Differential Graded Commutative Algebra

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## Definition

## Examples of DG-algebras

## Other facts about DG-algebras

## See also

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Differential Graded Commutative 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.

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 (but esoteric) 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 DG morphism between DG-algebras is a graded algebra homomorphism which respects the differential *d*.

A **differential graded augmented algebra** (also called a **DGA-algebra**,
an augmented DG-algebra or simply a **DGA**) is a DG-algebra 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 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 is a DG-algebra: the differential is given by the Bockstein homomorphism associated to the short exact sequence , 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*of a DG-algebra is a graded algebra. The homology of a DGA-algebra is an augmented algebra.

- 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.)
- Differential graded module

**^**Cartan, Henri (1954). "Sur les groupes d'Eilenberg-Mac Lane ".*Proceedings of the National Academy of Sciences of the United States of America*.**40**: 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

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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