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:

${\displaystyle M\otimes _{R}^{L}N}$ 

whose i-th homotopy is the i-th Tor:

${\displaystyle \pi _{i}(M\otimes _{R}^{L}N)=\operatorname {Tor} _{i}^{R}(M,N)}$ .

It is called the derived tensor product of M and N. In particular, ${\displaystyle \pi _{0}(M\otimes _{R}^{L}N)}$  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 ${\displaystyle \Omega _{Q(R)}^{1}}$  be the module of Kähler differentials. Then

${\displaystyle \mathbb {L} _{R}=\Omega _{Q(R)}^{1}\otimes _{Q(R)}^{L}R}$ 

is an R-module called the cotangent complex of R. It is functorial in R: each R → S gives rise to ${\displaystyle \mathbb {L} _{R}\to \mathbb {L} _{S}}$ . Then, for each R → S, there is the cofiber sequence of S-modules

${\displaystyle \mathbb {L} _{S/R}\to \mathbb {L} _{R}\otimes _{R}^{L}S\to \mathbb {L} _{S}.}$ 

The cofiber ${\displaystyle \mathbb {L} _{S/R}}$  is called the relative cotangent complex.

• derived scheme (derived tensor product gives a derived version of a scheme-theoretic intersection.)

• 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