The Stacks project

Lemma 24.25.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. There exists a set $S$ and for each $s$ an acyclic differential graded $\mathcal{A}$-module $\mathcal{M}_ s$ such that for every nonzero acyclic differential graded $\mathcal{A}$-module $\mathcal{M}$ there is an $s \in S$ and an injective map $\mathcal{M}_ s \to \mathcal{M}$ in $\textit{Mod}(\mathcal{A}, \text{d})$.

Proof. Before we start recall that our conventions guarantee the site $\mathcal{C}$ has a set of objects and morphisms and a set $\text{Cov}(\mathcal{C})$ of coverings. If $\mathcal{F}$ is a differential graded $\mathcal{A}$-module, let us define $|\mathcal{F}|$ to be the sum of the cardinality of

\[ \coprod \nolimits _{(U, n)} \mathcal{F}^ n(U) \]

as $U$ ranges over the objects of $\mathcal{C}$ and $n \in \mathbf{Z}$. Choose an infinite cardinal $\kappa $ bigger than the cardinals $|\mathop{\mathrm{Ob}}\nolimits (\mathcal{C})|$, $|\text{Arrows}(\mathcal{C})|$, $|\text{Cov}(\mathcal{C})|$, $\sup |I|$ for $\{ U_ i \to U\} _{i \in I} \in \text{Cov}(\mathcal{C})$, and $|\mathcal{A}|$.

Let $\mathcal{F} \subset \mathcal{M}$ be an inclusion of differential graded $\mathcal{A}$-modules. Suppose given a set $K$ and for each $k \in K$ a triple $(U_ k, n_ k, x_ k)$ consisting of an object $U_ k$ of $\mathcal{C}$, integer $n_ k$, and a section $x_ k \in \mathcal{M}^{n_ k}(U_ k)$. Then we can consider the smallest differential graded $\mathcal{A}$-submodule $\mathcal{F}' \subset \mathcal{M}$ containing $\mathcal{F}$ and the sections $x_ k$ for $k \in K$. We can describe

\[ (\mathcal{F}')^ n(U) \subset \mathcal{M}^ n(U) \]

as the set of elements $x \in \mathcal{M}^ n(U)$ such that there exists $\{ f_ i : U_ i \to U\} _{i \in I} \in \text{Cov}(\mathcal{C})$ such that for each $i \in I$ there is a finite set $T_ i$ and morphisms $g_ t : U_ i \to U_{k_ t}$

\[ f_ i^*x = y_ i + \sum \nolimits _{t \in T_ i} a_{it}g_ t^*x_{k_ t} + b_{it}g_ t^*\text{d}(x_{k_ t}) \]

for some section $y_ i \in \mathcal{F}^ n(U)$ and sections $a_{it} \in \mathcal{A}^{n - n_{k_ t}}(U_ i)$ and $b_{it} \in \mathcal{A}^{n - n_{k_ t} - 1}(U_ i)$. (Details omitted; hints: these sections are certainly in $\mathcal{F}'$ and you show conversely that this rule defines a differential graded $\mathcal{A}$-submodule.) It follows from this description that $|\mathcal{F}'| \leq \max (|\mathcal{F}|, |K|, \kappa )$.

Let $\mathcal{M}$ be a nonzero acyclic differential graded $\mathcal{A}$-module. Then we can find an integer $n$ and a nonzero section $x$ of $\mathcal{M}^ n$ over some object $U$ of $\mathcal{C}$. Let

\[ \mathcal{F}_0 \subset \mathcal{M} \]

be the smallest differential graded $\mathcal{A}$-submodule containing $x$. By the previous paragraph we have $|\mathcal{F}_0| \leq \kappa $. By induction, given $\mathcal{F}_0, \ldots , \mathcal{F}_ n$ define $\mathcal{F}_{n + 1}$ as follows. Consider the set

\[ L = \{ (U, n, x)\} \{ U_ i \to U\} _{i \in I}, (x_ i)_{i \in I})\} \]

of triples where $U$ is an object of $\mathcal{C}$, $n \in \mathbf{Z}$, and $x \in \mathcal{F}_ n(U)$ with $\text{d}(x) = 0$. Since $\mathcal{M}$ is acyclic for each triple $l = (U_ l, n_ l, x_ l) \in L$ we can choose $\{ (U_{l, i} \to U_ l\} _{i \in I_ l} \in \text{Cov}(\mathcal{C})$ and $x_{l, i} \in \mathcal{M}^{n_ l - 1}(U_{l, i})$ such that $\text{d}(x_{l, i}) = x|_{U_{l, i}}$. Then we set

\[ K = \{ (U_{l, i}, n_ l - 1, x_{l, i}) \mid l \in L, i \in I_ l\} \]

and we let $\mathcal{F}_{n + 1}$ be the smallest differential graded $\mathcal{A}$-submodule of $\mathcal{M}$ containing $\mathcal{F}_ n$ and the sections $x_{l, i}$. Since $|K| \leq \max (\kappa , |\mathcal{F}_ n|)$ we conclude that $|\mathcal{F}_{n + 1}| \leq \kappa $ by induction.

By construction the inclusion $\mathcal{F}_ n \to \mathcal{F}_{n + 1}$ induces the zero map on cohomology sheaves. Hence we see that $\mathcal{F} = \bigcup \mathcal{F}_ n$ is a nonzero acyclic submodule with $|\mathcal{F}| \leq \kappa $. Since there is only a set of isomorphism classes of differential graded $\mathcal{A}$-modules $\mathcal{F}$ with $|\mathcal{F}|$ bounded, we conclude. $\square$

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