hcju/stacksqa huggingface dataset:
Let $X' \to X$ be a morphism of good algebraic spaces over $B$. Let $\alpha \in \CH_k(X')$. Write $\alpha_j = c_j(\mathcal{E}) \cap \alpha$, so $\alpha_0 = \alpha$. By Lemma \ref{lemma-segre-classes} we have $$ \sum\nolimits_{i = 0}^r (-1)^i c_1(\mathcal{O}_{P'}(1))^i \cap (\pi')^*(\alpha_{r - i}) = 0 $$ in the chow group of the projective bundle $(\pi' : P' \to X', \mathcal{O}_{P'}(1))$ associated to $(X' \to X)^*\mathcal{E}$. Applying $c \cap -$ and using Lemma \ref{lemma-c1-center} and the properties of bivariant classes we obtain $$ \sum\nolimits_{i = 0}^r (-1)^i c_1(\mathcal{O}_{P'}(1))^i \cap \pi^*(c \cap \alpha_{r - i}) = 0 $$ in the Chow group of $P'$. Hence we see that $c \cap \alpha_j$ is equal to $c_j(\mathcal{E}) \cap (c \cap \alpha)$ by the uniqueness in Lemma \ref{lemma-segre-classes}. This proves the lemma.
The proof is identical to the proof of Chow Homology, Lemma \ref{chow-lemma-get-rid-of-trivial-subbundle} replacing the lemmas used there by Lemmas \ref{lemma-bivariant-zero}, \ref{lemma-relative-effective-cartier}, \ref{lemma-pushforward-cap-c1}, and \ref{lemma-segre-classes}.
We prove this by induction on the integer $r = \sum r_i$. If $r = 0$ we can take $\pi = \text{id}_X$. If $r_i = 1$ for all $i$, then we can also take $\pi = \text{id}_X$. Assume that $r_{i_0} > 1$ for some $i_0$. Let $(\pi : P \to X, \mathcal{O}_P(1))$ be the projective bundle associated to $\mathcal{E}_{i_0}$. The canonical map $\pi^*\mathcal{E}_{i_0} \to \mathcal{O}_P(1)$ is surjective and hence its kernel $\mathcal{E}'_{i_0}$ is finite locally free of rank $r_{i_0} - 1$. Observe that $\pi_Y^*$ is injective for any morphism $f : Y \to X$ of good algebraic spaces over $B$, see Lemma \ref{lemma-chow-ring-projective-bundle}. Thus it suffices to prove the lemma for $P$ and the locally free sheaves $\pi^*\mathcal{E}_i$. However, because we have the subbundle $\mathcal{E}_{i_0} \subset \pi^*\mathcal{E}_{i_0}$ with invertible quotient, it now suffices to prove the lemma for the collection $\{\mathcal{E}_i\}_{i \not = i_0} \cup \{\mathcal{E}'_{i_0}\}$. This decreases $r$ by $1$ and we win by induction hypothesis.
Let $Z_i \subset Z$, $i = 1, \ldots, t$ be the irreducible components of dimension $d$. Let $m_i$ be the multiplicity of $Z_i$ in $Z$. Then $[Z]_d = \sum m_i[Z_i]$ and $c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_d) \cap [Z]_d$ is the sum of the cycles $m_i c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_d) \cap [Z_i]$. Since we have a similar decomposition for $(\mathcal{L}_1 \cdots \mathcal{L}_d \cdot Z)$ by Spaces over Fields, Lemma \ref{spaces-over-fields-lemma-numerical-polynomial-leading-term} it suffices to prove the lemma in case $Z = X$ is a proper integral algebraic space over $k$. \medskip\noindent By Chow's lemma there exists a proper morphism $f : X' \to X$ which is an isomorphism over a dense open $U \subset X$ such that $X'$ is a scheme. See More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-chow-noetherian-separated}. Then $X'$ is a proper scheme over $k$. After replacing $X'$ by the scheme theoretic closure of $f^{-1}(U)$ we may assume that $X'$ is integral. Then $$ (f^*\mathcal{L}_1 \cdots f^*\mathcal{L}_d \cdot X') = (\mathcal{L}_1 \cdots \mathcal{L}_d \cdot X) $$ by Spaces over Fields, Lemma \ref{spaces-over-fields-lemma-intersection-number-and-pullback} and we have $$ f_*(c_1(f^*\mathcal{L}_1) \cap \ldots \cap c_1(f^*\mathcal{L}_d) \cap [Y]) = c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_d) \cap [X] $$ by Lemma \ref{lemma-pushforward-cap-c1}. Thus we may replace $X$ by $X'$ and assume that $X$ is a proper scheme over $k$. This case was proven in Chow Homology, Lemma \ref{chow-lemma-degrees-and-numerical-intersections}.
