasi-isomorphism. Namely, the maps $H^i(f) = H^0(f[i])$ and $H^i(g) = H^0(g[i])$ are the same by Lemma \ref{lemma-cohomology-homological}. Thus it is unambiguous to say that a morphism of the homotopy category $K(\text{Mod}_{(A, \text{d})})$ is a quasi-isomorphism. For definitions of ``multiplicative system'', ``saturated'', and ``compatible with the triangulated structure'' see Derived Categories, Definition \ref{derived-definition-localization} and Categories, Definitions \ref{categories-definition-multiplicative-system} and \ref{categories-definition-saturated-multiplicative-system}. \medskip\noindent To actually prove the lemma consider the composition of exact functors of triangulated categories $$ K(\text{Mod}_{(\mathcal{A}, \text{d})}) \longrightarrow K(\textit{Mod}(\mathcal{O})) \longrightarrow D(\mathcal{O}) $$ and observe that a morphism $f : \mathcal{M} \to \mathcal{N}$ of $K(\text{Mod}_{(\mathcal{A}, \text{d})})$ is in $\text{Qis}$ if and only if it maps to an isomorphism in $D(\mathcal{O})$. Thus the lemma follows from Derived Categories, Lemma \ref{derived-lemma-triangle-functor-localize}.
Since $\mathcal{N} \to \mathcal{I}$ is a quasi-isomorphism we see that $$ \Hom_{D(\mathcal{A}, \text{d})}(\mathcal{M}, \mathcal{N}) = \Hom_{D(\mathcal{A}, \text{d})}(\mathcal{M}, \mathcal{I}) $$ In the discussion preceding Definition \ref{definition-derived-category} we found, using Lemma \ref{lemma-second-property-dg-injective}, that any morphism $\mathcal{M} \to \mathcal{I}$ in $D(\mathcal{A}, \text{d})$ can be represented by a morphism $f : \mathcal{M} \to \mathcal{I}$ in $K(\text{Mod}_{(\mathcal{A}, \text{d})})$. Now, if $f, f' :  \mathcal{M} \to \mathcal{I}$ are two morphism in $K(\text{Mod}_{(\mathcal{A}, \text{d})})$, then they define the same morphism in $D(\mathcal{A}, \text{d})$ if and only if there exists a quasi-isomorphism $g : \mathcal{I} \to \mathcal{K}$ in $K(\text{Mod}_{(\mathcal{A}, \text{d})})$ such that $g \circ f = g \circ f'$, see Categories, Lemma \ref{categories-lemma-equality-morphisms-left-localization}. However, by Lemma \ref{lemma-second-property-dg-injective} there exists a map $h : \mathcal{K} \to \mathcal{I}$ such that $h \circ g = \text{id}_\mathcal{I}$ in in $K(\text{Mod}_{(\mathcal{A}, \text{d})})$. Thus $g \circ f = g \circ f'$ implies $f = f'$ and the proof is complete.
