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.

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


 * $$\rho\colon M \to M \otimes C$$

such that where Δ is the comultiplication for C, and ε is the counit.
 * 1) $$(\mathrm{id} \otimes \Delta) \circ \rho = (\rho \otimes \mathrm{id}) \circ \rho$$
 * 2) $$(\mathrm{id} \otimes \varepsilon) \circ \rho = \mathrm{id}$$,

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 finite-dimensional module over a finite-dimensional 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 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 $$v$$.

In algebraic topology
One important result in algebraic topology is the fact that homology $$H_*(X)$$ over the dual Steenrod algebra $$\mathcal{A}^*$$ forms a comodule. This comes from the fact the Steenrod algebra $$\mathcal{A}$$ has a canonical action on the cohomology"$\mu: \mathcal{A}\otimes H^*(X) \to H^*(X)$"When we dualize to the dual Steenrod algebra, this gives a comodule structure"$\mu^*:H_*(X) \to \mathcal{A}^*\otimes H_*(X)$"This result extends to other cohomology theories as well, such as complex cobordism and is instrumental in computing its cohomology ring $$\Omega_U^*(\{pt\})$$. The main reason for considering the comodule structure on homology instead of the module structure on cohomology lies in the fact the dual Steenrod algebra $$\mathcal{A}^*$$ is a commutative ring, and the setting of commutative algebra provides more tools for studying its structure.

Rational comodule
If M is a (right) comodule over the coalgebra C, then M is a (left) module over the dual algebra C&lowast;, but the converse is not true in general: a module over C&lowast; is not necessarily a comodule over C. A rational comodule is a module over C&lowast; which becomes a comodule over C in the natural way.

Comodule morphisms
Let R be a ring, M, N, and C be R-modules, and $$\rho_M: M \rightarrow M \otimes C,\ \rho_N: N \rightarrow N \otimes C$$ 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.