differential graded algebra

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


that is unitary: a1=a=1a, and is associative: a(bc)=(ab)c. We also will stipulate that a DG algebra is graded commutativePlanetmathPlanetmathPlanetmath; that is for each x,yA, we have


where |x| means the degree of x. Also, we assume that Ai=0 for i<0. Without these final assumptionsPlanetmathPlanetmath, we will say that A is an associative DG algebra.

The fact that the productPlanetmathPlanetmath is a chain map of degree zero is best described by the Leibniz Rule; that is, for each x,yA, we have

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