We will use that $\text{Mod}_{(A, \text{d})}$ is an abelian category with arbitrary direct sums and products, and that these give rise to direct sums and products in $K(\text{Mod}_{(A, \text{d})})$. See Lemmas \ref{lemma-dgm-abelian} and \ref{lemma-homotopy-direct-sums}. \medskip\noindent Let $\mathcal{M}_j$ be a family of differential graded $\mathcal{A}$-modules. Consider the direct sum $\mathcal{M} = \bigoplus \mathcal{M}_j$ as a differential graded $\mathcal{A}$-module. For a differential graded $\mathcal{A}$-module $\mathcal{N}$ choose a quasi-isomorphism $\mathcal{N} \to \mathcal{I}$ where $\mathcal{I}$ is graded injective and K-injective as a differential graded $\mathcal{A}$-module. See Theorem \ref{theorem-qis-into-dg-injective}. Using Lemma \ref{lemma-hom-derived} we have \begin{align*} \Hom_{D(\mathcal{A}, \text{d})}(\mathcal{M}, \mathcal{N}) & = \Hom_{K(\mathcal{A}, \text{d})}(\mathcal{M}, \mathcal{I}) \\ & = \prod \Hom_{K(\mathcal{A}, \text{d})}(\mathcal{M}_j, \mathcal{I}) \\ & = \prod \Hom_{D(\mathcal{A}, \text{d})}(\mathcal{M}_j, \mathcal{I}) \end{align*} whence the existence of direct sums in $D(A, \text{d})$ as given in part (2) of the lemma. \medskip\noindent Let $\mathcal{M}_j$ be a family of differential graded $\mathcal{A}$-modules. For each $j$ choose a quasi-isomorphism $\mathcal{M} \to \mathcal{I}_j$ where $\mathcal{I}_j$ is graded injective and K-injective as a differential graded $\mathcal{A}$-module. Consider the product $\mathcal{I} = \prod \mathcal{I}_j$ of differential graded $\mathcal{A}$-modules. By Lemmas \ref{lemma-product-K-injective} and \ref{lemma-product-graded-injective} we see that $\mathcal{I}$ is graded injective and K-injective as a differential graded $\mathcal{A}$-module. For a differential graded $\mathcal{A}$-module $\mathcal{N}$ using Lemma \ref{lemma-hom-derived} we have \begin{align*} \Hom_{D(\mathcal{A}, \text{d})}(\mathcal{N}, \mathcal{I}) & = \Hom_{K(\mathcal{A}, \text{d})}(\mathcal{N}, \mathcal{I}) \\ & = \prod \Hom_{K(\mathcal{A}, \text{d})}(\mathcal{N}, \mathcal{I}_j) \\ & = \prod \Hom_{D(\mathcal{A}, \text{d})}(\mathcal{N}, \mathcal{M}_j) \end{align*} whence the existence of products in $D(\mathcal{A}, \text{d})$ as given in part (3) of the lemma.
We have already seen that this choice leads to a distinguished triangle whenever given a short exact sequence of complexes. We have to show functoriality of this construction, see Derived Categories, Definition \ref{derived-definition-delta-functor}. This follows from Differential Graded Algebra, Lemma \ref{dga-lemma-cone} (which we may use by the discussion in Section \ref{section-conclude-triangulated}) with a bit of work. Compare with Derived Categories, Lemma \ref{derived-lemma-derived-canonical-delta-functor}.
Set $\mathcal{M} = \colim \mathcal{M}_n$. We have an exact sequence of differential graded $\mathcal{A}$-modules $$ 0 \to \bigoplus \mathcal{M}_n \to \bigoplus \mathcal{M}_n \to \mathcal{M} \to 0 $$ by Derived Categories, Lemma \ref{derived-lemma-compute-colimit} (applied the underlying complexes of $\mathcal{O}$-modules). The direct sums are direct sums in $D(\mathcal{A}, \text{d})$ by Lemma \ref{lemma-derived-products}. Thus the result follows from the definition of derived colimits in Derived Categories, Definition \ref{derived-definition-derived-colimit} and the fact that a short exact sequence of complexes gives a distinguished triangle (Lemma \ref{lemma-derived-canonical-delta-functor}).
