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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