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

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

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:

:# 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 $\mathcal{A}^*$ forms a comodule.{{Cite journal|last=Liulevicius|first=Arunas|date=1968|title=Homology Comodules|url=https://www.ams.org/journals/tran/1968-134-02/S0002-9947-1968-0251720-X/S0002-9947-1968-0251720-X.pdf|journal=Transactions of the American Mathematical Society|volume=134|issue=2|pages=375–382|doi=10.2307/1994750|jstor=1994750 |issn=0002-9947|doi-access=free}} 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\})

.{{Cite web|last=Mueller|first=Michael|title=Calculating Cobordism Rings|url=https://www.brown.edu/academics/math/sites/math/files/Mueller,%20Michael.pdf|url-status=live|archive-url=https://web.archive.org/web/20210102194203/https://www.brown.edu/academics/math/sites/math/files/Mueller%2C%20Michael.pdf|archive-date=2 Jan 2021}} 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.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

References

{{Citation| last=Gómez-Torrecillas| first=José| title=Coalgebras and comodules over a commutative ring| year=1998 | journal= Revue Roumaine de Mathématiques Pures et Appliquées| volume=43 | pages=591–603}}
{{Cite book | last = Montgomery | first = Susan | author-link=Susan Montgomery | year = 1993 | series=Regional Conference Series in Mathematics | volume=82 | title = Hopf algebras and their actions on rings | zbl=0793.16029 | publisher = American Mathematical Society | location = Providence, RI | isbn=0-8218-0738-2 }}
{{Citation| last=Sweedler| first=Moss| title = Hopf Algebras| year=1969 | publisher = W.A.Benjamin| location = New York}}

Category:Module theory

Category:Coalgebras