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.

A differential graded algebra (or simply DG-algebra) A is a graded algebra equipped with a map ${\displaystyle d\colon A\to A}$ which is either degree 1 (cochain complex convention) or degree ${\displaystyle -1}$ (chain complex convention) that satisfies two conditions:

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).

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

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