Derived tensor product
where and 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 .
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.
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 . Then, for each R → S, there is the cofiber sequence of S-modules
The cofiber is called the relative cotangent complex.
- Lurie, J., Spectral Algebraic Geometry (under construction)
- Lecture 4 of Part II of Moerdijk-Toen, Simplicial Methods for Operads and Algebraic Geometry
- Ch. 2.2. of Toen-Vezzosi's HAG II