Recall that for any (right) differential graded $\mathcal{B}$-module $\mathcal{M}$ there exists a quasi-isomorphism $\mathcal{P} \to \mathcal{M}$ with $\mathcal{P}$ a good differential graded $\mathcal{B}$-module. See Lemma \ref{lemma-resolve}. Hence by Derived Categories, Lemma \ref{derived-lemma-find-existence-computes} it suffices to show that given a quasi-isomorphism $\mathcal{P} \to \mathcal{P}'$ of good differential graded $\mathcal{B}$-modules the induced map $$ f^*\mathcal{P} \otimes_\mathcal{A} \mathcal{N} \longrightarrow f^*\mathcal{P}' \otimes_\mathcal{A} \mathcal{N} $$ is a quasi-isomorphism. The cone $\mathcal{P}''$ on $\mathcal{P} \to \mathcal{P}'$ is a good differential graded $\mathcal{A}$-module by Lemma \ref{lemma-good-admissible-ses}. Since we have a distinguished triangle $$ \mathcal{P} \to \mathcal{P}' \to \mathcal{P}'' \to \mathcal{P}[1] $$ in $K(\text{Mod}_{(\mathcal{B}, \text{d})})$ we obtain a distinguished triangle $$ f^*\mathcal{P} \otimes_\mathcal{A} \mathcal{N} \to f^*\mathcal{P}' \otimes_\mathcal{A} \mathcal{N} \to f^*\mathcal{P}'' \otimes_\mathcal{A} \mathcal{N} \to f^*\mathcal{P}[1] \otimes_\mathcal{A} \mathcal{N} $$ in $K(\text{Mod}_{(\mathcal{A}', \text{d})})$. By Lemma \ref{lemma-acyclic-good} the differential graded module $f^*\mathcal{P}'' \otimes_\mathcal{A} \mathcal{N}$ is acyclic and the proof is complete.
It is enough to show that $\mathcal{P} \otimes_\mathcal{A} \mathcal{N} \to \mathcal{P} \otimes_\mathcal{A} \mathcal{N}'$ is an isomorphism on cohomology sheaves for any good differential graded $\mathcal{A}$-module $\mathcal{P}$. To do this, let $\mathcal{N}''$ be the cone on the map $\mathcal{N} \to \mathcal{N}'$ as a left differential graded $\mathcal{A}$-module, see Definition \ref{definition-cone}. (To be sure, $\mathcal{N}''$ is a bimodule too but we don't need this.) By functoriality of the tensor construction (it is a functor of differential graded categories) we see that $\mathcal{P} \otimes_\mathcal{A} \mathcal{N}''$ is the cone (as a complex of $\mathcal{O}$-modules) on the map $\mathcal{P} \otimes_\mathcal{A} \mathcal{N} \to \mathcal{P} \otimes_\mathcal{A} \mathcal{N}'$. Hence it suffices to show that $\mathcal{P} \otimes_\mathcal{A} \mathcal{N}''$ is acyclic. This follows from the fact that $\mathcal{P}$ is good and the fact that $\mathcal{N}''$ is acyclic as a cone on a quasi-isomorphism.
Choose a good differential graded $\mathcal{A}$-module $\mathcal{P}$ and a quasi-isomorphism $\mathcal{P} \to \mathcal{M}$, see Lemma \ref{lemma-resolve}. Then $$ \mathcal{M} \otimes_\mathcal{A}^\mathbf{L} (\mathcal{N} \otimes_{\mathcal{A}'} \mathcal{N}') = \mathcal{P} \otimes_\mathcal{A} \mathcal{N} \otimes_{\mathcal{A}'} \mathcal{N}' $$ and we have $$ (\mathcal{M} \otimes_\mathcal{A}^\mathbf{L} \mathcal{N}) \otimes_{\mathcal{A}'}^\mathbf{L} \mathcal{N}' = (\mathcal{P} \otimes_\mathcal{A} \mathcal{N}) \otimes_{\mathcal{A}'}^\mathbf{L} \mathcal{N}' $$ Thus we have to show the canonical map $$ (\mathcal{P} \otimes_\mathcal{A} \mathcal{N}) \otimes_{\mathcal{A}'}^\mathbf{L} \mathcal{N}' \longrightarrow \mathcal{P} \otimes_\mathcal{A} \mathcal{N} \otimes_{\mathcal{A}'} \mathcal{N}' $$ is a quasi-isomorphism. Choose a quasi-isomorphism $\mathcal{Q} \to \mathcal{N}'$ where $\mathcal{Q}$ is a good left differential graded $\mathcal{A}'$-module (Lemma \ref{lemma-resolve}). By Lemma \ref{lemma-good-on-other-side} the map above as a map in the derived category of $\mathcal{O}$-modules is the map $$ \mathcal{P} \otimes_\mathcal{A} \mathcal{N} \otimes_{\mathcal{A}'} \mathcal{Q} \longrightarrow \mathcal{P} \otimes_\mathcal{A} \mathcal{N} \otimes_{\mathcal{A}'} \mathcal{N}' $$ Since $\mathcal{N} \otimes_{\mathcal{A}'} \mathcal{Q} \to \mathcal{N} \otimes_{\mathcal{A}'} \mathcal{N}'$ is a quasi-isomorphism by assumption and $\mathcal{P}$ is a good differential graded $\mathcal{A}$-module this map is an quasi-isomorphism by Lemma \ref{lemma-tensor-symmetry} (the left and right hand side compute $\mathcal{P} \otimes_\mathcal{A}^\mathbf{L} (\mathcal{N} \otimes_{\mathcal{A}'} \mathcal{Q})$ and $\mathcal{P} \otimes_\mathcal{A}^\mathbf{L} (\mathcal{N} \otimes_{\mathcal{A}'} \mathcal{N}')$ or you can just repeat the argument in the proof of the lemma).
