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:

  1. .
    This says that d gives A the structure of a chain complex or cochain complex (accordingly as the differential reduces or raises degree).
  2. , 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.

Examples of DG-algebras

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

References

  1. 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.
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