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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 $d:A\to A$ which is either degree 1 (cochain complex convention) or degree $-1$ (chain complex convention) that satisfies two conditions:

(i) $d\circ d=0$

This says that d gives A the structure of a chain complex or cochain complex (accordingly as the differential reduces or raises degree).

(ii) $d(a\cdot b)=(da)\cdot b+(-1{)}^{\left|a\right|}a\cdot (db)$.<Graded Leibniz rule>...</Graded Leibniz rule>

This says that the differential d respects the graded Leibniz rule.

A DGA is an augmented DG-algebra, or differential graded augmented algebraTemplate:Clarify (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.

Other facts about DGAs

• The homology $H_{*}(A)=\ker (d)/\mathrm{im}(d)$ of a

DG-algebra $(A,d)$ is a graded algerba. The homology of a DGA is an augmented algebra.

See also

• Chain complex
• Commutative ring spectrum
• Derived scheme
• Differential graded category
• Differential graded Lie algebra
• Graded (mathematics)
• Graded algebra

References

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