Derived tensor product

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

$-\underset{A}{\overset{\mathbf{\text{L}}}{\otimes }}-:D({𝖬}_A)\times D({}_A𝖬)\to D({}_R𝖬)$

where ${𝖬}_A$ and ${}_A𝖬$ 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 $-{\otimes }_A-:{𝖬}_A\times {}_A𝖬\to {}_R𝖬$.

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:

$M\underset{R}{\overset{L}{\otimes }}N$

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

${\pi }_i(M\underset{R}{\overset{L}{\otimes }}N)={\mathrm{Tor}}_i^R(M,N)$.

It is called the derived tensor product of M and N. In particular, ${\pi }_0(M\underset{R}{\overset{L}{\otimes }}N)$ 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 ${\mathrm{\Omega }}_{Q(R)}^1$ be the module of Kähler differentials. Then

${𝕃}_R={\mathrm{\Omega }}_{Q(R)}^1\underset{Q(R)}{\overset{L}{\otimes }}R$

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-modules

${𝕃}_{S/R}\to {𝕃}_R\underset{R}{\overset{L}{\otimes }}S\to {𝕃}_S.$

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

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