Derived tensor product


In algebra, given a differential graded algebra A over a commutative ring R, the derived tensor product functor is
where and are the categories of right A-modules and left A-modules and D refers to the homotopy category. By definition, it is the left derived functor of the tensor product functor.

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:
whose i-th homotopy is the i-th Tor:
It is called the derived tensor product of M and N. In particular, is the usual tensor product of modules M and N over R.
Geometrically, the derived tensor product corresponds to the intersection product.
Example: Let R be a simplicial commutative ring, Q → R be a cofibrant replacement, and be the module of Kähler differentials. Then
is an R-module called the cotangent complex of R. It is functorial in R: each R → S gives rise to. Then, for each R → S, there is the cofiber sequence of S-modules
The cofiber is called the relative cotangent complex.