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
# $(\mathrm \otimes \Delta) \circ \rho = (\rho \otimes \mathrm) \circ \rho$
# $(\mathrm \otimes \varepsilon) \circ \rho = \mathrm$,
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:
:# Let the comultiplication on $C_I$ be given by $\Delta(e_i) = e_i \otimes e_i$.
:# Let the counit on $C_I$ be given by $\varepsilon(e_i) = 1\$.
:# 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 $\mathcal^*$ forms a comodule. This comes from the fact the Steenrod algebra $\mathcal$ has a canonical action on the cohomology

$\mu: \mathcal\otimes H^*(X) \to H^*(X)$

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

$\mu^*:H_*(X) \to \mathcal^*\otimes H_*(X)$

This result extends to other cohomology theories as well, such as complex cobordism and is instrumental in computing its cohomology ring $\Omega_U^*(\)$. 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 $\mathcal^*$ 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.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

See also

* Divided power structure

References

*
*
*{{Citation, last=Sweedler, first=Moss, title = Hopf Algebras, year=1969 , publisher = W.A.Benjamin, location = New York
Module theory
Coalgebras