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
formula_1
such that
where Δ is the comultiplication for "C", and ε is the counit.
Note that in the second rule we have identified formula_4 with formula_5.
# Let the comultiplication on formula_6 be given by formula_12.
# Let the counit on formula_6 be given by formula_14.
# Let the map formula_15 on "V" be given by formula_16, where formula_17 is the "i"-th homogeneous piece of formula_18.
Examples.
In algebraic topology.
One important result in algebraic topology is the fact that homology formula_19 over the dual Steenrod algebra formula_20 forms a comodule. This comes from the fact the Steenrod algebra formula_21 has a canonical action on the cohomologyformula_22When we dualize to the dual Steenrod algebra, this gives a comodule structureformula_23This result extends to other cohomology theories as well, such as complex cobordism and is instrumental in computing its cohomology ring formula_24. 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 formula_20 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
formula_26
be right "C"-comodules. Then an "R"-linear map formula_27 is called a (right) comodule morphism, or (right) C-colinear, if
formula_28
This notion is dual to the notion of a linear map between vector spaces, or, more generally, of a homomorphism between "R"-modules.
References.
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