Comodule

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In mathematics, a comodule 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.

[edit] 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. 
2. ,

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

Note that in the second rule we have identified with .

[edit] 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 C_I be the vector space with basis e_i for . We turn C_I into a coalgebra and V into a C_I-comodule, as follows:

1. Let the comultiplication on C_I be given by .
2. Let the counit on C_I be given by .
3. Let the map ρ on V be given by , where v_i is the i-th homogeneous piece of v.