# 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

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

Title | differential graded algebra |
---|---|

Canonical name | DifferentialGradedAlgebra |

Date of creation | 2013-03-22 15:34:43 |

Last modified on | 2013-03-22 15:34:43 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 8 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 16E45 |

Synonym | DG Algebra |