Lemma 22.3.5. Let $R$ be a ring. Let $(A, \text{d})$, $(B, \text{d})$ be differential graded algebras over $R$. Denote $A^\bullet$, $B^\bullet$ the underlying cochain complexes. As cochain complexes of $R$-modules we have

$(A \otimes _ R B)^\bullet = \text{Tot}(A^\bullet \otimes _ R B^\bullet ).$

Proof. Recall that the differential of the total complex is given by $\text{d}_1^{p, q} + (-1)^ p \text{d}_2^{p, q}$ on $A^ p \otimes _ R B^ q$. And this is exactly the same as the rule for the differential on $A \otimes _ R B$ in Definition 22.3.4. $\square$

Comment #283 by arp on

Typo: In the statement of the lemma, the tensor product in $\text{Tot}(A^\bullet \otimes_A B^\bullet)$ should be over $R$ not $A$.

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