Let $R$ and $\mathcal{M}_r \to \mathcal{M}'_r$ be a set of morphisms of $\text{Mod}_{(\mathcal{A}, \text{d})}$ found in Lemma \ref{lemma-better-set-of-monos}. Let $M$ with transformation $\text{id} \to M$ be as constructed in Lemma \ref{lemma-functor-set-of-monos} using $R$ and $\mathcal{M}_r \to \mathcal{M}'_r$. By transfinite induction on $\alpha$ we define a sequence of functors $M_\alpha$ and natural transformations $M_\beta \to M_\alpha$ for $\alpha < \beta$ by setting \begin{enumerate} \item $M_0 = \text{id}$, \item $M_{\alpha + 1} = M \circ M_\alpha$ with natural transformation $M_\beta \to M_{\alpha + 1}$ for $\beta < \alpha + 1$ coming from the already constructed $M_\beta \to M_\alpha$ and the maps $M_\alpha \to M \circ M_\alpha$ coming from $\text{id} \to M$, and \item $M_\alpha = \colim_{\beta < \alpha} M_\beta$ if $\alpha$ is a limit ordinal with the coprojections as transformations $M_\beta \to M_\alpha$ for $\alpha < \beta$. \end{enumerate} Observe that for every differential graded $\mathcal{A}$-module the maps $\mathcal{M} \to M_\beta(\mathcal{M}) \to M_\alpha(\mathcal{M})$ are injective quasi-isomorphisms (as filtered colimits are exact). \medskip\noindent Recall that $\text{Mod}_{(\mathcal{A}, \text{d})}$ is a Grothendieck abelian category. Thus by Injectives, Proposition \ref{injectives-proposition-objects-are-small} (applied to the direct sum of $\mathcal{M}_r$ for all $r \in R$) there is a limit ordinal $\alpha$ such that $\mathcal{M}_r$ is $\alpha$-small with respect to injections for every $r \in R$. We claim that $\mathcal{M} \to M_\alpha(\mathcal{M})$ is the desired functorial embedding of $\mathcal{M}$ into a graded injective K-injective module. \medskip\noindent Namely, any map $\mathcal{M}_r \to M_\alpha(\mathcal{M})$ factors through $M_\beta(\mathcal{M})$ for some $\beta < \alpha$. However, by the construction of $M$ we see that this means that $\mathcal{M}_r \to M_{\beta + 1}(\mathcal{M}) = M(M_\beta(\mathcal{M}))$ factors through $\mathcal{M}'_r$. Since $M_\beta(\mathcal{M}) \subset  M_{\beta + 1}(\mathcal{M}) \subset M_\alpha(\mathcal{M})$ we get the desired factorizaton into $M_\alpha(\mathcal{M})$. We conclude by our choice of $R$ and $\mathcal{M}_r \to \mathcal{M}'_r$ in Lemma \ref{lemma-better-set-of-monos}.
