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 (i.e., derived category).[1] By definition, it is the left derived functor of the tensor product functor .

Derived tensor product in derived ring theoryEdit

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 RS gives rise to  . Then, for each RS, there is the cofiber sequence of S-modules


The cofiber   is called the relative cotangent complex.

See alsoEdit


  1. ^ Hinich, Vladimir (1997-02-11). "Homological algebra of homotopy algebras". arXiv:q-alg/9702015.