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Comodule

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.

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

\rho: M \to M \otimes C

such that

  1. (id \otimes \Delta) \circ \rho = (\rho \otimes id) \circ \rho
  2. (id \otimes \epsilon) \circ \rho = id,

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

Note that in the second rule we have identified M \otimes K with M\,.

Examples

  • A coalgebra is a comodule over itself.
  • If M is a module over a 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 CI be the vector space with basis ei for i \in I. We turn CI into a coalgebra and V into a CI-comodule, as follows:
  1. Let the comultiplication on CI be given by \Delta(e_i) = e_i \otimes e_i.
  2. Let the counit on CI be given by \epsilon(e_i) = 1\.
  3. Let the map ρ on V be given by \rho(v) = \sum v_i \otimes e_i, where vi is the i-th homogeneous piece of v.
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