By Theorem \ref{theorem-qis-into-dg-injective} for any (right) differential graded $\mathcal{A}$-module $\mathcal{M}$ there exists a quasi-isomorphism $\mathcal{M} \to \mathcal{I}$ where $\mathcal{I}$ is a graded injective and K-injective differential graded $\mathcal{A}$-module. Hence by Derived Categories, Lemma \ref{derived-lemma-find-existence-computes} it suffices to show that given a quasi-isomorphism $\mathcal{I} \to \mathcal{I}'$ of differential graded $\mathcal{A}$-modules which are both graded injective and K-injective then $T(\mathcal{I}) \to T(\mathcal{I}')$ is an isomorphism. This is true because the map $\mathcal{I} \to \mathcal{I}'$ is an isomorphism in $K(\text{Mod}_{(\mathcal{A}, \text{d})})$ as follows for example from Lemma \ref{lemma-hom-derived} (or one can deduce it from Lemma \ref{lemma-second-property-dg-injective}).
By Lemmas \ref{lemma-tensor-hom-adjunction-dg} and \ref{lemma-adjunction-push-pull-dg} the functor (\ref{equation-pushforward}) is right adjoint to the functor (\ref{equation-pullback}). By Derived Categories, Lemma \ref{derived-lemma-pre-derived-adjoint-functors-general} the functor $RT$ is right adjoint to the functor of Lemma \ref{lemma-derived-tensor-product} which is equal to $Lf^*(-) \otimes_\mathcal{A}^\mathbf{L} \mathcal{N}$ by Lemma \ref{lemma-compose-pullback-tensor}. By Lemmas \ref{lemma-derived-adjoint-tensor-hom} and \ref{lemma-derived-adjoint-push-pull} the functor $Lf^*(-) \otimes_\mathcal{A}^\mathbf{L} \mathcal{N}$ is left adjoint to $Rf_* R\SheafHom(\mathcal{N}, -)$ Thus we conclude by uniqueness of adjoints.
