Differential graded algebra
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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.
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[edit] Definition
A differential graded algebra (or simply DGA) A is a graded algebra equipped with a degree -1 map 
that satisfies two conditions:
- (i)
- (ii)
.
Condition (i) says that d gives A the structure of a chain complex. Condition (ii) says that the differential d respects the graded Leibniz rule.
[edit] 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.
[edit] Other facts about DGAs
- The homology H * (A) = ker(d) / im(d) of a DGA (A,d) is a graded ring.
