Graded derivation

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 (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).

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