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 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)^{\operatorname{deg}(a)}a \cdot (db), where deg is the degree.
This says that the differential d respects the graded Leibniz rule.

A DGA is an augmented DG-algebra, or differential graded augmented algebra[clarification needed : “augmented”?, synonyms for “DG-algebra”?] (the terminology is due to Henri Cartan).[1] Many sources use the term DGAlgebra for a DG-algebra.

Examples of DGAs[edit]

Other facts about DGAs[edit]

DG-algebra (A,d) is a graded algebra. The homology of a DGA is an augmented algebra.

See also[edit]

References[edit]

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