Editing Comodule (section)

=== 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.