Let us denote $\text{gr}^n : \text{Mod}_\mathcal{A} \to \textit{Mod}(\mathcal{O})$ the functor in the statement of the lemma. Consider a homomorphism $f : \mathcal{M} \to \mathcal{N}$ of graded $\mathcal{A}$-modules. The kernel and cokernel of $f$ as maps of graded $\mathcal{O}$-modules are additionally endowed with multiplication maps as in Definition \ref{definition-gm}. Hence these are also the kernel and cokernel in $\text{Mod}_\mathcal{A}$. Thus $\text{Mod}_\mathcal{A}$ is an abelian category and taking kernels and cokernels commutes with $\text{gr}^n$. \medskip\noindent To prove the existence of limits and colimits it is sufficient to prove the existence of products and direct sums, see Categories, Lemmas \ref{categories-lemma-limits-products-equalizers} and \ref{categories-lemma-colimits-coproducts-coequalizers}. The same lemmas show that proving the commutation of limits and colimits with $\text{gr}^n$ follows if $\text{gr}^n$ commutes with direct sums and products. \medskip\noindent Let $\mathcal{M}_t$, $t \in T$ be a set of graded $\mathcal{A}$-modules. Then we can consider the graded $\mathcal{A}$-module whose degree $n$ term is $\bigoplus_{t \in T} \mathcal{M}_t^n$ (with obvious multiplication maps). The reader easily verifies that this is a direct sum in $\text{Mod}_\mathcal{A}$. Similarly for products. \medskip\noindent Observe that $\text{gr}^n$ is an exact functor for all $n$ and that a complex $\mathcal{M}_1 \to \mathcal{M}_2 \to \mathcal{M}_3$ of $\text{Mod}_\mathcal{A}$ is exact if and only if $\text{gr}^n\mathcal{M}_1 \to \text{gr}^n\mathcal{M}_2 \to \text{gr}^n\mathcal{M}_3$ is exact in $\textit{Mod}(\mathcal{O})$ for all $n$. Hence we conclude that (3) holds as filtered colimits are exact in $\textit{Mod}(\mathcal{O})$; it is a Grothendieck abelian category, see Cohomology on Sites, Section \ref{sites-cohomology-section-unbounded}.
By the discussion in Modules on Sites, Section \ref{sites-modules-section-localize} the functors $j_!$ and $j^*$ on $\mathcal{O}$-modules are adjoint. Thus if we only look at the $\mathcal{O}$-module structures we know that $$ \Hom_{\text{Mod}_\mathcal{O}^{gr}}( j_!\mathcal{M}, \mathcal{N}) = \Hom_{\text{Mod}_{\mathcal{O}_U}^{gr}}( \mathcal{M}, j^*\mathcal{N}) $$ Then one has to check that these identifications map the $\mathcal{A}$-module maps on the left hand side to the $\mathcal{A}_U$-module maps on the right hand side. To check this, given $\mathcal{O}_U$-linear maps $f^n : \mathcal{M}^n \to j^*\mathcal{N}^{n + d}$ corresponding to $\mathcal{O}$-linear maps $g^n : j_!\mathcal{M}^n \to \mathcal{N}^{n + d}$ it suffices to show that $$ \xymatrix{ \mathcal{M}^n \otimes_{\mathcal{O}_U} \mathcal{A}_U^m \ar[r]_{f^n \otimes 1} \ar[d] & j^*\mathcal{N}^{n + d} \otimes_{\mathcal{O}_U} \mathcal{A}_U^m \ar[d] \\ \mathcal{M}^{n + m} \ar[r]^{f^{n + m}} & j^*\mathcal{N}^{n + m + d} } $$ commutes if and only if $$ \xymatrix{ j_!\mathcal{M}^n \otimes_\mathcal{O} \mathcal{A}^m \ar[r]_{g^n \otimes 1} \ar[d] & \mathcal{N}^{n + d} \otimes_\mathcal{O} \mathcal{A}_U^m \ar[d] \\ j_!\mathcal{M}^{n + m} \ar[r]^{g^{n + m}} & \mathcal{N}^{n + m + d} } $$ commutes. However, we know that \begin{align*} \Hom_{\mathcal{O}_U}(\mathcal{M}^n \otimes_{\mathcal{O}_U} \mathcal{A}_U^m, j^*\mathcal{N}^{n + d + m}) & = \Hom_\mathcal{O}(j_!(\mathcal{M}^n \otimes_{\mathcal{O}_U} \mathcal{A}_U^m), \mathcal{N}^{n + d + m}) \\ & = \Hom_\mathcal{O}(j_!\mathcal{M}^n \otimes_\mathcal{O} \mathcal{A}^m, \mathcal{N}^{n + d + m}) \end{align*} by the already used Modules on Sites, Lemma \ref{sites-modules-lemma-j-shriek-and-tensor}. We omit the verification that shows that the obstruction to the commutativity of the first diagram in the first group maps to the obstruction to the commutativity of the second diagram in the last group.
