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.
A differential graded algebra (or simply DG-algebra) A is a graded algebra equipped with a map d : A → A which is either degree 1 (cochain complex convention) or degree − 1 (chain complex convention) that satisfies two conditions:
A more succinct way to state the same definition is to say that a DG-algebra is a monoid object in the monoidal category of chain complexes.
A differential graded augmented algebra (or simply DGA-algebra) or an augmented DG-algebra is a DG-algebra equipped with a morphism to the ground ring (the terminology is due to Henri Cartan).
Many sources use the term DGAlgebra for a DG-algebra.
The Koszul complex is a DG-algebra.The tensor algebra is a DG-algebra with differential similar to that of the Koszul complex.The singular cohomology of a topological space with coefficients in Z/pZ is a DG-algebra: the differential is given by the Bockstein homomorphism associated to the short exact sequence 0 → Z/pZ → Z/p2Z → Z/pZ → 0, and the product is given by the cup product.Differential forms on a manifold, together with the exterior derivation and the wedge product form a DG-algebra. See also de Rham cohomology.The homology H ∗ ( A ) = ker ( d ) / im ( d ) of a DG-algebra ( A , d ) is a graded algebra. The homology of a DGA-algebra is an augmented algebra.