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

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

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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