Differential Graded Commutative Algebra
Get Differential Graded Commutative Algebra essential facts below. View Videos or join the Differential Graded Commutative Algebra discussion. Add Differential Graded Commutative Algebra to your PopFlock.com topic list for future reference or share this resource on social media.
Differential Graded Commutative 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.

Definition

A differential graded algebra (or simply DG-algebra) A is a graded algebra equipped with a map which is 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 (but esoteric) 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-algebras

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.

See also

References

  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: 467-471.

  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Differential_graded_commutative_algebra
 



 



 
Music Scenes