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.^[1] This comes from the fact the Steenrod algebra ${\mathcal {A}}$ has a canonical action on the cohomology

${\displaystyle \mu$

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 $\Omega _{U}^{*}(\{pt\})$ .^[2] 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.^[3]

References

↑ Liulevicius, Arunas (1968). "Homology Comodules" (PDF). Transactions of the American Mathematical Society. 134 (2): 375–382. doi:10.2307/1994750. ISSN 0002-9947. JSTOR 1994750.
↑ Mueller, Michael. "Calculating Cobordism Rings" (PDF). Archived (PDF) from the original on 2 Jan 2021.
↑ 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

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

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[1] Liulevicius, Arunas (1968). "Homology Comodules" (PDF). Transactions of the American Mathematical Society. 134 (2): 375–382. doi:10.2307/1994750. ISSN 0002-9947. JSTOR 1994750.

[2] Mueller, Michael. "Calculating Cobordism Rings" (PDF). Archived (PDF) from the original on 2 Jan 2021.

[3] 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

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