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 DGA) A is a graded algebra equipped with a map $d\colon A \to A$ which is either degree 1 (cochain complex convention) or degree − 1 (chain complex convention) that satisfies two conditions:

(i) $d \circ d=0$

This says that d gives A the structure of a chain complex or cochain complex (accordingly as the differential reduces or raises degree).

(ii) $d(a \cdot b)=(da) \cdot b + (-1)^{|a|}a \cdot (db)$.

This says that the differential d respects the graded Leibniz rule.

Examples of DGAs

• The Koszul complex is a DGA.
• The Tensor algebra is a DGA with differential similar to that of the Koszul complex.
• The Singular cohomology with coefficients in a ring is a DGA; the differential is given by the Bockstein homomorphism, and the product given by the cup product.
• Differential forms on a manifold, together with the exterior derivation and the wedge-product form a DGA.

Other facts about DGAs

• The homology H _* (A) = ker(d) / im(d) of a DGA (A,d) is a graded ring.

See also

• graded algebra
• chain complex
• differential graded category

References

• Manin, Yuri Ivanovich; Gelfand, Sergei I. (2003), Methods of Homological Algebra, Berlin, New York: Springer-Verlag, ISBN 978-3-540-43583-9 , see chapter V.3