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.
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
One important result in algebraic topology is the fact that homology
over the dual Steenrod algebra
[1] This comes from the fact the Steenrod algebra
has a canonical action on the cohomology
When we dualize to the dual Steenrod algebra, this gives a comodule structure
This result extends to other cohomology theories as well, such as complex cobordism and is instrumental in computing its cohomology ring
[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
is a commutative ring, and the setting of commutative algebra provides more tools for studying its structure.
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.
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.