# Derived tensor product

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

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

where ${\displaystyle {\textbf {Mod}}-A}$ and ${\displaystyle A-{\textbf {Mod}}}$ 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 ${\displaystyle -\otimes _{A}-:{\textbf {Mod}}-A\to A-{\textbf {Mod}}\to R-{\textbf {Mod}}}$.