• Source: 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




ρ
:
M
→
M
⊗
C


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


such that




(

i
d

⊗
Δ
)
∘
ρ
=
(
ρ
⊗

i
d

)
∘
ρ


{\displaystyle (\mathrm {id} \otimes \Delta )\circ \rho =(\rho \otimes \mathrm {id} )\circ \rho }





(

i
d

⊗
ε
)
∘
ρ
=

i
d



{\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



M
⊗
K


{\displaystyle M\otimes K}

with



M



{\displaystyle 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




{\displaystyle C_{I}}

be the vector space with basis




e

i




{\displaystyle e_{i}}

for



i
∈
I


{\displaystyle i\in I}

. We turn




C

I




{\displaystyle C_{I}}

into a coalgebra and V into a




C

I




{\displaystyle C_{I}}

-comodule, as follows:
Let the comultiplication on




C

I




{\displaystyle C_{I}}

be given by



Δ
(

e

i


)
=

e

i


⊗

e

i




{\displaystyle \Delta (e_{i})=e_{i}\otimes e_{i}}

.
Let the counit on




C

I




{\displaystyle C_{I}}

be given by



ε
(

e

i


)
=
1



{\displaystyle \varepsilon (e_{i})=1\ }

.
Let the map



ρ


{\displaystyle \rho }

on V be given by



ρ
(
v
)
=
∑

v

i


⊗

e

i




{\displaystyle \rho (v)=\sum v_{i}\otimes e_{i}}

, where




v

i




{\displaystyle v_{i}}

is the i-th homogeneous piece of



v


{\displaystyle v}

.




= In algebraic topology

=
One important result in algebraic topology is the fact that homology




H

∗


(
X
)


{\displaystyle H_{*}(X)}

over the dual Steenrod algebra






A



∗




{\displaystyle {\mathcal {A}}^{*}}

forms a comodule. This comes from the fact the Steenrod algebra





A




{\displaystyle {\mathcal {A}}}

has a canonical action on the cohomology



μ
:


A


⊗

H

∗


(
X
)
→

H

∗


(
X
)


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

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




μ

∗


:

H

∗


(
X
)
→



A



∗


⊗

H

∗


(
X
)


{\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




Ω

U


∗


(
{
p
t
}
)


{\displaystyle \Omega _{U}^{*}(\{pt\})}

. 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






A



∗




{\displaystyle {\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





ρ

M


:
M
→
M
⊗
C
,


ρ

N


:
N
→
N
⊗
C


{\displaystyle \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
→
N


{\displaystyle f:M\rightarrow N}

is called a (right) comodule morphism, or (right) C-colinear, if





ρ

N


∘
f
=
(
f
⊗
1
)
∘

ρ

M


.


{\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.




See also


Divided power structure




References



Gómez-Torrecillas, José (1998), "Coalgebras and comodules over a commutative ring", Revue Roumaine de Mathématiques Pures et Appliquées, 43: 591–603
Montgomery, Susan (1993). Hopf algebras and their actions on rings. Regional Conference Series in Mathematics. Vol. 82. Providence, RI: American Mathematical Society. ISBN 0-8218-0738-2. Zbl 0793.16029.
Sweedler, Moss (1969), Hopf Algebras, New York: W.A.Benjamin

Kata Kunci Pencarian:

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