comodules

Comodules are the dual notion to modules in the theory of coalgebras. Let k be a field and C a coalgebra with comultiplication Δ: C → C ⊗ C and counit ε: C → k. A left C-comodule is a vector space M equipped with a structure (coaction) map ρ: M → C ⊗ M satisfying coassociativity (Δ ⊗ id_M) ∘ ρ = (id_C ⊗ ρ) ∘ ρ and counit (ε ⊗ id_M) ∘ ρ = id_M. A right C-comodule is similarly a vector space N with ρ: N → N ⊗ C satisfying (id_N ⊗ Δ) ∘ ρ = (ρ ⊗ id_C) ∘ ρ and (id_N ⊗ ε) ∘ ρ = id_N. A C-comodule morphism f: M → N is a linear map that commutes with the coactions, i.e., (id_C ⊗ f) ∘ ρ_M = ρ_N ∘ f for left comodules; the right case uses the appropriate tensor factor order.

Comodules form categories, denoted ^C Mod for left comodules and Mod^C for right comodules. If C is finite dimensional, there is a close relationship with modules over the dual algebra C*, because finite-dimensional comodules correspond to finite-dimensional C*-modules via the action f • m = ∑ f(c_i) m_i for ρ(m) = ∑ c_i ⊗ m_i.

Examples include the base field C = k, where every vector space has a trivial coaction, and coalgebras arising from group-like elements, where the comodules encode group representations in the coaction framework.

Properties and use: the category of left C-comodules is abelian and has enough injectives; semisimplicity of C (cosemisimplicity) implies that every comodule is semisimple. Comodules play a central role in representation theory of coalgebras and Hopf algebras, Tannaka–Krein duality, and the study of corepresentations in quantum groups and algebraic geometry.