Definition 22.3.4. Let R be a ring. Let (A, \text{d}), (B, \text{d}) be differential graded algebras over R. The tensor product differential graded algebra of A and B is the algebra A \otimes _ R B with multiplication defined by
(a \otimes b)(a' \otimes b') = (-1)^{\deg (a')\deg (b)} aa' \otimes bb'
endowed with differential \text{d} defined by the rule \text{d}(a \otimes b) = \text{d}(a) \otimes b + (-1)^ m a \otimes \text{d}(b) where m = \deg (a).
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