Derived tensor product

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

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

where ${\displaystyle {\mathsf {M}}_{A}}$ and ${\displaystyle {}_{A}{\mathsf {M}}}$ are the categories of right A-modules and left A-modules and D refers to the homotopy category (i.e., derived category).[1] By definition, it is the left derived functor of the tensor product functor ${\displaystyle -\otimes _{A}-:{\mathsf {M}}_{A}\times {}_{A}{\mathsf {M}}\to {}_{R}{\mathsf {M}}}$.