A differential graded algebra (or simply DG-algebra) A is a graded algebra equipped with a map which has either degree 1 (cochain complex convention) or degree (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 monoidal category of chain complexes. 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.
Examples of DG-algebras
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
This has a canonical product given by tensoring elements
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. This differential graded algebra was used to help compute the cohomology of Eilenberg–MacLane spaces in the Cartan seminar.
Other facts about DG-algebras
- The homology of a DG-algebra is a graded algebra. The homology of a DGA-algebra is an augmented algebra.
- Homotopy associative algebra
- Differential graded category
- Differential graded Lie algebra
- Differential graded scheme (which is obtained by gluing the spectra of graded-commutative differential graded algebras with respect to the étale topology.)
- Differential graded module
- Cartan, Henri (1954). "Sur les groupes d'Eilenberg-Mac Lane ". Proceedings of the National Academy of Sciences of the United States of America. 40 (6): 467–471. doi:10.1073/pnas.40.6.467. PMC 534072. PMID 16589508.
- Manetti. "Differential graded Lie algebras and formal deformation theory" (PDF). Archived (PDF) from the original on 16 Jun 2013.
- Cartan, H. (1954–1955). "DGA-algèbres et DGA-modules". Séminaire Henri Cartan. 7 (1): 1–9.CS1 maint: date format (link)
- Cartan, H. (1954–1955). "DGA-modules (suite), notion de construction". Séminaire Henri Cartan. 7 (1): 1–11.CS1 maint: date format (link)