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The Stacks project

Lemma 22.36.3. 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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