Editing Comodule

In [[mathematics]], a '''comodule''' or corepresentation is a concept [[duality (mathematics)|dual]] to a [[module (mathematics)|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 (mathematics)|field]], and ''C'' be a [[coalgebra]] over ''K''.  A (right) '''comodule''' over ''C'' is a ''K''-[[vector space]] ''M'' together with a [[linear map]]

:<math>\rho\colon M \to M \otimes C</math>

such that
# <math>(\mathrm{id} \otimes \Delta) \circ \rho = (\rho \otimes \mathrm{id}) \circ \rho</math>
# <math>(\mathrm{id} \otimes \varepsilon) \circ \rho = \mathrm{id}</math>,
where Δ is the comultiplication for ''C'', and ε is the counit.

Note that in the second rule we have identified <math>M \otimes K</math> with <math>M\,</math>.

== 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 function]]s 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 <math>C_I</math> be the vector space with basis <math>e_i</math> for <math>i \in I</math>.  We turn <math>C_I</math> into a coalgebra and ''V'' into a <math>C_I</math>-comodule, as follows:
:# Let the comultiplication on <math>C_I</math> be given by <math>\Delta(e_i) = e_i \otimes e_i</math>.
:# Let the counit on <math>C_I</math> be given by <math>\varepsilon(e_i) = 1\ </math>.
:# Let the map <math>\rho</math> on ''V'' be given by <math>\rho(v) = \sum v_i \otimes e_i</math>, where <math>v_i</math> is the ''i''-th homogeneous piece of <math>v</math>.

=== In algebraic topology ===
One important result in [[algebraic topology]] is the fact that homology <math>H_*(X)</math> over the dual [[Steenrod algebra]] <math>\mathcal{A}^*</math> forms a comodule.<ref>{{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}}</ref> This comes from the fact the Steenrod algebra <math>\mathcal{A}</math> has a canonical action on the cohomology<blockquote><math>\mu: \mathcal{A}\otimes H^*(X) \to H^*(X)</math></blockquote>When we dualize to the [[dual Steenrod algebra]], this gives a comodule structure<blockquote><math>\mu^*:H_*(X) \to \mathcal{A}^*\otimes H_*(X)</math></blockquote>This result extends to other cohomology theories as well, such as [[complex cobordism]] and is instrumental in computing its cohomology ring <math>\Omega_U^*(\{pt\})</math>.<ref>{{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}}</ref> 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 <math>\mathcal{A}^*</math> 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''<sup>&lowast;</sup>, but the converse is not true in general: a module over ''C''<sup>&lowast;</sup> is not necessarily a comodule over ''C''.  A '''rational comodule''' is a module over ''C''<sup>&lowast;</sup> which becomes a comodule over ''C'' in the natural way.

== Comodule morphisms ==
Let ''R'' be a [[Ring (mathematics)|ring]], ''M'', ''N'', and ''C'' be ''R''-modules, and
<math display="block">\rho_M: M \rightarrow M \otimes C,\ \rho_N: N \rightarrow N \otimes C</math>
be right ''C''-comodules. Then an ''R''-linear map <math>f: M \rightarrow N</math> is called a '''(right) comodule morphism''', or '''(right) C-colinear''', if
<math display="block">\rho_N \circ f = (f \otimes 1) \circ \rho_M.</math>
This notion is dual to the notion of a [[linear map]] between [[vector space]]s, or, more generally, of a [[module homomorphism|homomorphism]] between [[module (mathematics)|''R''-modules]].<ref>Khaled AL-Takhman, ''Equivalences of Comodule Categories for Coalgebras over Rings'', J. Pure Appl. Algebra,.V. 173, Issue: 3, September 7, 2002, pp.&nbsp;245–271</ref>
== See also ==

* [[Divided power structure]]

== References ==
{{Reflist}}

*{{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]]