Differential graded algebras

Definition 8.3.2 (Differential graded algebra). Let \(R\) be a commutative ring. An associative differential graded \(R\)-algebra, or associative DGA over \(R\), is an associative algebra object in the symmetric monoidal category \(\Ch (R)\) of chain complexes of \(R\)-modules. Concretely, it consists of a chain complex \(A_\bullet \), a unit \(R[0]\to A_\bullet \), and an associative multiplication \[ A_p\otimes _R A_q\longrightarrow A_{p+q},\qquad x\otimes y\longmapsto xy, \] which satisfies the Leibniz rule \[ d(xy)=d(x)y+(-1)^p x d(y) \] for homogeneous \(x\in A_p\) and \(y\in A_q\).

A commutative DGA over \(R\) is a commutative algebra object in \(\Ch (R)\). Equivalently, it is an associative DGA whose multiplication is graded commutative: \[ xy=(-1)^{pq}yx \] for homogeneous \(x\in A_p\) and \(y\in A_q\).