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 with .
One important result in algebraic topology is the fact that homology over the dual Steenrod algebra forms a comodule.[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.
Let R be a ring, M, N, and C be R-modules, and