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 $i<0$. 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

 $\partial^{A}(xy)=\partial^{A}(x)y+(-1)^{|x|}x\partial^{A}(y).$
Title differential graded algebra DifferentialGradedAlgebra 2013-03-22 15:34:43 2013-03-22 15:34:43 CWoo (3771) CWoo (3771) 8 CWoo (3771) Definition msc 16E45 DG Algebra