nLab Milnor's theorem

Idea

Milnor’s theorem describes the dual mod-$p$ Steenrod algebra in terms of generators and relations.

Statement

In the following, we use for $p = 2$ the notation

$$P^n \coloneqq Sq^{2n}$$

$$\beta \coloneqq Sq^1 \,.$$

This serves to unify the expressions for $p = 2$ and for $p \gt 2$ in the following. Notice that for all $p$

• $P^n$ has even degree $deg(P^n) = 2n(p-1)$;

• $\beta$ has odd degree $deg(\beta) = 1$.

Theorem

(Milnor’s theorem)

The dual mod $p$-Steenrod algebra $\mathcal{A}^\ast_{\mathbb{F}_p}$ (def. ) is, as an associative algebra, the free graded commutative algebra

$$\mathcal{A}^\ast_{\mathbb{F}_p} \simeq Sym_{\mathbb{F}_p}(\xi_1, \xi_2, \cdots, \;\tau_0, \tau_1, \cdots)$$

on generators:

• $\xi_n$ being the linear dual to $P^{p^{n-1}} P^{p^{n-2}} \cdots P^p P^1$;

• $\tau_n$ being linear dual to $P^{p^{n-1}} P^{p^{n-2}} \cdots P^p P^1\beta$.

Moreover, the coproduct on $\mathcal{A}^\ast_{\mathbb{F}_p}$ is given by

$$\Psi(\xi_n) = \underoverset{k = 0}{n}{\sum} \xi_{n-k}^{p^k} \otimes \xi_k$$

and

$$\Psi(\tau_n) = \tau_n \otimes 1 + \underoverset{k=0}{n}{\sum} \xi_{n-k}^{p^k} \xi_{n-k}^{p^k}\otimes \tau_k \,,$$

where we set $\xi_0 \coloneqq 1$.

This is due to (Milnor 58). See for instance (Kochman 96, theorem 2.5.1)

References

• John Milnor, The Steenrod algebra and its dual, Ann. of Math. 67 (1958), 150–171.

• Stanley Kochman, section 2.5 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996