differential graded algebra

Let $R$ be a commutative ring. A differential graded algebra (or DG algebra) over $R$ is a complex $(A,{\partial }^A)$ of $R$-modules with an element $1\in A$ (the unit) and a degree zero chain map

$$A{\otimes }_{R}A\to A$$

that is unitary: $a1=a=1a$, and is associative: $a(bc)=(ab)c$. We also will stipulate that a DG algebra is graded commutative; that is for each $x,y\in A$, we have

$$xy={(-1)}^{|x||y|}yx$$

where $|x|$ means the degree of $x$. Also, we assume that $A_i=0$ for . Without these final assumptions, we will say that $A$ is an associative DG algebra.

The fact that the product is a chain map of degree zero is best described by the Leibniz Rule; that is, for each $x,y\in A$, we have

$${\partial }^A(xy)={\partial }^A(x)y+{(-1)}^{|x|}x{\partial }^A(y).$$