Comodule

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

1. (\mathrm{id} \otimes \Delta) \circ \rho = (\rho \otimes \mathrm{id}) \circ \rho
2. (\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. This comes from the fact the Steenrod algebra \mathcal{A} has a canonical action on the cohomology\mu: \mathcal{A}\otimes H^(X) \to H^(X)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}). 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.

References

References

1. Liulevicius, Arunas. (1968). "Homology Comodules". Transactions of the American Mathematical Society.
2. Mueller, Michael. "Calculating Cobordism Rings".
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