Derived tensor product

In algebra, given a differential graded algebra "A" over a commutative ring "R", the derived tensor product functor is
where formula_2 and formula_3 are the categories of right "A"-modules and left "A"-modules and "D" refers to the homotopy category (i.e., derived category). By definition, it is the left derived functor of the tensor product functor formula_4.
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, formula_7 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 formula_8 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 formula_10. Then, for each "R" → "S", there is the cofiber sequence of "S"-modules
The cofiber formula_12 is called the relative cotangent complex.