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

$\rho :M\to M\otimes C$

such that

1. $(\mathrm{i}\mathrm{d}\otimes \mathrm{\Delta })\circ \rho =(\rho \otimes \mathrm{i}\mathrm{d})\circ \rho$
2. $(\mathrm{i}\mathrm{d}\otimes \epsilon )\circ \rho =\mathrm{i}\mathrm{d}$,

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 $\mathrm{\Delta }(e_i)=e_i\otimes e_i$.
2. Let the counit on $C_I$ be given by $\epsilon (e_i)=1$.
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 $H_{\ast }(X)$ over the dual Steenrod algebra ${𝒜}^{\ast }$ forms a comodule. This comes from the fact the Steenrod algebra $𝒜$ has a canonical action on the cohomology

$\mu :𝒜\otimes H^{\ast }(X)\to H^{\ast }(X)$

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

${\mu }^{\ast }:H_{\ast }(X)\to {𝒜}^{\ast }\otimes H_{\ast }(X)$

This result extends to other cohomology theories as well, such as complex cobordism and is instrumental in computing its cohomology ring ${\mathrm{\Omega }}_U^{\ast }(\{pt\})$. 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 ${𝒜}^{\ast }$ 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 $${\rho }_M:M\to M\otimes C,{\rho }_N:N\to N\otimes C$$ be right C-comodules. Then an R-linear map $f:M\to N$ 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

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