Inalgebraic topology, through an algebraic operation (dualization), there is an associated commutative algebra^[1] from the noncommutative Steenrod algebras called thedual 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${\displaystyle {\pi }_{\ast }(MU)}$^{[2]: 61–62}) with much ease.

Recall^[2]: 59 that the Steenrod algebra${\displaystyle {\mathcal{A}}_p^{\ast }}$ (also denoted${\displaystyle {\mathcal{A}}^{\ast }}$) is a graded noncommutativeHopf algebra which is cocommutative, meaning its comultiplication is cocommutative. This implies if we take the dual Hopf algebra, denoted${\displaystyle {\mathcal{A}}_{p,\ast }}$, or just${\displaystyle {\mathcal{A}}_{\ast }}$, 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:

${\displaystyle {\mathcal{A}}_p^{\ast }\stackrel{{\psi }^{\ast }}{\to }{\mathcal{A}}_p^{\ast }\otimes {\mathcal{A}}_p^{\ast }\stackrel{{\varphi }^{\ast }}{\to }{\mathcal{A}}_p^{\ast }}$

If we dualize we get maps

${\displaystyle {\mathcal{A}}_{p,\ast }\stackrel{{\psi }_{\ast }}{\leftarrow }{\mathcal{A}}_{p,\ast }\otimes {\mathcal{A}}_{p,\ast }\stackrel{{\varphi }_{\ast }}{\leftarrow }{\mathcal{A}}_{p,\ast }}$

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${\displaystyle 2}$ or odd.

In this case, the dual Steenrod algebra is a graded commutative polynomial algebra${\displaystyle {\mathcal{A}}_{\ast }=\mathbb{Z}/2[{\xi }_1,{\xi }_2,\dots ]}$ where the degree${\displaystyle \mathrm{deg}({\xi }_n)=2^n-1}$. Then, the coproduct map is given by

${\displaystyle \mathrm{\Delta }:{\mathcal{A}}_{\ast }\to {\mathcal{A}}_{\ast }\otimes {\mathcal{A}}_{\ast }}$

sending

${\displaystyle \mathrm{\Delta }{\xi }_n=\sum _{0\le i\le n}{\xi }_{n-i}^{2^i}\otimes {\xi }_i}$

where${\displaystyle {\xi }_0=1}$.

For all other prime numbers, the dual Steenrod algebra is slightly more complex and involves a graded-commutativeexterior algebra in addition to a graded-commutative polynomial algebra. If we let${\displaystyle \mathrm{\Lambda }(x,y)}$ denote an exterior algebra over${\displaystyle \mathbb{Z}/p}$ with generators${\displaystyle x}$ and${\displaystyle y}$, then the dual Steenrod algebra has the presentation

${\displaystyle {\mathcal{A}}_{\ast }=\mathbb{Z}/p[{\xi }_1,{\xi }_2,\dots ]\otimes \mathrm{\Lambda }({\tau }_0,{\tau }_1,\dots )}$

where

In addition, it has the comultiplication${\displaystyle \mathrm{\Delta }:{\mathcal{A}}_{\ast }\to {\mathcal{A}}_{\ast }\otimes {\mathcal{A}}_{\ast }}$ defined by

where again${\displaystyle {\xi }_0=1}$.

The rest of the Hopf algebra structures can be described exactly the same in both cases. There is both a unit map${\displaystyle \eta }$ and counit map${\displaystyle \epsilon }$

which are both isomorphisms in degree${\displaystyle 0}$: these come from the original Steenrod algebra. In addition, there is also a conjugation map${\displaystyle c:{\mathcal{A}}_{\ast }\to {\mathcal{A}}_{\ast }}$ defined recursively by the equations

In addition, we will denote${\displaystyle \overline{{\mathcal{A}}_{\ast }}}$ as the kernel of the counit map${\displaystyle \epsilon }$ which is isomorphic to${\displaystyle {\mathcal{A}}_{\ast }}$ in degrees${\displaystyle >1}$.

• Adams-Novikov spectral sequence

• Algebraic topology
• Hopf algebras
• Homological algebra