In algebra, given a differential graded algebra A over a commutative ring R, the derived tensor product functor is where
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 (of derived schemes).
Example: Let R be a simplicial commutative ring, Q(R) → 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
is called the relative cotangent complex.