Recall that the degree $n$ component of $j_!\mathcal{M} \otimes_\mathcal{A} \mathcal{N}$ is the cokernel of the canonical map $$ \bigoplus\nolimits_{r + s + t = n} j_!\mathcal{M}^r \otimes_\mathcal{O} \mathcal{A}^s \otimes_\mathcal{O} \mathcal{N}^t \longrightarrow \bigoplus\nolimits_{p + q = n} j_!\mathcal{M}^p \otimes_\mathcal{O} \mathcal{N}^q $$ See Section \ref{section-tensor-product}. By Modules on Sites, Lemma \ref{sites-modules-lemma-j-shriek-and-tensor} this is the same thing as the cokernel of $$ \bigoplus\nolimits_{r + s + t = n} j_!(\mathcal{M}^r \otimes_{\mathcal{O}_U} \mathcal{A}^s|_U \otimes_{\mathcal{O}_U} \mathcal{N}^t|_U) \longrightarrow \bigoplus\nolimits_{p + q = n} j_!(\mathcal{M}^p \otimes_{\mathcal{O}_U} \mathcal{N}^q|_U) $$ and we win. An alternative proof would be to redo the Yoneda argument given in the proof of the lemma cited above.
By Lemma \ref{lemma-gm-abelian} and the definition of a Grothendieck abelian category (Injectives, Definition \ref{injectives-definition-grothendieck-conditions}) it suffices to show that $\text{Mod}_\mathcal{A}$ has a generator. We claim that $$ \mathcal{G} = \bigoplus\nolimits_{k, U} j_{U!}\mathcal{A}_U[k] $$ is a generator where the sum is over all objects $U$ of $\mathcal{C}$ and $k \in \mathbf{Z}$. Indeed, given a graded $\mathcal{A}$-module $\mathcal{M}$ if there are no nonzero maps from $\mathcal{G}$ to $\mathcal{M}$, then we see that for all $k$ and $U$ we have $$ \Hom_{\text{Mod}_\mathcal{A}}(j_{U!}\mathcal{A}_U[k], \mathcal{M}) = \Hom_{\text{Mod}_{\mathcal{A}_U}}(\mathcal{A}_U[k], \mathcal{M}|_U) = \Gamma(U, \mathcal{M}^{-k}) $$ is equal to zero. Hence $\mathcal{M}$ is zero.
Let us denote $F : \text{Mod}_{(\mathcal{A}, \text{d})} \to \text{Mod}_\mathcal{A}$ the functor in the statement of the lemma. Observe that the category $\text{Mod}_\mathcal{A}$ has properties (1) -- (5), see Lemma \ref{lemma-gm-abelian}. \medskip\noindent Consider a homomorphism $f : \mathcal{M} \to \mathcal{N}$ of graded $\mathcal{A}$-modules. The kernel and cokernel of $f$ as maps of graded $\mathcal{A}$-modules are additionally endowed with differentials as in Definition \ref{definition-dgm}. Hence these are also the kernel and cokernel in $\text{Mod}_{(\mathcal{A}, \text{d})}$. Thus $\text{Mod}_{(\mathcal{A}, \text{d})}$ is an abelian category and taking kernels and cokernels commutes with $F$. \medskip\noindent To prove the existence of limits and colimits it is sufficient to prove the existence of products and direct sums, see Categories, Lemmas \ref{categories-lemma-limits-products-equalizers} and \ref{categories-lemma-colimits-coproducts-coequalizers}. The same lemmas show that proving the commutation of limits and colimits with $F$ follows if $F$ commutes with direct sums and products. \medskip\noindent Let $\mathcal{M}_t$, $t \in T$ be a set of differential graded $\mathcal{A}$-modules. Then we can consider the direct sum $\bigoplus \mathcal{M}_t$ as a graded $\mathcal{A}$-module. Since the direct sum of graded modules is done termwise, it is clear that $\bigoplus \mathcal{M}_t$ comes endowed with a differential. The reader easily verifies that this is a direct sum in $\text{Mod}_{(\mathcal{A}, \text{d})}$. Similarly for products. \medskip\noindent Observe that $F$ is an exact functor and that a complex $\mathcal{M}_1 \to \mathcal{M}_2 \to \mathcal{M}_3$ of $\text{Mod}_{(\mathcal{A}, \text{d})}$ is exact if and only if $F(\mathcal{M}_1) \to F(\mathcal{M}_2) \to F(\mathcal{M}_3)$ is exact in $\text{Mod}_\mathcal{A}$. Hence we conclude that (3) holds as filtered colimits are exact in $\textit{Mod}_\mathcal{A})$.
