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Concept
Differential graded algebra
Summary
In mathematics, in particular in homological algebra, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure.
TOC
A differential graded algebra (or DG-algebra for short) A is a graded algebra equipped with a map which has either degree 1 (cochain complex convention) or degree −1 (chain complex convention) that satisfies two conditions:
A more succinct way to state the same definition is to say that a DG-algebra is a monoid object in the .
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.
The tensor algebra is a DG-algebra with differential similar to that of the Koszul complex. For a vector space over a field there is a graded vector space defined as
where .
If is a basis for there is a differential on the tensor algebra defined component-wise
sending basis elements to
In particular we have and so
One of the foundational examples of a differential graded algebra, widely used in commutative algebra and algebraic geometry, is the Koszul complex. This is because of its wide array of applications, including constructing flat resolutions of complete intersections, and from a derived perspective, they give the derived algebra representing a derived critical locus.
Differential forms on a manifold, together with the exterior derivation and the exterior product form a DG-algebra. These have wide applications, including in derived deformation theory. See also de Rham cohomology.
The singular cohomology of a topological space with coefficients in is a DG-algebra: the differential is given by the Bockstein homomorphism associated to the short exact sequence , and the product is given by the cup product.
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