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:
- This says that d gives A the structure of a chain complex or cochain complex (accordingly as the differential reduces or raises degree).
- (ii) , where deg is the degree.
- This says that the differential d respects the graded Leibniz rule.
A DGA is an augmented DG-algebra, or differential graded augmented algebra[clarification needed : “augmented”?, synonyms for “DG-algebra”?] (the terminology is due to Henri Cartan). Many sources use the term DGAlgebra for a DG-algebra.
Examples of DGAs
- The Koszul complex is a DGA.
- The Tensor algebra is a DGA with differential similar to that of the Koszul complex.
- The Singular cohomology with coefficients in a ring is a DGA; the differential is given by the Bockstein homomorphism, and the product given by the cup product.
- Differential forms on a manifold, together with the exterior derivation and the wedge-product form a DGA. See also de Rham cohomology.
Other facts about DGAs
- The homology of a DG-algebra is a graded algebra. The homology of a DGA is an augmented algebra.
- Chain complex
- Commutative ring spectrum
- Derived scheme
- Differential graded category
- Differential graded Lie algebra
- Graded (mathematics)
- Graded algebra
- H. Cartan, Sur les groupes d'Eilenberg-Mac Lane H(Π,n), Proc. Nat. Acad. Sci. U. S. A. 40, (1954). 467–471