Suppose given differential graded $\mathcal{A}$-modules $\mathcal{M}$ and $\mathcal{N}$. Consider the differential graded $\mathcal{A}$-module $\mathcal{M} \oplus \mathcal{N}$ defined in the obvious manner. Then the coprojections $i : \mathcal{M} \to \mathcal{M} \oplus \mathcal{N}$ and $j : \mathcal{N} \to \mathcal{M} \oplus \mathcal{N}$ and the projections $p : \mathcal{M} \oplus \mathcal{N} \to \mathcal{N}$ and $q : \mathcal{M} \oplus \mathcal{N} \to \mathcal{M}$ are morphisms of differential graded $\mathcal{A}$-modules. Hence $i, j, p, q$ are homogeneous of degree $0$ and closed, i.e., $\text{d}(i) = 0$, etc. Thus this direct sum is a differential graded sum in the sense of Differential Graded Algebra, Definition \ref{dga-definition-dg-direct-sum}. This proves axiom (A). \medskip\noindent Axiom (B) was shown in Section \ref{section-shift-dg}.
By Lemma \ref{lemma-dgm-abelian} and the definition of a Grothendieck abelian category (Injectives, Definition \ref{injectives-definition-grothendieck-conditions}) it suffices to show that $\text{Mod}_{(\mathcal{A}, \text{d})}$ has a generator. For every object $U$ of $\mathcal{C}$ we denote $C_U$ the cone on the identity map $\mathcal{A}_U \to \mathcal{A}_U$ as in Remark \ref{remark-cone-identity}. We claim that $$ \mathcal{G} = \bigoplus\nolimits_{k, U} j_{U!}C_U[k] $$ is a generator where the sum is over all objects $U$ of $\mathcal{C}$ and $k \in \mathbf{Z}$. Indeed, given a differential graded $\mathcal{A}$-module $\mathcal{M}$ if there are no nonzero maps from $\mathcal{G}$ to $\mathcal{M}$, then we see that for all $k$ and $U$ we have \begin{align*} \Hom_{\text{Mod}_\mathcal{A}}(j_{U!}C_U[k], \mathcal{M}) \\ & = \Hom_{\text{Mod}_{\mathcal{A}_U}}(C_U[k], \mathcal{M}|_U) \\ & = \{(x, y) \in \mathcal{M}^{-k}(U) \times \mathcal{M}^{-k - 1}(U) \mid x = \text{d}(y)\} \end{align*} is equal to zero. Hence $\mathcal{M}$ is zero.
Let $\mathcal{N}$ be a left graded $\mathcal{A}$-module. By Lemma \ref{lemma-tensor-with-extension-by-zero} we have $$ j_!\mathcal{A}_U \otimes_\mathcal{A} \mathcal{N} = j_!(\mathcal{A}_U \otimes_{\mathcal{A}_U} \mathcal{N}|_U) = j_!(\mathcal{N}_U) $$ as graded modules. Since both restriction to $U$ and $j_!$ are exact this proves condition (1). The same argument works for (2) using Lemma \ref{lemma-tensor-with-extension-by-zero-dg}. \medskip\noindent Consider a morphism $(f, f^\sharp) : (\Sh(\mathcal{C}'), \mathcal{O}') \to (\Sh(\mathcal{C}), \mathcal{O})$ of ringed topoi, a differential graded $\mathcal{O}'$-algebra $\mathcal{A}'$, and a map $\varphi : f^{-1}\mathcal{A} \to \mathcal{A}'$ of differential graded $f^{-1}\mathcal{O}$-algebras. We have to show that $$ f^*j_!\mathcal{A}_U = f^{-1}j_!\mathcal{A}_U \otimes_{f^{-1}\mathcal{A}} \mathcal{A}' $$ satisfies (1) and (2) for the ringed topos $(\Sh(\mathcal{C}'), \mathcal{O}')$ endowed with the sheaf of differential graded $\mathcal{O}'$-algebras $\mathcal{A}'$. To prove this we may replace $(\Sh(\mathcal{C}), \mathcal{O})$ and $(\Sh(\mathcal{C}'), \mathcal{O}')$ by equivalent ringed topoi. Thus by Modules on Sites, Lemma \ref{sites-modules-lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites} we may assume that $f$ comes from a morphism of sites $f : \mathcal{C} \to \mathcal{C}'$ given by the continuous functor $u : \mathcal{C} \to \mathcal{C}'$. In this case, set $U' = u(U)$ and denote $j' : \Sh(\mathcal{C}'/U') \to \Sh(\mathcal{C}')$ the corresponding localization morphism. We obtain a commutative square of morphisms of ringed topoi $$ \xymatrix{ (\Sh(\mathcal{C}'/U'), \mathcal{O}'_{U'}) \ar[rr]_{(j', (j')^\sharp)} \ar[d]_{(f', (f')^\sharp)} & & (\Sh(\mathcal{C}'), \mathcal{O}') \ar[d]^{(f, f^\sharp)} \\ (\Sh(\mathcal{C}/U), \mathcal{O}_U) \ar[rr]^{(j, j^\sharp)} & & (\Sh(\mathcal{C}), \mathcal{O}). } $$ and we have $f'_*(j')^{-1} = j^{-1}f_*$. See Modules on Sites, Lemma \ref{sites-modules-lemma-localize-morphism-ringed-sites}. By uniqueness of adjoints we obtain $f^{-1}j_! = j'_!