The Stacks project

Lemma 22.27.3. Let $(A, \text{d})$ be a differential graded algebra. Let $E$ be a compact object of $D(A, \text{d})$. Let $P$ be a differential graded $A$-module which has a finite filtration

\[ 0 = F_{-1}P \subset F_0P \subset F_1P \subset \ldots \subset F_ nP = P \]

by differential graded submodules such that

\[ F_{i + 1}P/F_ iP \cong \bigoplus \nolimits _{j \in J_ i} A[k_{i, j}] \]

as differential graded $A$-modules for some sets $J_ i$ and integers $k_{i, j}$. Let $E \to P$ be a morphism of $D(A, \text{d})$. Then there exists a differential graded submodule $P' \subset P$ such that $F_{i + 1}P \cap P'/(F_ iP \cap P')$ is equal to $\bigoplus _{j \in J'_ i} A[k_{i, j}]$ for some finite subsets $J'_ i \subset J_ i$ and such that $E \to P$ factors through $P'$.

Proof. We will prove by induction on $-1 \leq m \leq n$ that there exists a differential graded submodule $P' \subset P$ such that

  1. $F_ mP \subset P'$,

  2. for $i \geq m$ the quotient $F_{i + 1}P \cap P'/(F_ iP \cap P')$ is isomorphic to $\bigoplus _{j \in J'_ i} A[k_{i, j}]$ for some finite subsets $J'_ i \subset J_ i$, and

  3. $E \to P$ factors through $P'$.

The base case is $m = n$ where we can take $P' = P$.

Induction step. Assume $P'$ works for $m$. For $i \geq m$ and $j \in J'_ i$ let $x_{i, j} \in F_{i + 1}P \cap P'$ be a homogeneous element of degree $k_{i, j}$ whose image in $F_{i + 1}P \cap P'/(F_ iP \cap P')$ is the generator in the summand corresponding to $j \in J_ i$. The $x_{i, j}$ generate $P'/F_ mP$ as an $A$-module. Write

\[ \text{d}(x_{i, j}) = \sum x_{i', j'} a_{i, j}^{i', j'} + y_{i, j} \]

with $y_{i, j} \in F_ mP$ and $a_{i, j}^{i', j'} \in A$. There exists a finite subset $J'_{m - 1} \subset J_{m - 1}$ such that each $y_{i, j}$ maps to an element of the submodule $\bigoplus _{j \in J'_{m - 1}} A[k_{m - 1, j}]$ of $F_ mP/F_{m - 1}P$. Let $P'' \subset F_ mP$ be the inverse image of $\bigoplus _{j \in J'_{m - 1}} A[k_{m - 1, j}]$ under the map $F_ mP \to F_ mP/F_{m - 1}P$. Then we see that the $A$-submodule

\[ P'' + \sum x_{i, j}A \]

is a differential graded submodule of the type we are looking for. Moreover

\[ P'/(P'' + \sum x_{i, j}A) = \bigoplus \nolimits _{j \in J_{m - 1} \setminus J'_{m - 1}} A[k_{m - 1, j}] \]

Since $E$ is compact, the composition of the given map $E \to P'$ with the quotient map, factors through a finite direct subsum of the module displayed above. Hence after enlarging $J'_{m - 1}$ we may assume $E \to P'$ factors through $P'' + \sum x_{i, j}A$ as desired. $\square$


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