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.

DefinitionEdit

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:

  1.  .
    This says that d gives A the structure of a chain complex or cochain complex (accordingly as the differential reduces or raises degree).
  2.  , where deg is the degree of homogeneous elements.
    This says that the differential d respects the graded Leibniz rule.

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).[1]

Warning: some sources use the term DGA for a DG-algebra.

Examples of DG-algebrasEdit

Tensor algebraEdit

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

 

Koszul complexEdit

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.

De-Rham algebraEdit

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.[2] See also de Rham cohomology.

Singular cohomologyEdit

  • 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.[3][4]

Other facts about DG-algebrasEdit

  • The homology   of a DG-algebra   is a graded algebra. The homology of a DGA-algebra is an augmented algebra.

See alsoEdit

ReferencesEdit

  1. ^ 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.
  2. ^ Manetti. "Differential graded Lie algebras and formal deformation theory" (PDF). Archived (PDF) from the original on 16 Jun 2013.
  3. ^ Cartan, H. (1954–1955). "DGA-algèbres et DGA-modules". Séminaire Henri Cartan. 7 (1): 1–9.
  4. ^ Cartan, H. (1954–1955). "DGA-modules (suite), notion de construction". Séminaire Henri Cartan. 7 (1): 1–11.