Differential graded algebra

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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[edit]

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 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[clarification needed] to the ground ring (the terminology is due to Henri Cartan).[1]

Many sources use the term DGAlgebra for a DG-algebra.[citation needed]

Examples of DG-algebras[edit]

Other facts about DG-algebras[edit]

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

See also[edit]

References[edit]

  1. ^ H. Cartan, Sur les groupes d'Eilenberg-Mac Lane H(Π,n), Proc. Natl. Acad. Sci. U.S.A. 40, (1954). 467–471