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.

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

${\displaystyle \rho \colon M\to M\otimes C}$

such that

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

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

Note that in the second rule we have identified ${\displaystyle M\otimes K}$ with ${\displaystyle M\,}$.

• 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 ${\displaystyle C_{I}}$ be the vector space with basis ${\displaystyle e_{i}}$ for ${\displaystyle i\in I}$. We turn ${\displaystyle C_{I}}$ into a coalgebra and V into a ${\displaystyle C_{I}}$-comodule, as follows:

1. Let the comultiplication on ${\displaystyle C_{I}}$ be given by ${\displaystyle \Delta (e_{i})=e_{i}\otimes e_{i}}$.
2. Let the counit on ${\displaystyle C_{I}}$ be given by ${\displaystyle \varepsilon (e_{i})=1\ }$.
3. Let the map ${\displaystyle \rho }$ on V be given by ${\displaystyle \rho (v)=\sum v_{i}\otimes e_{i}}$, where ${\displaystyle v_{i}}$ is the i-th homogeneous piece of ${\displaystyle v}$.

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

${\displaystyle \mu :{\mathcal {A}}\otimes H^{*}(X)\to H^{*}(X)}$

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

${\displaystyle \mu ^{*}:H_{*}(X)\to {\mathcal {A}}^{*}\otimes H_{*}(X)}$

This result extends to other cohomology theories as well, such as complex cobordism and is instrumental in computing its cohomology ring ${\displaystyle \Omega _{U}^{*}(\{pt\})}$.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 ${\displaystyle {\mathcal {A}}^{*}}$ is a commutative ring, and the setting of commutative algebra provides more tools for studying its structure.

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.

Let R be a ring, M, N, and C be R-modules, and $${\displaystyle \rho _{M}:M\rightarrow M\otimes C,\ \rho _{N}:N\rightarrow N\otimes C}$$ be right C-comodules. Then an R-linear map ${\displaystyle f:M\rightarrow N}$ is called a (right) comodule morphism, or (right) C-colinear, if $${\displaystyle \rho _{N}\circ f=(f\otimes 1)\circ \rho _{M}.}$$ This notion is dual to the notion of a linear map between vector spaces, or, more generally, of a homomorphism between R-modules.³