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 d\colon A \to A which is either degree 1 (cochain complex convention) or degree -1 (chain complex convention) that satisfies two conditions:

  1. 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. d(a \cdot b)=(da) \cdot b + (-1)^{\operatorname{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 differential graded augmented algebra (or simply DGA-algebra) or an augmented DG-algebra is a DG-algebra equipped with a morphism to the ground ring (the terminology is due to Henri Cartan).[1]

Many sources use the term DGAlgebra for a DG-algebra.

Examples of DG-algebras

Other facts about DG-algebras

See also

References

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