Английская Википедия: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
- <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 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 <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.[1] This comes from the fact the Steenrod algebra <math>\mathcal{A}</math> has a canonical action on the cohomology
<math>\mu: \mathcal{A}\otimes H^*(X) \to H^*(X)</math>
When we dualize to the dual Steenrod algebra, this gives a comodule structure
<math>\mu^*:H_*(X) \to \mathcal{A}^*\otimes H_*(X)</math>
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>.[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 <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∗, 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 <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 spaces, or, more generally, of a homomorphism between R-modules.[3]
See also
References
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite web
- ↑ 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
