The Stacks project

Lemma 24.23.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{M}$ be a differential graded $\mathcal{A}$-module. There exists a homomorphism $\mathcal{P} \to \mathcal{M}$ of differential graded $\mathcal{A}$-modules with the following properties

  1. $\mathcal{P} \to \mathcal{M}$ is surjective,

  2. $\mathop{\mathrm{Ker}}(\text{d}_\mathcal {P}) \to \mathop{\mathrm{Ker}}(\text{d}_\mathcal {M})$ is surjective, and

  3. $\mathcal{P}$ is good.

Proof. 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 24.23.1) and any direct sum of good objects is good (Lemma 24.23.3).

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 24.22.5. The element $x$ will determine a map $\varphi _ x : C_ U[-k - 1] \to \mathcal{A}_ U$, see Remark 24.22.5. 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 24.23.2 and the already used Lemma 24.23.1. 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. $\square$

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