# Comodule

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In [mathematics](/source/Mathematics), a **comodule** or corepresentation is a concept [dual](/source/Duality_(mathematics)) to a [module](/source/Module_(mathematics)). The definition of a comodule over a [coalgebra](/source/Coalgebra) is formed by dualizing the definition of a module over an [associative algebra](/source/Associative_algebra).

## Formal definition

Let *K* be a [field](/source/Field_(mathematics)), and *C* be a [coalgebra](/source/Coalgebra) over *K*. A (right) **comodule** over *C* is a *K*-[vector space](/source/Vector_space) *M* together with a [linear map](/source/Linear_map)

- ρ : M → M ⊗ C {\displaystyle \rho \colon M\to M\otimes C}

such that

1. ( i d ⊗ Δ ) ∘ ρ = ( ρ ⊗ i d ) ∘ ρ {\displaystyle (\mathrm {id} \otimes \Delta )\circ \rho =(\rho \otimes \mathrm {id} )\circ \rho }

1. ( i d ⊗ ε ) ∘ ρ = i d {\displaystyle (\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 ⊗ K {\displaystyle M\otimes K} with M {\displaystyle 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](/source/Linear_function) 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](/source/Graded_vector_space) *V* can be made into a comodule. Let *I* be the [index set](/source/Index_set) for the graded vector space, and let C I {\displaystyle C_{I}} be the vector space with basis e i {\displaystyle e_{i}} for i ∈ I {\displaystyle i\in I} . We turn C I {\displaystyle C_{I}} into a coalgebra and *V* into a C I {\displaystyle C_{I}} -comodule, as follows:

- 1. Let the comultiplication on C I {\displaystyle C_{I}} be given by Δ ( e i ) = e i ⊗ e i {\displaystyle \Delta (e_{i})=e_{i}\otimes e_{i}} . 1. Let the counit on C I {\displaystyle C_{I}} be given by ε ( e i ) = 1 {\displaystyle \varepsilon (e_{i})=1\ } . 1. Let the map ρ {\displaystyle \rho } on *V* be given by ρ ( v ) = ∑ v i ⊗ e i {\displaystyle \rho (v)=\sum v_{i}\otimes e_{i}} , where v i {\displaystyle v_{i}} is the *i*-th homogeneous piece of v {\displaystyle v} .

### In algebraic topology

One important result in [algebraic topology](/source/Algebraic_topology) is the fact that homology H ∗ ( X ) {\displaystyle H_{*}(X)} over the dual [Steenrod algebra](/source/Steenrod_algebra) A ∗ {\displaystyle {\mathcal {A}}^{*}} forms a comodule.[1] This comes from the fact the Steenrod algebra A {\displaystyle {\mathcal {A}}} has a canonical action on the cohomology

μ : A ⊗ H ∗ ( X ) → H ∗ ( X ) {\displaystyle \mu :{\mathcal {A}}\otimes H^{*}(X)\to H^{*}(X)}

When we dualize to the [dual Steenrod algebra](/source/Dual_Steenrod_algebra), this gives a comodule structure

μ ∗ : H ∗ ( X ) → A ∗ ⊗ H ∗ ( X ) {\displaystyle \mu ^{*}:H_{*}(X)\to {\mathcal {A}}^{*}\otimes H_{*}(X)}

This result extends to other cohomology theories as well, such as [complex cobordism](/source/Complex_cobordism) and is instrumental in computing its cohomology ring Ω U ∗ ( { p t } ) {\displaystyle \Omega _{U}^{*}(\{pt\})} .[2] 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 A ∗ {\displaystyle {\mathcal {A}}^{*}} is a [commutative ring](/source/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](/source/Ring_(mathematics)), *M*, *N*, and *C* be *R*-modules, and ρ M : M → M ⊗ C , ρ N : N → N ⊗ C {\displaystyle \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 → N {\displaystyle f:M\rightarrow N} is called a **(right) comodule morphism**, or **(right) C-colinear**, if ρ N ∘ f = ( f ⊗ 1 ) ∘ ρ M . {\displaystyle \rho _{N}\circ f=(f\otimes 1)\circ \rho _{M}.} This notion is dual to the notion of a [linear map](/source/Linear_map) between [vector spaces](/source/Vector_space), or, more generally, of a [homomorphism](/source/Module_homomorphism) between [*R*-modules](/source/Module_(mathematics)).[3]

## See also

- [Divided power structure](/source/Divided_power_structure)

## References

1. **[^](#cite_ref-1)** Liulevicius, Arunas (1968). ["Homology Comodules"](https://www.ams.org/journals/tran/1968-134-02/S0002-9947-1968-0251720-X/S0002-9947-1968-0251720-X.pdf) (PDF). *Transactions of the American Mathematical Society*. **134** (2): 375–382. [doi](/source/Doi_(identifier)):[10.2307/1994750](https://doi.org/10.2307%2F1994750). [ISSN](/source/ISSN_(identifier)) [0002-9947](https://search.worldcat.org/issn/0002-9947). [JSTOR](/source/JSTOR_(identifier)) [1994750](https://www.jstor.org/stable/1994750).

1. **[^](#cite_ref-2)** Mueller, Michael. ["Calculating Cobordism Rings"](https://www.brown.edu/academics/math/sites/math/files/Mueller,%20Michael.pdf) (PDF). [Archived](https://web.archive.org/web/20210102194203/https://www.brown.edu/academics/math/sites/math/files/Mueller%2C%20Michael.pdf) (PDF) from the original on 2 Jan 2021.

1. **[^](#cite_ref-3)** Khaled AL-Takhman, *Equivalences of Comodule Categories for Coalgebras over Rings*, J. Pure Appl. Algebra,.V. 173, Issue: 3, September 7, 2002, pp. 245–271

- Gómez-Torrecillas, José (1998), "Coalgebras and comodules over a commutative ring", *Revue Roumaine de Mathématiques Pures et Appliquées*, **43**: 591–603

- [Montgomery, Susan](/source/Susan_Montgomery) (1993). *Hopf algebras and their actions on rings*. Regional Conference Series in Mathematics. Vol. 82. Providence, RI: [American Mathematical Society](/source/American_Mathematical_Society). [ISBN](/source/ISBN_(identifier)) [0-8218-0738-2](https://en.wikipedia.org/wiki/Special:BookSources/0-8218-0738-2). [Zbl](/source/Zbl_(identifier)) [0793.16029](https://zbmath.org/?format=complete&q=an:0793.16029).

- Sweedler, Moss (1969), *Hopf Algebras*, New York: [W.A.Benjamin](https://en.wikipedia.org/w/index.php?title=W.A.Benjamin&action=edit&redlink=1)

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