The Stacks project

Proof. Observe that $f^*\mathcal{A} = \mathop{\mathrm{colim}}\nolimits f^*\mathcal{A}_ i$ and that

with differential given by the inductive procedure above using $f^{-1}\delta _{i + 1}$. Thus it suffices to prove that $\mathcal{A}$ is flat as a graded $\mathcal{O}$-module and is K-flat as a complex of $\mathcal{O}$-modules. For this it suffices to prove that each $\mathcal{A}_ i$ is flat as a graded $\mathcal{O}$-module and is K-flat as a complex of $\mathcal{O}$-modules, compare with Lemma 24.23.3.

For $i \geq 1$ write $\mathcal{S} = \mathcal{S}_0 \amalg \ldots \amalg \mathcal{S}_ i$ so that we have $\mathcal{A}_ i = \mathcal{O}\langle \mathcal{S} \rangle $ as a graded $\mathcal{O}$-algebra. We are going to construct a filtration of this algebra by differential graded $\mathcal{O}$-submodules.

Set $W = \mathbf{Z}_{\geq 0}^{i + 1}$ considered with lexicographical ordering. Namely, given $w = (w_0, \ldots w_ i)$ and $w' = (w'_0, \ldots , w'_ i)$ in $W$ we say

and so on. Suppose given a section $s = s_1 \cdot \ldots \cdot s_ r$ of $\mathcal{S} \times \ldots \times \mathcal{S}$ over $U$. We say that the weight of $s$ is defined if we have $s_ a \in \mathcal{S}_{j_ a}(U)$ for a unique $0 \leq j_ a \leq i$. In this case we define the weight

The weight of any section of $\mathcal{S} \times \ldots \times \mathcal{S}$ is defined locally. The reader checks easily that we obtain a disjoint union decomposition

into the subsheaves of sections of a given weight. Of course only $w \in W$ with $\sum _{0 \leq j \leq i} w_ j = r$ show up for a given $r$. We correspondingly obtain a decomposition

The rest of the proof relies on the following trivial observation: given $r$, $w$ and local section $s = s_1 \cdot \ldots \cdot s_ r$ of $\left(\mathcal{S} \times \ldots \times \mathcal{S}\right)_ w$ we have

The reason is that in each of the expressions

whose sum give the element $\text{d}(s)$ the element $\delta (s_ a)$ is locally a $\mathcal{O}$-linear combination of elements $s'_1 \cdot \ldots \cdot s'_{r'}$ with $s'_{a'}$ in $\mathcal{S}_{j'_ a}$ for some $0 \leq j'_{a'} < j_ a$ where $j_ a$ is such that $s_ a$ is section of $\mathcal{S}_{j_ a}$.

What this means is the following. Suppose for $w \in W$ we set

By the observation above this is a differential graded $\mathcal{O}$-submodule. We get admissible short exact sequences

of differential graded $\mathcal{A}$-modules where the differential on the right hand side is zero.

Now we finish the proof by transfinite induction over the ordered set $W$. The differential graded complex $F_0\mathcal{A}_0$ is the summand $\mathcal{O}$ and this is K-flat and graded flat. For $w \in W$ if the result is true for $F_{w'}\mathcal{A}_ i$ for $w' < w$, then by Lemmas 24.23.3, 24.23.2, and 24.23.6 we obtain the result for $w$. Finally, we have $\mathcal{A}_ i = \mathop{\mathrm{colim}}\nolimits _{w \in W} F_ w\mathcal{A}_ i$ and we conclude. $\square$

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. $\square$