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In
mathematics, a '''comodule''' 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
:<math>\rho: M \to M \otimes C</math>
such that
# <math>\rho \circ (id \otimes \Delta) = \rho \circ (\rho \otimes id)</math>
# <math>\rho \circ (id \otimes \epsilon) = id \otimes 1</math>,
where Δ is the comultiplication for ''C'', and ε is the counit.
== Examples ==
* A coalgebra is a comodule over itself.
* If ''M'' is a module over a ''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 <math>C_I</math> be the vector space with basis <math>e_i</math> for <math>i \in I</math>. We turn <math>C_I</math> into a coalgebra and ''V'' into a <math>C_I</math>-comodule, as follows:
:# Let the comultiplication on <math>C_I</math> be given by <math>\Delta(e_i) = e_i \otimes e_i</math>.
:# Let the counit on <math>C_I</math> be given by <math>\epsilon(e_i) = 1\ </math>.
:# Let the map <math>\rho</math> on ''V'' be given by <math>\rho(v) = \sum v_i \otimes e_i</math>, where <math>v_i</math> is the ''i''-th homogeneous piece of <math>v</math>.
