Dual Steenrod algebra

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

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

${\displaystyle {\mathcal {A}}_{p}^{*}\xrightarrow {\psi ^{*}} {\mathcal {A}}_{p}^{*}\otimes {\mathcal {A}}_{p}^{*}\xrightarrow {\phi ^{*}} {\mathcal {A}}_{p}^{*}}$

If we dualize we get maps

${\displaystyle {\mathcal {A}}_{p,*}\xleftarrow {\psi _{*}} {\mathcal {A}}_{p,*}\otimes {\mathcal {A}}_{p,*}\xleftarrow {\phi _{*}} {\mathcal {A}}_{p,*}}$

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 ${\displaystyle {\mathcal {A}}_{*}=\mathbb {Z} /2[\xi _{1},\xi _{2},\ldots ]}$ where the degree ${\displaystyle \deg(\xi _{n})=2^{n}-1}$. Then, the coproduct map is given by

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

sending

${\displaystyle \Delta \xi _{n}=\sum _{0\leq i\leq n}\xi _{n-i}^{2^{i}}\otimes \xi _{i}}$

where ${\displaystyle \xi _{0}=1}$.

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

${\displaystyle {\mathcal {A}}_{*}=\mathbb {Z} /p[\xi _{1},\xi _{2},\ldots ]\otimes \Lambda (\tau _{0},\tau _{1},\ldots )}$

where

${\displaystyle {\begin{aligned}\deg(\xi _{n})&=2(p^{n}-1)\\\deg(\tau _{n})&=2p^{n}-1\end{aligned}}}$

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

${\displaystyle {\begin{aligned}\Delta (\xi _{n})&=\sum _{0\leq i\leq n}\xi _{n-i}^{p^{i}}\otimes \xi _{i}\\\Delta (\tau _{n})&=\tau _{n}\otimes 1+\sum _{0\leq i\leq n}\xi _{n-i}^{p^{i}}\otimes \tau _{i}\end{aligned}}}$

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 ${\displaystyle \varepsilon }$

${\displaystyle {\begin{aligned}\eta &:\mathbb {Z} /p\to {\mathcal {A}}_{*}\\\varepsilon &:{\mathcal {A}}_{*}\to \mathbb {Z} /p\end{aligned}}}$

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

${\displaystyle {\begin{aligned}c(\xi _{0})&=1\\\sum _{0\leq i\leq n}\xi _{n-i}^{p^{i}}c(\xi _{i})&=0\end{aligned}}}$

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

• Adams-Novikov spectral sequence