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

− ⊗ A L − : D ( M A ) × D ( A M ) → D ( R M ) {\displaystyle -\otimes _{A}^{\textbf {L}}-:D({\mathsf {M}}_{A})\times D({}_{A}{\mathsf {M}})\to D({}_{R}{\mathsf {M}})}

where M A {\displaystyle {\mathsf {M}}_{A}} and A M {\displaystyle {}_{A}{\mathsf {M}}} 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 − ⊗ A − : M A × A M → R M {\displaystyle -\otimes _{A}-:{\mathsf {M}}_{A}\times {}_{A}{\mathsf {M}}\to {}_{R}{\mathsf {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 ⊗ R L N {\displaystyle M\otimes _{R}^{L}N}

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

π i ( M ⊗ R L N ) = Tor i R ⁡ ( M , N ) {\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, π 0 ( M ⊗ R L N ) {\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 Ω Q ( R ) 1 {\displaystyle \Omega _{Q(R)}^{1}} be the module of Kähler differentials. Then

L R = Ω Q ( R ) 1 ⊗ Q ( R ) L R {\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 L R → L S {\displaystyle \mathbb {L} _{R}\to \mathbb {L} _{S}}. Then, for each R → S, there is the cofiber sequence of S-modules

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

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

See also

Notes

  • Lurie, J.,
  • Lecture 4 of Part II of Moerdijk-Toen, Simplicial Methods for Operads and Algebraic Geometry
  • Ch. 2.2. of