(f')^{-1}$. Thus we obtain \begin{align*} f^*j_!\mathcal{A}_U & = f^{-1}j_!\mathcal{A}_U \otimes_{f^{-1}\mathcal{A}} \mathcal{A}' \\ & = j'_!(f')^{-1}\mathcal{A}_U \otimes_{f^{-1}\mathcal{A}} \mathcal{A}' \\ & = j'_!\left( (f')^{-1}\mathcal{A}_U \otimes_{f^{-1}\mathcal{A}|_{U'}} \mathcal{A}'|_{U'}\right) \\ & = j'_!\mathcal{A}'_{U'} \end{align*} The first equation is the definition of the pullback of $j_!\mathcal{A}_U$ to a differential graded module over $\mathcal{A}'$. The second equation because $f^{-1}j_! = j'_!(f')^{-1}$. The third equation by Lemma \ref{lemma-tensor-with-extension-by-zero-dg} applied to the ringed site $(\mathcal{C}', f^{-1}\mathcal{O})$ with sheaf of differential graded algebras $f^{-1}\mathcal{A}$ and with differential graded modules $(f')^{-1}\mathcal{A}_U$ on $\mathcal{C}'/U'$ and $\mathcal{A}'$ on $\mathcal{C}'$. The fourth equation holds because of course we have $(f')^{-1}\mathcal{A}_U = f^{-1}\mathcal{A}|_{U'}$. Hence we see that the pullback is another module of the same kind and we've proven conditions (1) and (2) for it above.
Consider triples $(U, k, x)$ where $U$ is an object of $\mathcal{C}$, $k \in \mathbf{Z}$, and $x$ is a section of $\mathcal{M}^k$ over $U$ with $\text{d}_\mathcal{M}(x) = 0$. Then we obtain a unique morphism of differential graded $\mathcal{A}_U$-modules $\varphi_x : \mathcal{A}_U[-k] \to \mathcal{M}|_U$ mapping $1$ to $x$. This is adjoint to a morphism $\psi_x : j_{U!}\mathcal{A}_U[-k] \to \mathcal{M}$. Observe that $1 \in \mathcal{A}_U(U)$ corresponds to a section $1 \in j_{U!}\mathcal{A}_U[-k](U)$ of degree $k$ whose differential is zero and which is mapped to $x$ by $\psi_x$. Thus if we consider the map $$ \bigoplus\nolimits_{(U, k, x)} j_{U!}\mathcal{A}_U[-k] \longrightarrow \mathcal{M} $$ then we will have conditions (2) and (3). Namely, the objects $j_{U!}\mathcal{A}_U[-k]$ are good (Lemma \ref{lemma-supply-good}) and any direct sum of good objects is good (Lemma \ref{lemma-good-direct-sum}). \medskip\noindent Next, consider triples $(U, k, x)$ where $U$ is an object of $\mathcal{C}$, $k \in \mathbf{Z}$, and $x$ is a section of $\mathcal{M}^k$ (not necessarily annihilated by the differential). Then we can consider the cone $C_U$ on the identity map $\mathcal{A}_U \to \mathcal{A}_U$ as in Remark \ref{remark-cone-identity}. The element $x$ will determine a map $\varphi_x : C_U[-k - 1] \to \mathcal{A}_U$, see Remark \ref{remark-cone-identity}. Now, since we have an admissible short exact sequence $$ 0 \to \mathcal{A}_U \to C_U \to \mathcal{A}_U[1] \to 0 $$ we conclude that $j_{U!}C_U$ is a good module by Lemma \ref{lemma-good-admissible-ses} and the already used Lemma \ref{lemma-supply-good}. As above we conclude that the direct sum of the maps $\psi_x : j_{U!}C_U \to \mathcal{M}$ adjoint to the $\varphi_x$ $$ \bigoplus\nolimits_{(U, k, x)} j_{U!}C_U \longrightarrow \mathcal{M} $$ is surjective. Taking the direct sum with the map produced in the first paragraph we conclude.
