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 (right) 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:

# Let the comultiplication on be given by .

# Let the counit on be given by .

# Let the map on V be given by , where is the i-th homogeneous piece of .

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<blockquote></blockquote>When we dualize to the dual Steenrod algebra, this gives a comodule structure<blockquote></blockquote>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 (right) comodule over the coalgebra C, then M is a (left) module over the dual algebra C^&lowast;, but the converse is not true in general: a module over C^&lowast; is not necessarily a comodule over C. A rational comodule is a module over C^&lowast; 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 (right) comodule morphism, or (right) 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.

See also

• Divided power structure

References