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

1. ,

where Δ is the comultiplication for C, and ε is the counit.

.

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 into a coalgebra and V-comodule, as follows:

• .

• .

• on V is the i.

In algebraic topology

One important result in algebraic topology over the dual Steenrod algebra forms a comodule. has a canonical action on the cohomologyWhen we dualize to the dual Steenrod algebra, this gives a comodule structureThis result extends to other cohomology theories as well, such as complex cobordism. 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, 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 be right C-comodules. Then an R is called a (right) comodule morphism, or (right) C-colinear 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