[First proof] Let $\mathcal{S}_0$ be the sheaf of graded sets (Remark \ref{remark-sheaf-graded-sets}) whose degree $n$ part is $\Ker(\text{d}_\mathcal{M}^n)$. Consider the homomorphism of differential graded modules $$ \mathcal{P}_0 = \mathcal{A}[\mathcal{S}_0] \longrightarrow \mathcal{M} $$ where the left hand side is as in Remark \ref{remark-sheaf-graded-sets} and the map sends a local section $s$ of $\mathcal{S}_0$ to the corresponding local section of $\mathcal{M}^{\deg(s)}$ (which is in the kernel of the differential, so our map is a map of differential graded modules indeed). By construction the induced maps on cohomology sheaves $H^n(\mathcal{P}_0) \to H^n(\mathcal{M})$ are surjective. We are going to inductively construct maps $$ \mathcal{P}_0 \to \mathcal{P}_1 \to \mathcal{P}_2 \to \ldots \to \mathcal{M} $$ Observe that of course $H^*(\mathcal{P}_i) \to H^*(\mathcal{M})$ will be surjective for all $i$. Given $\mathcal{P}_i \to \mathcal{M}$ denote $\mathcal{S}_{i + 1}$ the sheaf of graded sets whose degree $n$ part is $$ \Ker(\text{d}_{\mathcal{P}_i}^{n + 1}) \times_{\mathcal{M}^{n + 1}, \text{d}} \mathcal{M}^n $$ Then we set $$ \mathcal{P}_{i + 1} = \mathcal{P}_i \oplus \mathcal{A}[\mathcal{S}_{i + 1}] $$ as graded $\mathcal{A}$-module with differential and map to $\mathcal{M}$ defined as follows \begin{enumerate} \item for local sections of $\mathcal{P}_i$ use the differential on $\mathcal{P}_i$ and the given map to $\mathcal{M}$, \item for a local section $s = (p, m)$ of $\mathcal{S}_{i + 1}$ we set $\text{d}(s)$ equal to $p$ viewed as a section of $\mathcal{P}_i$ of degree $\deg(s) + 1$ and we map $s$ to $m$ in $\mathcal{M}$, and \item extend the differential uniquely so that the Leibniz rule holds. \end{enumerate} This makes sense because $\text{d}(m)$ is the image of $p$ and $\text{d}(p) = 0$. Finally, we set $\mathcal{P} = \colim \mathcal{P}_i$ with the induced map to $\mathcal{M}$. \medskip\noindent The map $\mathcal{P} \to \mathcal{M}$ is a quasi-isomorphism: we have $H^n(\mathcal{P}) = \colim H^n(\mathcal{P}_i)$ and for each $i$ the map $H^n(\mathcal{P}_i) \to H^n(\mathcal{M})$ is surjective with kernel annihilated  by the map $H^n(\mathcal{P}_i) \to H^n(\mathcal{P}_{i + 1})$ by construction. Each $\mathcal{P}_i$ is good because $\mathcal{P}_0$ is good by Lemma \ref{lemma-free-graded-module-good} and each $\mathcal{P}_{i + 1}$ is in the middle of the admissible short exact sequence $0 \to \mathcal{P}_i \to \mathcal{P}_{i + 1} \to \mathcal{A}[\mathcal{S}_{i + 1}] \to 0$ whose outer terms are good by induction. Hence $\mathcal{P}_{i + 1}$ is good by Lemma \ref{lemma-good-admissible-ses}. Finally, we conclude that $\mathcal{P}$ is good by Lemma \ref{lemma-good-direct-sum}.