Derived tensor product

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

AL:D(MA)×D(AM)D(RM)

where MA and AM 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:MA×AMRM.

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:

MRLN

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

πi(MRLN)=ToriR(M,N).

It is called the derived tensor product of M and N. In particular, π0(MRLN) 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 ΩQ(R)1 be the module of Kähler differentials. Then

𝕃R=ΩQ(R)1Q(R)LR

is an R-module called the cotangent complex of R. It is functorial in R: each RS gives rise to 𝕃R𝕃S. Then, for each RS, there is the cofiber sequence of S-modules

𝕃S/R𝕃RRLS𝕃S.

The cofiber 𝕃S/R is called the relative cotangent complex.

See also

Notes

  1. Hinich, Vladimir (1997-02-11). "Homological algebra of homotopy algebras". arXiv:q-alg/9702015.

References