In algebra, given a differential graded algebra A over a commutative ring R, the derived tensor product functor is

${\displaystyle -\otimes _{A}^{\textbf {L))-:D({\mathsf {M))_{A})\times D({}_{A}{\mathsf {M)))\to D({}_{R}{\mathsf {M)))}$

where ${\displaystyle {\mathsf {M))_{A))$ and ${\displaystyle {}_{A}{\mathsf {M))}$ are the categories of right A-modules and left A-modules and D refers to the homotopy category (i.e., derived category).^[1] By definition, it is the left derived functor of the tensor product functor ${\displaystyle -\otimes _{A}-:{\mathsf {M))_{A}\times {}_{A}{\mathsf {M))\to {}_{R}{\mathsf {M))}$.

Derived tensor product in derived ring theory

If R is an ordinary ring and M, N right and left modules over it, then, regarding them as discrete spectra, one can form the smash product of them:

${\displaystyle M\otimes _{R}^{L}N}$

whose i-th homotopy is the i-th Tor:

${\displaystyle \pi _{i}(M\otimes _{R}^{L}N)=\operatorname {Tor} _{i}^{R}(M,N)}$.

It is called the derived tensor product of M and N. In particular, ${\displaystyle \pi _{0}(M\otimes _{R}^{L}N)}$ is the usual tensor product of modules M and N over R.

Geometrically, the derived tensor product corresponds to the intersection product (of derived schemes).

Example: Let R be a simplicial commutative ring, Q(R) → R be a cofibrant replacement, and ${\displaystyle \Omega _{Q(R)}^{1))$ be the module of Kähler differentials. Then

${\displaystyle \mathbb {L} _{R}=\Omega _{Q(R)}^{1}\otimes _{Q(R)}^{L}R}$

is an R-module called the cotangent complex of R. It is functorial in R: each R → S gives rise to ${\displaystyle \mathbb {L} _{R}\to \mathbb {L} _{S))$. Then, for each R → S, there is the cofiber sequence of S-modules

${\displaystyle \mathbb {L} _{S/R}\to \mathbb {L} _{R}\otimes _{R}^{L}S\to \mathbb {L} _{S}.}$

The cofiber ${\displaystyle \mathbb {L} _{S/R))$ is called the relative cotangent complex.