Derived tensor product explained

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

- ⊗ ^{bf{L} _{A} } - : D(\mathsf_A) \times D(_A \mathsf) \to D(_R \mathsf)where M _{A} and {} _{A} M 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 - ⊗ _{A} - : M _{A} x {} _{A} M \to {} _{R} M .

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:

M ⊗ ^{L} _{R} N whose i-th homotopy is the i-th Tor:\pi _{i} (M ⊗ ^{L} _{R} N) = \operatorname{Tor} ^{R} _{i(M,} N) .It is called the derived tensor product of M and N. In particular, \pi _{0} (M ⊗ ^{L} _{R} 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

\Omega ^{1} _{Q(R)} be the module of Kähler differentials. ThenL _{R} = \Omega ^{1} _{Q(R)} ⊗ ^{L} _{Q(R)} R is an R-module called the cotangent complex of R. It is functorial in R: each R → S gives rise to L _{R} \to L _{S} . Then, for each R → S, there is the cofiber sequence of S-modulesL _{S/R} \to L _{R} ⊗ ^{L} _{R} S \to L _{S.} The cofiber L _{S/R} 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

Notes and References

1. Hinich. Vladimir. 1997-02-11. Homological algebra of homotopy algebras. q-alg/9702015.