Differential on tensor product of $A_\infty$ modules

Question

Let $A$ be an $A_\infty$-algebra over a commutative ring $k$, and $M$ is a left $A$-module and $N$ is a right $A$ module. By modules I mean $A_\infty$-modules. Then we can define $A_\infty$ tensor product $M \otimes_A^{\infty} N$. The construction is briefly described in Keller's "Introduction to $A_\infty$ algebras and modules", where it is claimed that it is $M \otimes_k T(A) \otimes_k N$ with some differential. What is the formula for the differential? I'm also interested in special case: $A$ is an associative algebra or a DG algebra. Are there any papers where I can find detailed description of the $A_\infty$ tensor product?

Answer 1

See Section 2.7 in this paper of Rinna Anno and Timothy Logvinenko. The construction is given explicitly.

Recall that an $A_\infty$-algebra $A$ is the same as the quasi-free dg coalgebra $B_\infty A$, and if $M$ is a complex, then an $A_\infty$-module structure on $M$ is the same as a differential on the free comodule $B_\infty A \otimes M$.

Now you can take cotensor products of comodules, just like tensor products. The end result is that one $B_\infty A$ cancels leaving you with $N \otimes B_\infty A \otimes M$, and if you follow the isomorphism, you should get an explicit form of the differential making $M\otimes N$ an $A_\infty$-comodule.