Comodule over a Coalgebra
Let \(R\) be a commutative semiring and let \(A\) be a coalgebra over \(R\), with comultiplication \(\Delta : A \to A \otimes_R A\) and counit \(\varepsilon : A \to R\). A (right) \(A\)-comodule is an \(R\)-module \(V\) together with an \(R\)-linear map (the coaction)
\[ \rho : V \to V \otimes_R A \]
satisfying:
- Coassociativity. The two paths from \(V\) to \(V \otimes_R A \otimes_R A\) agree:
\[ (\rho \otimes \operatorname{id}_A) \circ \rho \;=\; (\operatorname{id}_V \otimes \Delta) \circ \rho, \]
after the canonical identification \((V \otimes A) \otimes A \simeq V \otimes (A \otimes A)\).
- Counit. The composition
\[ V \to V \otimes_R A \to V \otimes_R R \cong V, \]
where the first arrow is \(\rho\) and the second is \(\operatorname{id}_V \otimes \varepsilon\), is the identity map of \(V\).
A morphism of \(A\)-comodules is an \(R\)-linear map of underlying modules commuting with the coactions.
When \(A\) is a Hopf algebra, \(A\)-comodules play the role of representations of the dual group scheme; see representations equal comodules.