Besides identifying some categories, this lemma follows immediately from Lemma \ref{lemma-compose-pushforward}. \medskip\noindent We may view $(\mathcal{O}_\mathcal{C}, 0)$ as a differential graded $\mathcal{O}_\mathcal{C}$-algebra by placing $\mathcal{O}_\mathcal{C}$ in degree $0$ and endowing it with the zero differential. It is clear that we have $$ \text{Mod}_{(\mathcal{O}_\mathcal{C}, 0)} = \text{Comp}(\mathcal{O}_\mathcal{C}) \quad\text{and}\quad D(\mathcal{O}_\mathcal{C}, 0) = D(\mathcal{O}_\mathcal{C}) $$ Via this identification the forgetful functor $\text{Mod}_{(\mathcal{A}, \text{d})} \to \text{Comp}(\mathcal{O}_\mathcal{C})$ is the ``pushforward'' $\text{id}_{\mathcal{C}, *}$ defined in Section \ref{section-functoriality-dg} corresponding to the identity morphism $\text{id}_\mathcal{C} : (\mathcal{C}, \mathcal{O}_\mathcal{C}) \to (\mathcal{C}, \mathcal{O}_\mathcal{C})$ of ringed topoi and the map $(\mathcal{O}_\mathcal{C}, 0) \to (\mathcal{A}, \text{d})$ of differential graded $\mathcal{O}_\mathcal{C}$-algebras. Since $\text{id}_{\mathcal{C}, *}$ is exact, we immediately see that $$ R\text{id}_{\mathcal{C}, *} = forget : D(\mathcal{A}, \text{d}) \longrightarrow D(\mathcal{O}_\mathcal{C}, 0) = D(\mathcal{O}_\mathcal{C}) $$ The exact same reasoning shows that $$ R\text{id}_{\mathcal{D}, *} = forget : D(\mathcal{B}, \text{d}) \longrightarrow D(\mathcal{O}_\mathcal{D}, 0) = D(\mathcal{O}_\mathcal{D}) $$ Moreover, the construction of $Rf_* : D(\mathcal{O}_\mathcal{C}) \to D(\mathcal{O}_\mathcal{D})$ of Cohomology on Sites, Section \ref{sites-cohomology-section-unbounded} agrees with the construction of $Rf_* : D(\mathcal{O}_\mathcal{C}, 0) \to D(\mathcal{O}_\mathcal{D}, 0)$ in Definition \ref{definition-pushforward} as both functors are defined as the right derived extension of pushforward on underlying complexes of modules. By Lemma \ref{lemma-compose-pushforward} we see that both $Rf_* \circ R\text{id}_{\mathcal{C}, *}$ and $R\text{id}_{\mathcal{D}, *} \circ Rf_*$ are the derived functors of $f_* \circ forget = forget \circ f_*$ and hence equal by uniqueness of adjoints.
To prove the formula, observe that $$ R\Gamma(\mathcal{C}, \mathcal{M}) = \Gamma(\mathcal{C}, \mathcal{I}) $$ where $\mathcal{M} \to \mathcal{I}$ is a quasi-isomorphism to a graded injective and K-injective differential graded $\mathcal{A}$-module $\mathcal{I}$ (combine Lemmas \ref{lemma-right-derived} and \ref{lemma-pushforward-agrees}). By Lemma \ref{lemma-hom-derived} we have $$ \Hom_{D(\mathcal{A}, \text{d})}(\mathcal{A}, \mathcal{M}[n]) = \Hom_{K(\text{Mod}_{(\mathcal{A}, \text{d})})}(\mathcal{M}, \mathcal{I}[n]) = H^0(\Gamma(\mathcal{C}, \mathcal{I}[n])) = H^n(\Gamma(\mathcal{C}, \mathcal{I})) $$ Combining these two results we obtain our equality.
Recall that the restriction functor $$ \text{Mod}^{dg}_{(\mathcal{B}, \text{d})} \to \text{Mod}^{dg}_{(\mathcal{A}, \text{d})},\quad \mathcal{N} \mapsto res_\varphi \mathcal{N} $$ is a right adjoint to $$ \text{Mod}^{dg}_{(\mathcal{A}, \text{d})} \to \text{Mod}^{dg}_{(\mathcal{B}, \text{d})},\quad \mathcal{M} \mapsto \mathcal{M} \otimes_\mathcal{A} \mathcal{B} $$ See Section \ref{section-dg-bimodules}. Since restriction sends quasi-isomorphisms to quasi-isomorphisms, we see that it trivially has a left derived extension (given by restriction). This functor will be right adjoint to $- \otimes_\mathcal{A}^\mathbf{L} \mathcal{B}$ by Derived Categories, Lemma \ref{derived-lemma-pre-derived-adjoint-functors-general}. The adjunction map $$ \mathcal{M} \to res_\varphi(\mathcal{M} \otimes_\mathcal{A}^\mathbf{L} \mathcal{B}) $$ is an isomorphism in $D(\mathcal{A}, \text{d})$ by our assumption that $\mathcal{A} \to \mathcal{B}$ is a quasi-isomorphism of (left) differential graded $\mathcal{A}$-modules. In particular, the functor of the lemma is fully faithful, see Categories, Lemma \ref{categories-lemma-adjoint-fully-faithful}. It is clear that the kernel of the restriction functor $D(\mathcal{B}, \text{d}) \to D(\mathcal{A}, \text{d})$ is zero. Thus we conclude by Derived Categories, Lemma \ref{derived-lemma-fully-faithful-adjoint-kernel-zero}.
