# 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.

## Definition

A **differential graded algebra** (or simply **DG-algebra**) *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 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).^{[1]}

Many sources use the term *DGAlgebra* for a DG-algebra.

## Examples of DG-algebras

- 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**/*p***Z**is a DG-algebra: the differential is given by the Bockstein homomorphism associated to the short exact sequence 0 →**Z**/*p***Z**→**Z**/*p*^{2}**Z**→**Z**/*p***Z**→ 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.

## Other facts about DG-algebras

- The
*homology*of a DG-algebra is a graded algebra. The homology of a DGA-algebra is an augmented algebra.

## See also

- Differential graded category
- Differential graded Lie algebra
- Differential graded scheme (which is obtained by gluing the spectra of graded-commutative differential graded algebras with respect to the étale topology.)

## References

- ↑ H. Cartan, Sur les groupes d'Eilenberg-Mac 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: Springer-Verlag, ISBN 978-3-540-43583-9, see sections V.3 and V.5.6