Differential graded algebra
A differential graded algebra (or simply DG-algebra) A is a graded algebra equipped with a map which is either degree 1 (cochain complex convention) or degree (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).
Warning: some sources use the term DGA for a DG-algebra.
Examples of DG-algebras
- 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.
Other facts about DG-algebras
- The homology of a DG-algebra is a graded algebra. The homology of a DGA-algebra is an augmented algebra.
- Differential graded category
- Differential graded Lie algebra
- Differential graded scheme (which is obtained by gluing the spectra of graded-commutative differential graded algebras with respect to the étale topology.)
- H. Cartan, Sur les groupes d'Eilenberg-Mac Lane H(Π,n), Proc. Natl. Acad. Sci. U.S.A. 40, (1954). 467–471