The proof is exactly the same as the first proof of Lemma \ref{lemma-resolve} but now working with free graded algebras instead of free graded modules. \medskip\noindent We will construct $\mathcal{A} = \colim \mathcal{A}_i$ as in Lemma \ref{lemma-special-good} by constructing $$ \mathcal{A}_0 \to \mathcal{A}_1 \to \mathcal{A}_2 \to \ldots \to \mathcal{B} $$ Let $\mathcal{S}_0$ be the sheaf of graded sets (Remark \ref{remark-sheaf-graded-sets}) whose degree $n$ part is $\Ker(\text{d}_\mathcal{B}^n)$. Consider the homomorphism of differential graded modules $$ \mathcal{A}_0 = \mathcal{O}\langle \mathcal{S}_0 \rangle \longrightarrow \mathcal{B} $$ where map sends a local section $s$ of $\mathcal{S}_0$ to the corresponding local section of $\mathcal{A}^{\deg(s)}$ (which is in the kernel of the differential, so our map is a map of differential graded algebras indeed). By construction the induced maps on cohomology sheaves $H^n(\mathcal{A}_0) \to H^n(\mathcal{B})$ are surjective and hence the same will remain true for all $i$. \medskip\noindent Induction step of the construction. Given $\mathcal{A}_i \to \mathcal{B}$ denote $\mathcal{S}_{i + 1}$ the sheaf of graded sets whose degree $n$ part is $$ \Ker(\text{d}_{\mathcal{A}_i}^{n + 1}) \times_{\mathcal{B}^{n + 1}, \text{d}} \mathcal{B}^n $$ This comes equipped with a canonical map $$ \delta_{i + 1} : \mathcal{S}_{i + 1} \longrightarrow \mathcal{A}_i $$ whose image is contained in the kernel of $\text{d}_{\mathcal{A}_i}$ by construction. Hence $\mathcal{A}_{i + 1} = \mathcal{O}\langle \mathcal{S}_0 \amalg \ldots \mathcal{S}_{i + 1}\rangle$ has a differential exteding the differential on $\mathcal{A}_i$, see discussion at the start of this section. The map from $\mathcal{A}_{i + 1}$ to $\mathcal{B}$ is the unique map of graded algebras which restricts to the given map on $\mathcal{A}_i$ and sends a local section $s = (a, b)$ of $\mathcal{S}_{i + 1}$ to $b$ in $\mathcal{B}$. This is compatible with differentials exactly because $\text{d}(b)$ is the image of $a$ in $\mathcal{B}$. \medskip\noindent The map $\mathcal{A} \to \mathcal{B}$ is a quasi-isomorphism: we have $H^n(\mathcal{A}) = \colim H^n(\mathcal{A}_i)$ and for each $i$ the map $H^n(\mathcal{A}_i) \to H^n(\mathcal{B})$ is surjective with kernel annihilated by the map $H^n(\mathcal{A}_i) \to H^n(\mathcal{A}_{i + 1})$ by construction. Finally, the flatness condition for $\mathcal{A}$ where shown in Lemma \ref{lemma-special-good}.
Recall that $K(\text{Mod}_{(\mathcal{A}, \text{d})}) = K(\text{Mod}^{dg}_{(\mathcal{A}, \text{d})})$, see Section \ref{section-homotopy}. Having said this, the proposition follows from Lemmas \ref{lemma-axioms-AB} and \ref{lemma-axiom-C} and Differential Graded Algebra, Proposition \ref{dga-proposition-ABC-homotopy-category-triangulated}.