Комодульдер және дуалды алгебралар: алгебралық топологиядағы қолданыстары

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.

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 .

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.

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.