Differential graded algebra

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.

[edit] Differential Graded Algebra

A differential graded algebra (or simply DGA) A is a graded algebra equipped with a degree -1 map $d:A \to A$ that satisfies two conditions:

(i) $d \circ d=0$

(ii) $d(a \cdot b)=(da) \cdot b + (-1)^{|a|}a \cdot (db)$ .

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.

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