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Homedifferential graded algebra

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# 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 $i<0$. 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

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

Synonym:

DG Algebra

Type of Math Object:

Definition

Major Section:

Reference

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## Mathematics Subject Classification

16E45*no label found*

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