Comodule

In mathematics, a comodule or corepresentation 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.

In mathematics, a comodule or corepresentation 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. (\mathrm{id} \otimes \Delta) \circ \rho = (\rho \otimes \mathrm{id}) \circ \rho (\mathrm{id} \otimes \varepsilon) \circ \rho = \mathrm{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\, . Let the comultiplication on C_I be given by \Delta(e_i) = e_i \otimes e_i . Let the counit on C_I be given by \varepsilon(e_i) = 1\ . 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 . \rho_M: M \rightarrow M \otimes C,\ \rho_N: N \rightarrow N \otimes C be right C-comodules. Then an R-linear map f: M \rightarrow N is called a (right) comodule morphism, or (right) C-colinear, if \rho_N \circ f = (f \otimes 1) \circ \rho_M. This notion is dual to the notion of a linear map between vector spaces, or, more generally, of a homomorphism between R-modules.

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