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
d
:
A
→
A
which is either degree 1 (cochain complex convention) or degree
−
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.
The Koszul complex is a DG-algebra.
The tensor algebra is a DG-algebra with differential similar to that of the Koszul complex.
The singular cohomology of a topological space with coefficients in Z/pZ is a DG-algebra: the differential is given by the Bockstein homomorphism associated to the short exact sequence 0 → Z/pZ → Z/p2Z → Z/pZ → 0, and the product is given by the cup product.
Differential forms on a manifold, together with the exterior derivation and the wedge product form a DG-algebra. See also de Rham cohomology.
The homology
H
∗
(
A
)
=
ker
(
d
)
/
im
(
d
)
of a DG-algebra
(
A
,
d
)
is a graded algebra. The homology of a DGA-algebra is an augmented algebra.
