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

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\,

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 $C_I$ be the vector space with basis $e_i$ for $i \in I$ . We turn $C_I$ into a coalgebra and V into a $C_I$ -comodule, as follows:

Let the map $\rho$ on V be given by $\rho(v) = \sum v_i \otimes e_i$ , where $v_i$ is the i-th homogeneous piece of $v$ .

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