Differential graded algebra
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.
Contents
Definition
A differential graded algebra (or simply DGalgebra) A is a graded algebra equipped with a map which is either degree 1 (cochain complex convention) or degree (chain complex convention) that satisfies two conditions:

.
This says that d gives A the structure of a chain complex or cochain complex (accordingly as the differential reduces or raises degree). 
, where deg is the degree of homogeneous elements.
This says that the differential d respects the graded Leibniz rule.
A more succinct (but esoteric) way to state the same definition is to say that a DGalgebra is a monoid object in the monoidal category of chain complexes. A DG morphism between DGalgebras is a graded algebra homomorphism which respects the differential d.
A differential graded augmented algebra (also called a DGAalgebra, an augmented DGalgebra or simply a DGA) is a DGalgebra equipped with a DG morphism to the ground ring (the terminology is due to Henri Cartan).^{[1]}
Warning: some sources use the term DGA for a DGalgebra.
Examples of DGalgebras
 The Koszul complex is a DGalgebra.
 The tensor algebra is a DGalgebra with differential similar to that of the Koszul complex.
 The singular cohomology of a topological space with coefficients in Z/pZ is a DGalgebra: the differential is given by the Bockstein homomorphism associated to the short exact sequence 0 → Z/pZ → Z/p^{2}Z → 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 DGalgebra. See also de Rham cohomology.
Other facts about DGalgebras
 The homology of a DGalgebra is a graded algebra. The homology of a DGAalgebra is an augmented algebra.
See also
 Differential graded category
 Differential graded Lie algebra
 Differential graded scheme (which is obtained by gluing the spectra of gradedcommutative differential graded algebras with respect to the étale topology.)
References
 ^ H. Cartan, Sur les groupes d'EilenbergMac Lane H(Π,n), Proc. Natl. Acad. Sci. U.S.A. 40, (1954). 467–471
 Manin, Yuri Ivanovich; Gelfand, Sergei I. (2003), Methods of Homological Algebra, Berlin, New York: SpringerVerlag, ISBN 9783540435839, see sections V.3 and V.5.6