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
such that
, where Δ is the comultiplication for C, and ε is the counit.
Note that in the second rule we have identified with .
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 be the vector space with basis for . We turn into a coalgebra and V into a -comodule, as follows: Let the comultiplication on be given by . Let the counit on be given by . Let the map on V be given by , where is the i-th homogeneous piece of .
In algebraic topology One important result in algebraic topology is the fact that homology over the dual Steenrod algebra forms a comodule. This comes from the fact the Steenrod algebra has a canonical action on the cohomologyWhen we dualize to the dual Steenrod algebra, this gives a comodule structureThis result extends to other cohomology theories as well, such as complex cobordism and is instrumental in computing its cohomology ring . 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 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
be right C-comodules. Then an R-linear map is called a (right) comodule morphism, or (right) C-colinear, if
This notion is dual to the notion of a linear map between vector spaces, or, more generally, of a homomorphism between R-modules.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 See also
Divided power structure
References