In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a comodule or corepresentation is a concept
dual to a
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
. The definition of a comodule over a
coalgebra In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams ...
is formed by dualizing the definition of a module over an
associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
.
Formal definition
Let ''K'' be a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
, and ''C'' be a
coalgebra In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams ...
over ''K''. A (right) comodule over ''C'' is a ''K''-
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
''M'' together with a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
:
such that
#
#
,
where Δ is the comultiplication for ''C'', and ε is the counit.
Note that in the second rule we have identified
with
.
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
In mathematics, the term linear function refers to two distinct but related notions:
* In calculus and related areas, a linear function is a function (mathematics), function whose graph of a function, graph is a straight line, that is, a polynomia ...
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
In mathematics, a graded vector space is a vector space that has the extra structure of a '' grading'' or a ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces.
Integer gradation
Let \mathbb be th ...
''V'' can be made into a comodule. Let ''I'' be the
index set
In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consists ...
for the graded vector space, and let
be the vector space with basis
for
. We turn
into a coalgebra and ''V'' into a
-comodule, as follows:
:# Let the comultiplication on
be given by
.
:# Let the counit on
be given by
.
:# Let the map
on ''V'' be given by
, where
is the ''i''-th homogeneous piece of
.
In algebraic topology
One important result in algebraic topology is the fact that homology
over the dual
Steenrod algebra forms a comodule. This comes from the fact the Steenrod algebra
has a canonical action on the cohomology
When we dualize to the dual Steenrod algebra, this gives a comodule structure
This result extends to other cohomology theories as well, such as
complex cobordism In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it ...
and is instrumental in computing its cohomology ring
. 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
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
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
, ''M'', ''N'', and ''C'' be ''R''-modules, and
be right ''C''-
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, an ...
s. Then an ''R''-linear map
is called a (right) comodule morphism, or (right) C-colinear, if
This notion is dual to the notion of a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
between
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s, or, more generally, of a
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
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
In mathematics, specifically commutative algebra, a divided power structure is a way of making expressions of the form x^n / n! meaningful even when it is not possible to actually divide by n!.
Definition
Let ''A'' be a commutative ring with an ...
References
*
*
*{{Citation, last=Sweedler, first=Moss, title = Hopf Algebras, year=1969 , publisher =
W.A.Benjamin, location = New York
Module theory
Coalgebras