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 formula_1 which has either degree 1 (cochain complex convention) or degree formula_2 (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 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).
"Warning:" some sources use the term "DGA" for a DG-algebra.
Examples of DG-algebras.
Tensor algebra.
The tensor algebra is a DG-algebra with differential similar to that of the Koszul complex. For a vector space formula_3 over a field formula_4 there is a graded vector space formula_5 defined asformula_6where formula_7. If formula_8 is a basis for formula_3 there is a differential formula_10 on the tensor algebra defined component wiseformula_11sending basis elements toformula_12 This has a canonical product given by tensoring elementsformula_13
Koszul complex.
One of the foundational examples of a differential graded algebra, widely used in commutative algebra and algebraic geometry, is the Koszul complex. This is because of its wide array of applications, including constructing flat resolutions of complete intersections, and from a derived perspective, they give the derived algebra representing a derived critical locus.
De-Rham algebra.
Differential forms on a manifold, together with the exterior derivation and the exterior product form a DG-algebra. These have wide applications, including in derived deformation theory. See also de Rham cohomology.