Derived tensor product

In algebra, given a differential graded algebra A over a commutative ring R, the derived tensor product functor is

[math]\displaystyle{ - \otimes_A^{\textbf{L}} - : D(\mathsf{M}_A) \times D({}_A \mathsf{M}) \to D({}_R \mathsf{M}) }[/math]

where [math]\displaystyle{ \mathsf{M}_A }[/math] and [math]\displaystyle{ {}_A \mathsf{M} }[/math] 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 [math]\displaystyle{ - \otimes_A - : \mathsf{M}_A \times {}_A \mathsf{M} \to {}_R \mathsf{M} }[/math].

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:

[math]\displaystyle{ M \otimes_R^L N }[/math]

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

[math]\displaystyle{ \pi_i (M \otimes_R^L N) = \operatorname{Tor}^R_i(M, N) }[/math].

It is called the derived tensor product of M and N. In particular, [math]\displaystyle{ \pi_0 (M \otimes_R^L N) }[/math] 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 [math]\displaystyle{ \Omega_{Q(R)}^1 }[/math] be the module of Kähler differentials. Then

[math]\displaystyle{ \mathbb{L}_R = \Omega_{Q(R)}^1 \otimes^L_{Q(R)} R }[/math]

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

[math]\displaystyle{ \mathbb{L}_{S/R} \to \mathbb{L}_R \otimes_R^L S \to \mathbb{L}_S. }[/math]

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

See also

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

References

• 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