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.

This article is about (co)chain complexes. For an algebra with derivations, see differential algebra.

Definition

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

1. ${\displaystyle d\circ d=0}$.
   This says that d gives A the structure of a chain complex or cochain complex (accordingly as the differential reduces or raises degree).
2. ${\displaystyle d(a\cdot b)=(da)\cdot b+(-1)^{\deg(a)}a\cdot (db)}$, 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 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.

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 ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$ is a DG-algebra: the differential is given by the Bockstein homomorphism associated to the short exact sequence ${\displaystyle 0\to \mathbb {Z} /p\mathbb {Z} \to \mathbb {Z} /p^{2}\mathbb {Z} \to \mathbb {Z} /p\mathbb {Z} \to 0}$, 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 ${\displaystyle H_{*}(A)=\ker(d)/\operatorname {im} (d)}$ of a DG-algebra ${\displaystyle (A,d)}$ is a graded algebra. The homology of a DGA-algebra is an augmented algebra.

See also

• Homotopy associative 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

References

1. Cartan, Henri (1954). "Sur les groupes d'Eilenberg-Mac Lane ${\displaystyle H(\Pi ,n)}$". 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