In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded algebra with an added chain complex structure that respects the algebra structure.

## Definition

A differential graded algebra (or simply DG-algebra) A is a graded algebra equipped with a map $d\colon A \to A$ which is either degree 1 (cochain complex convention) or degree $-1$ (chain complex convention) that satisfies two conditions:

(i) $d \circ d=0$
This says that d gives A the structure of a chain complex or cochain complex (accordingly as the differential reduces or raises degree).
(ii) $d(a \cdot b)=(da) \cdot b + (-1)^{\operatorname{deg}(a)}a \cdot (db)$, where deg is the degree.
This says that the differential d respects the graded Leibniz rule.

A DGA is an augmented DG-algebra, or differential graded augmented algebra[clarification needed : “augmented”?, synonyms for “DG-algebra”?] (the terminology is due to Henri Cartan).[1] Many sources use the term DGAlgebra for a DG-algebra.

## Examples of DGAs

• The homology $H_*(A) = \ker(d) / \operatorname{im}(d)$ of a
DG-algebra $(A,d)$ is a graded algebra. The homology of a DGA is an augmented algebra.