Comodule


In mathematics, a comodule or corepresentation is a concept dual to a module. The definition of a comodule over a coalgebra is formed by dualizing the definition of a module over an associative algebra.

Formal definition

Let K be a field, and C be a coalgebra over K. A comodule over C is a K-vector space M together with a linear map
such that
  2. ,
where Δ is the comultiplication for C, and ε is the counit.
Note that in the second rule we have identified with.

Examples

  • A coalgebra is a comodule over itself.
  • If M is a finite-dimensional module over a finite-dimensional K-algebra A, then the set of linear functions from A to K forms a coalgebra, and the set of linear functions from M to K forms a comodule over that coalgebra.
  • A graded vector space V can be made into a comodule. Let I be the index set for the graded vector space, and let be the vector space with basis for. We turn into a coalgebra and V into a -comodule, as follows:

    In algebraic topology

One important result in algebraic topology is the fact that homology over the dual Steenrod algebra forms a comodule. This comes from the fact the Steenrod algebra has a canonical action on the cohomology
When we dualize to the dual Steenrod algebra, this gives a comodule structure
This result extends to other cohomology theories as well, such as complex cobordism and is instrumental in computing its cohomology ring. The main reason for considering the comodule structure on homology instead of the module structure on cohomology lies in the fact the dual Steenrod algebra is a commutative ring, and the setting of commutative algebra provides more tools for studying its structure.

Rational comodule

If M is a comodule over the coalgebra C, then M is a module over the dual algebra C∗, but the converse is not true in general: a module over C∗ is not necessarily a comodule over C. A rational comodule is a module over C∗ which becomes a comodule over C in the natural way.

Comodule morphisms

Let R be a ring, M, N, and C be R-modules, and
be right C-comodules. Then an R-linear map is called a comodule morphism, or C-colinear, if
This notion is dual to the notion of a linear map between vector spaces, or, more generally, of a homomorphism between R-modules.