Definition 22.3.1. Let $R$ be a commutative ring. A differential graded algebra over $R$ is either

1. a chain complex $A_\bullet$ of $R$-modules endowed with $R$-bilinear maps $A_ n \times A_ m \to A_{n + m}$, $(a, b) \mapsto ab$ such that

$\text{d}_{n + m}(ab) = \text{d}_ n(a)b + (-1)^ n a\text{d}_ m(b)$

and such that $\bigoplus A_ n$ becomes an associative and unital $R$-algebra, or

2. a cochain complex $A^\bullet$ of $R$-modules endowed with $R$-bilinear maps $A^ n \times A^ m \to A^{n + m}$, $(a, b) \mapsto ab$ such that

$\text{d}^{n + m}(ab) = \text{d}^ n(a)b + (-1)^ n a\text{d}^ m(b)$

and such that $\bigoplus A^ n$ becomes an associative and unital $R$-algebra.

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