Derived tensor product

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

$-\otimes _{A}^{\textbf {L}}-:D({\mathsf {M}}_{A})\times D({}_{A}{\mathsf {M}})\to D({}_{R}{\mathsf {M}})$

where ${\mathsf {M}}_{A}$ and ${}_{A}{\mathsf {M}}$ 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 $-\otimes _{A}-:{\mathsf {M}}_{A}\times {}_{A}{\mathsf {M}}\to {}_{R}{\mathsf {M}}$.

See also

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

References

• Lurie, J., Spectral Algebraic Geometry (under construction)