Lemma 115.9.1. Let (A, \text{d}) and (B, \text{d}) be differential graded algebras. Let N be a differential graded (A, B)-bimodule with property (P). Let M be a differential graded A-module with property (P). Then Q = M \otimes _ A N is a differential graded B-module which represents M \otimes _ A^\mathbf {L} N in D(B) and which has a filtration
0 = F_{-1}Q \subset F_0Q \subset F_1Q \subset \ldots \subset Q
by differential graded submodules such that Q = \bigcup F_ pQ, the inclusions F_ iQ \to F_{i + 1}Q are admissible monomorphisms, the quotients F_{i + 1}Q/F_ iQ are isomorphic as differential graded B-modules to a direct sum of (A \otimes _ R B)[k].
Proof.
Choose filtrations F_\bullet on M and N. Then consider the filtration on Q = M \otimes _ A N given by
F_ n(Q) = \sum \nolimits _{i + j = n} F_ i(M) \otimes _ A F_ j(N)
This is clearly a differential graded B-submodule. We see that
F_ n(Q)/F_{n - 1}(Q) = \bigoplus \nolimits _{i + j = n} F_ i(M)/F_{i - 1}(M) \otimes _ A F_ j(N)/F_{j - 1}(N)
for example because the filtration of M is split in the category of graded A-modules. Since by assumption the quotients on the right hand side are isomorphic to direct sums of shifts of A and A \otimes _ R B and since A \otimes _ A (A \otimes _ R B) = A \otimes _ R B, we conclude that the left hand side is a direct sum of shifts of A \otimes _ R B as a differential graded B-module. (Warning: Q does not have a structure of (A, B)-bimodule.) This proves the first statement of the lemma. The second statement is immediate from the definition of the functor in Differential Graded Algebra, Lemma 22.33.2.
\square
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