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

1. $(\mathrm {id} \otimes \Delta )\circ \rho =(\rho \otimes \mathrm {id} )\circ \rho$
2. $(\mathrm {id} \otimes \varepsilon )\circ \rho =\mathrm {id}$,

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 $M\,$.

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:

1. Let the comultiplication on $C_{I}$ be given by $\Delta (e_{i})=e_{i}\otimes e_{i}$.
2. Let the counit on $C_{I}$ be given by $\varepsilon (e_{i})=1\$.
3. 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^∗, 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

$$\rho _{M}:M\rightarrow M\otimes C,\ \rho _{N}:N\rightarrow N\otimes C$$

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.

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. Vol. 82. Providence, RI: American Mathematical Society. ISBN 0-8218-0738-2. Zbl 0793.16029.
• Sweedler, Moss (1969), Hopf Algebras, New York: W.A.Benjamin