Editing Comodule (section)

== Examples ==
* 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 function]]s 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>\varepsilon(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>.

=== In algebraic topology ===
One important result in [[algebraic topology]] is the fact that homology <math>H_*(X)</math> over the dual [[Steenrod algebra]] <math>\mathcal{A}^*</math> forms a comodule.<ref>{{Cite journal|last=Liulevicius|first=Arunas|date=1968|title=Homology Comodules|url=https://www.ams.org/journals/tran/1968-134-02/S0002-9947-1968-0251720-X/S0002-9947-1968-0251720-X.pdf|journal=Transactions of the American Mathematical Society|volume=134|issue=2|pages=375–382|doi=10.2307/1994750|jstor=1994750 |issn=0002-9947|doi-access=free}}</ref> This comes from the fact the Steenrod algebra <math>\mathcal{A}</math> has a canonical action on the cohomology<blockquote><math>\mu: \mathcal{A}\otimes H^*(X) \to H^*(X)</math></blockquote>When we dualize to the [[dual Steenrod algebra]], this gives a comodule structure<blockquote><math>\mu^*:H_*(X) \to \mathcal{A}^*\otimes H_*(X)</math></blockquote>This result extends to other cohomology theories as well, such as [[complex cobordism]] and is instrumental in computing its cohomology ring <math>\Omega_U^*(\{pt\})</math>.<ref>{{Cite web|last=Mueller|first=Michael|title=Calculating Cobordism Rings|url=https://www.brown.edu/academics/math/sites/math/files/Mueller,%20Michael.pdf|url-status=live|archive-url=https://web.archive.org/web/20210102194203/https://www.brown.edu/academics/math/sites/math/files/Mueller%2C%20Michael.pdf|archive-date=2 Jan 2021}}</ref> 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 <math>\mathcal{A}^*</math> is a [[commutative ring]], and the setting of commutative algebra provides more tools for studying its structure.