Lemma 24.31.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{B}, \text{d})$ be a differential graded $\mathcal{O}$-algebra. There exists a quasi-isomorphism of differential graded $\mathcal{O}$-algebras $(\mathcal{A}, \text{d}) \to (\mathcal{B}, \text{d})$ such that $\mathcal{A}$ is graded flat and K-flat as a complex of $\mathcal{O}$-modules and such that the same is true after pullback by any morphism of ringed topoi.

**Proof.**
The proof is exactly the same as the first proof of Lemma 24.23.7 but now working with free graded algebras instead of free graded modules.

We will construct $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i$ as in Lemma 24.31.1 by constructing

Let $\mathcal{S}_0$ be the sheaf of graded sets (Remark 24.23.5) whose degree $n$ part is $\mathop{\mathrm{Ker}}(\text{d}_\mathcal {B}^ n)$. Consider the homomorphism of differential graded modules

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 $i$.

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

This comes equipped with a canonical map

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 $\mathcal{B}$.

The map $\mathcal{A} \to \mathcal{B}$ is a quasi-isomorphism: we have $H^ n(\mathcal{A}) = \mathop{\mathrm{colim}}\nolimits H^ n(\mathcal{A}_ i)$ 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 24.31.1. $\square$

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