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In algebra, given a differential graded algebra A over a commutative ring R, the derived tensor product functor is
where M A {\displaystyle {\mathsf {M}}_{A}} and A M {\displaystyle {}_{A}{\mathsf {M}}} are the categories of right A-modules and left A-modules and D refers to the homotopy category. By definition, it is the left derived functor of the tensor product functor − ⊗ A − : M A × A M → R M {\displaystyle -\otimes _{A}-:{\mathsf {M}}_{A}\times {}_{A}{\mathsf {M}}\to {}_{R}{\mathsf {M}}}.