In [[algebraic topology]], through an algebraic operation (dualization), there is an associated commutative algebra<ref>{{Citation|last=Milnor|first=John|title=The Steenrod algebra and its dual|date=2012-03-29|url=https://www.worldscientific.com/doi/abs/10.1142/9789814401319_0006|work=Topological Library|volume=50|pages=357–382|series=Series on Knots and Everything|publisher=WORLD SCIENTIFIC|doi=10.1142/9789814401319_0006|isbn=978-981-4401-30-2|access-date=2021-01-05|url-access=subscription}}</ref> from the noncommutative [[Steenrod algebra]]s called the '''dual Steenrod algebra'''. This dual algebra has a number of surprising benefits, such as being commutative and provided technical tools for computing the Adams spectral sequence in many cases (such as <math>\pi_*(MU)</math><ref name="Ravenel">{{Cite book|last=Ravenel|first=Douglas C.|url=https://web.math.rochester.edu/people/faculty/doug/mu.html|title=Complex cobordism and stable homotopy groups of spheres|date=1986|publisher=Academic Press|isbn=978-0-08-087440-1|location=Orlando|pages=|oclc=316566772|author-link=Douglas Ravenel}}</ref>{{rp|pages=61–62}}) with much ease.
== Definition == Recall<ref name="Ravenel" />{{rp|page=59}} that the Steenrod algebra <math>\mathcal{A}_p^*</math> (also denoted <math>\mathcal{A}^*</math>) is a graded noncommutative [[Hopf algebra]] which is cocommutative, meaning its comultiplication is cocommutative. This implies if we take the dual Hopf algebra, denoted <math>\mathcal{A}_{p,*}</math>, or just <math>\mathcal{A}_*</math>, then this gives a graded-commutative algebra which has a noncommutative comultiplication. We can summarize this duality through dualizing a commutative diagram of the Steenrod's Hopf algebra structure:<blockquote><math>\mathcal{A}_p^* \xrightarrow{\psi^*} \mathcal{A}_p^* \otimes \mathcal{A}_p^* \xrightarrow{\phi^*} \mathcal{A}_p^*</math></blockquote>If we dualize we get maps<blockquote><math>\mathcal{A}_{p,*} \xleftarrow{\psi_*} \mathcal{A}_{p,*} \otimes \mathcal{A}_{p,*}\xleftarrow{\phi_*} \mathcal{A}_{p,*}</math></blockquote>giving the main structure maps for the dual Hopf algebra. It turns out there's a nice structure theorem for the dual Hopf algebra, separated by whether the prime is <math>2</math> or odd.
=== Case of p=2 === In this case, the dual Steenrod algebra is a graded commutative polynomial algebra <math>\mathcal{A}_* = \mathbb{Z}/2[\xi_1,\xi_2,\ldots]</math> where the degree <math>\deg(\xi_n) = 2^n-1</math>. Then, the coproduct map is given by<blockquote><math>\Delta:\mathcal{A}_* \to \mathcal{A}_*\otimes\mathcal{A}_*</math></blockquote>sending<blockquote><math>\Delta\xi_n = \sum_{0 \leq i \leq n} \xi_{n-i}^{2^i}\otimes \xi_i</math></blockquote>where <math>\xi_0 = 1</math>.
=== General case of p > 2 === For all other prime numbers, the dual Steenrod algebra is slightly more complex and involves a graded-commutative [[exterior algebra]] in addition to a graded-commutative polynomial algebra. If we let <math>\Lambda(x,y)</math> denote an exterior algebra over <math>\mathbb{Z}/p</math> with generators <math>x</math> and <math>y</math>, then the dual Steenrod algebra has the presentation<blockquote><math>\mathcal{A}_* = \mathbb{Z}/p[\xi_1,\xi_2,\ldots]\otimes \Lambda(\tau_0,\tau_1,\ldots)</math></blockquote>where<blockquote><math>\begin{align} \deg(\xi_n) &= 2(p^n - 1) \\ \deg(\tau_n) &= 2p^n - 1 \end{align}</math></blockquote>In addition, it has the comultiplication <math>\Delta:\mathcal{A}_* \to \mathcal{A}_*\otimes\mathcal{A}_*</math> defined by<blockquote><math>\begin{align} \Delta(\xi_n) &= \sum_{0 \leq i \leq n} \xi_{n-i}^{p^i}\otimes \xi_i \\ \Delta(\tau_n) &= \tau_n\otimes 1 + \sum_{0 \leq i \leq n}\xi_{n-i}^{p^i}\otimes \tau_i \end{align}</math></blockquote>where again <math>\xi_0 = 1</math>.
=== Rest of Hopf algebra structure in both cases === The rest of the Hopf algebra structures can be described exactly the same in both cases. There is both a unit map <math>\eta</math> and counit map <math>\varepsilon</math><blockquote><math>\begin{align} \eta&: \mathbb{Z}/p \to \mathcal{A}_* \\ \varepsilon&: \mathcal{A}_* \to \mathbb{Z}/p \end{align}</math></blockquote>which are both isomorphisms in degree <math>0</math>: these come from the original Steenrod algebra. In addition, there is also a conjugation map <math>c: \mathcal{A}_* \to \mathcal{A}_*</math> defined recursively by the equations<blockquote><math>\begin{align} c(\xi_0) &= 1 \\ \sum_{0 \leq i \leq n} \xi_{n-i}^{p^i}c(\xi_i)& = 0
\end{align}</math></blockquote>In addition, we will denote <math>\overline{\mathcal{A}_*}</math> as the kernel of the counit map <math>\varepsilon</math> which is isomorphic to <math>\mathcal{A}_*</math> in degrees <math>> 1</math>.
== See also ==
* [[Adams-Novikov spectral sequence]]
== References == {{Reflist}}
[[Category:Algebraic topology]] [[Category:Hopf algebras]] [[Category:Homological algebra]]