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Comodule

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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

  1. ( i d ⊗ Δ ) ∘ ρ = ( ρ ⊗ i d ) ∘ ρ {\displaystyle (\mathrm {id} \otimes \Delta )\circ \rho =(\rho \otimes \mathrm {id} )\circ \rho }
  2. ( 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:
  1. 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}} .
  2. Let the counit on C I {\displaystyle C_{I}} be given by ε ( e i ) = 1   {\displaystyle \varepsilon (e_{i})=1\ } .
  3. 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

References

  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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