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

[math]\displaystyle{ \rho\colon M \to M \otimes C }[/math]

such that

1. [math]\displaystyle{ (\mathrm{id} \otimes \Delta) \circ \rho = (\rho \otimes \mathrm{id}) \circ \rho }[/math]
2. [math]\displaystyle{ (\mathrm{id} \otimes \varepsilon) \circ \rho = \mathrm{id} }[/math],

where Δ is the comultiplication for C, and ε is the counit.

Note that in the second rule we have identified [math]\displaystyle{ M \otimes K }[/math] with [math]\displaystyle{ M\, }[/math].

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 [math]\displaystyle{ C_I }[/math] be the vector space with basis [math]\displaystyle{ e_i }[/math] for [math]\displaystyle{ i \in I }[/math]. We turn [math]\displaystyle{ C_I }[/math] into a coalgebra and V into a [math]\displaystyle{ C_I }[/math]-comodule, as follows:

1. Let the comultiplication on [math]\displaystyle{ C_I }[/math] be given by [math]\displaystyle{ \Delta(e_i) = e_i \otimes e_i }[/math].
2. Let the counit on [math]\displaystyle{ C_I }[/math] be given by [math]\displaystyle{ \varepsilon(e_i) = 1\ }[/math].
3. Let the map [math]\displaystyle{ \rho }[/math] on V be given by [math]\displaystyle{ \rho(v) = \sum v_i \otimes e_i }[/math], where [math]\displaystyle{ v_i }[/math] is the i-th homogeneous piece of [math]\displaystyle{ v }[/math].

In algebraic topology

One important result in algebraic topology is the fact that homology [math]\displaystyle{ H_*(X) }[/math] over the dual Steenrod algebra [math]\displaystyle{ \mathcal{A}^* }[/math] forms a comodule. This comes from the fact the Steenrod algebra [math]\displaystyle{ \mathcal{A} }[/math] has a canonical action on the cohomology

[math]\displaystyle{ \mu: \mathcal{A}\otimes H^*(X) \to H^*(X) }[/math]

When we dualize to the dual Steenrod algebra, this gives a comodule structure

[math]\displaystyle{ \mu^*:H_*(X) \to \mathcal{A}^*\otimes H_*(X) }[/math]

This result extends to other cohomology theories as well, such as complex cobordism and is instrumental in computing its cohomology ring [math]\displaystyle{ \Omega_U^*(\{pt\}) }[/math]. 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 [math]\displaystyle{ \mathcal{A}^* }[/math] 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^∗, but the converse is not true in general: a module over C^∗ is not necessarily a comodule over C. A rational comodule is a module over C^∗ 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 [math]\displaystyle{ \rho_M: M \rightarrow M \otimes C,\ \rho_N: N \rightarrow N \otimes C }[/math] be right C-comodules. Then an R-linear map [math]\displaystyle{ f: M \rightarrow N }[/math] is called a (right) comodule morphism, or (right) C-colinear, if [math]\displaystyle{ \rho_N \circ f = (f \otimes 1) \circ \rho_M. }[/math] This notion is dual to the notion of a linear map between vector spaces, or, more generally, of a homomorphism between R-modules.

See also

• Divided power structure

References

• Gómez-Torrecillas, José (1998), "Coalgebras and comodules over a commutative ring", Revue Roumaine de Mathématiques Pures et Appliquées 43: 591–603 
• Montgomery, Susan (1993). Hopf algebras and their actions on rings. Regional Conference Series in Mathematics. 82. Providence, RI: American Mathematical Society. ISBN 0-8218-0738-2. 
• Sweedler, Moss (1969), Hopf Algebras, New York: W.A.Benjamin