Reference request: Tensor product of DG-modules

Question

Let $A$ be a (skew-) commutative DG-algebra, and let $M,N$ be two DG $A$-modules.

I am looking for a reference which describes the functor $-\otimes_A -$ and its basic properties (associtivity, identity element, etc).

Assuming $A$ is a DG $K$-algebra, where $K$ is a commutative ring, most sources I find describe a tensor product which is simply a tensor product of complexes over $K$, but this is a wrong object from the point of view of DG-modules over $A$, because then we will not have $M\otimes A \cong M$.

Can anyone please point me to a reference where the "correct" notion is discussed?

Thanks!

Answer 1

A very good book on homological algebra is Osborne's "Basic Homological Algebra". You can find information of Hom-tensor products and projective/injective/flat objects in the very first chapters. If you want to focus your attention on algebras, I would start with Jacobson's "Basic Algebra I-II". The homological algebra book by Bourbaki is a good reference, as well.

On your concept of "correct notion". I really believe that everything depends on applications and theoretical framework. The isomorphism $M\otimes_A A\simeq M$ can be useful in your specific context, but there are others where the tensor product over the ground field is important as well. Consider for example the bar resolution $\mathcal B_M(A)$ of a left $A$-dg module $M$, where $A$ is a (possibly augmented) dg algebra $A$ over the ground field $\mathbb K$. Very briefly:

$$\mathcal B_M(A): A\otimes A[1]^{\otimes n}\otimes M \rightarrow A\otimes A[1]^{\otimes n-1}\otimes M \rightarrow\dots A\otimes M\rightarrow M,$$

with $\otimes=\otimes_{\mathbb K}$ and $[1]$ denotes suspension.

This construction is quite important in both algebraic topology and homological algebra.

Answer 2

The usual notion of tensor product over $A$ of modules $M_A$ and ${}_AN$ is just the obvious thing: $M\otimes_AN$, computed forgetting the gradings, has a differential such that $$d(m\otimes n)=d_M(m)\otimes n+(-1)^{\deg m}m\otimes d_N(n).$$