Steenrod squares as power operations vs. as cohomology operations

Question

There are already several excellent questions and answers on MO regarding Steenrod squares, understanding them in various ways and relating them to power operations and I think I get this. Still, I am confused about the sense in which the power operation perspective coincides with the stable cohomology operation perspective. Namely, there seems to be two very different constructions that for the Eilenberg-MacLane spectrum $H\mathbb{F}_2$ yield (almost) the same answer.

1) Given a spectrum $E$, we can consider the graded (non-commutative) ring $\mathcal{A}_E =[E,E]_*$ with multiplication induced by composition. For every spectrum $X$, the ring $\mathcal{A}_E$ acts on the graded abelian group $E^*(X)=[X,E]_*$. For $E=H\mathbb{F}_2$, we get the ordinary Steenrod algebra which is generated by the $Sq^i$ operations with $i\ge 0$. This can be calculated explicitly as

$$\mathcal{A}_2^*=\lim_nH^{*+n}(K(\mathbb{F}_2,n);\mathbb{F}_2)$$

and the Borel-Serre computation of the cohomology of Eilenberg-MacLane spaces (I mention this to emphasis that, as far as I understand, the presentation with $Sq^i$-s is obtained by computation)

2) Given an $E_\infty$-ring spectrum $R$ we can consider the homotopy groups of the endomorphism spectrum of the forgetful functor from $E_\infty$-algebras over $R$ to spectra. Less formally, the thing that naturally acts on the underlying spectrum of every $R$-algebra. This includes the power operations for example. For $R = H\mathbb{F}_2$, we get the big Steenrod algebra (more precisely, a certain completion of it. See Lurie's notes), which is generated (topologically) by all Steenrod squares $Sq^i$ with $i\in\mathbb{Z}$. The ordinary Steenrod algebra can be obtained from it by imposing the single relation $Sq^0 = Id$.

In general, the two constructions take different things as input and produce an algebra that in general acts on different things. Yet for $H\mathbb{F}_2$, we get closely related algebras.

Question: Is there a conceptual explanation of the relation between (1) and (2)? Is there anything more general we can say about this relation for (ring) spectra other then $H\mathbb{F}_p$?

Following Lurie's notes cited above, it seems that the main step in the comparison is the fact that the $\mathbb{F}_2$-cochains of $K(\mathbb{F}_2,n)$ as an algebra over $H\mathbb{F}_2$ is almost the free algebra on one generator of degree $n$, where the only relation we need to impose is precisely $Sq^0=Id$. But again, this is verified in a very computational way and from this perspective seems to me like a miracle. Another point is that power operations for other cohomology theories (like K-theory) are unstable in general, so this also seems rather special.

Answer 1

Here are some things that one can say for more general ring spectra. I choose to work with even periodic spectra, where $\pi_1(E)=0$ and $\pi_2(E)$ contains an invertible element, so that one can generally concentrate on $E_0(X)$ and $E^0(X)$. Key examples are $E=KU$ and $E=HP=\bigvee_{i\in\mathbb{Z}}\Sigma^{2i}H$ and $$ E=MP=\bigvee_{i\in\mathbb{Z}}\Sigma^{2i}MU=\text{ the Thom spectrum for } \mathbb{Z}\times BU. $$ (Here $H$ will be the Eilenberg-MacLane spectrum for mod $p$ (co)homology, for some fixed prime $p$.)

Stable operations $F^0(X)\to E^0(X)$ are controlled by $E^0F$, which in good cases is the $E_0$-linear dual of $E_0F$. This is most naturally described in terms of the scheme $\text{spec}(E_0F)$. Associated to $E$ we have a base scheme $S_E=\text{spec}(E_0)$ and a formal scheme $G_E=\text{spf}(E^0BS^1)$ which should be thought of as a bundle of groups over $S_E$. We have similar data for $F$, using which we construct the scheme $\text{Iso}(G_E,G_F)$ consisting of triples $(a,b,u)$ with $a\in S_E$ and $b\in S_F$ and $u$ an isomorphism between the corresponding fibres of $G_E$ and $G_F$. There is a natural map $\text{spec}(E_0F)\to\text{Iso}(G_E,G_F)$ which is an isomorphism when one of $E$ and $F$ is Landweber exact. In the case $E=F=HP$ we find that $\text{Iso}(G_E,G_F)$ is the polynomial part of the dual Steenrod algebra, but this picture loses the exterior part.

Unstable operation $F^0(X)\to E^0(X)$ (for spaces $X$) are controlled by $E^0(\Omega^\infty F)$. This has a natural Hopf algebra structure, and the additive unstable operations are the primitive elements. In good cases $E^0(\Omega^\infty F)$ is dual to $E_0(\Omega^\infty F)$, and the primitive elements are dual to the indecomposables in $E_0(\Omega^\infty F)$. Here $\Omega^\infty F$ is a ring up to homotopy, and the two ring operations give rise to two different products on $E_0(\Omega^\infty F)$. We take indecomposables with respect to the addition product, and the multiplication product passes to the quotient to give a ring structure on $\text{Ind}(E_0(\Omega^\infty F))$. This is again best described in terms of schemes. We can define $\text{Hom}(G_E,G_F)$ in the same style as $\text{Iso}(G_E,G_F)$ and we get a map $$\text{spec}(\text{Ind}(E_0(\Omega^\infty F))) \to \text{Hom}(G_E,G_F) $$ which is an isomorphism when $F$ is Landweber exact. The stabilization map $\Sigma^\infty\Omega^\infty F\to F$ gives rise to a map $\text{spec}(E_0F)\to\text{spec}(\text{Ind}(E_0\Omega^\infty F))$ which is compatible with the inclusion $\text{Iso}(G_E,G_F)\to\text{Hom}(G_E,G_F)$. In the case $E=F=HP$ this picture again sees the polynomial part of the cohomology of Eilenberg-MacLane spaces, but not the exterior part.

Now let $\text{Sub}(G_E)$ denote the scheme of pairs $(a,A)$ where $a\in S_E$ and $A$ is a finite flat subgroup scheme of $(G_E)_a$. Put $DS^0=\bigvee_nB\Sigma_n$. The group $E^0DS^0$ can be made into a ring using the transfer maps associated to the inclusions $\Sigma_i\times\Sigma_j\to\Sigma_{i+j}$. There is a natural map $\text{spf}(\text{Ind}(E^0DS^0))\to\text{Sub}(G)$, which is an isomorphism in good cases. In the case $E=HP/p$ this encodes the standard relationship between $H^*(B\Sigma_{p^d})$ and the Dickson invariant subalgebra of $H^*(BC_p^d)$, again ignoring the exterior parts. (The indecomposables are zero in $H^*(B\Sigma_m)$ if $m$ is not a power of $p$.)

If $E$ is an $H_\infty$ ring spectrum, we have the following picture. For each point $(a,A)\in\text{Sub}(G_E)$ there is a naturally associated point $b\in S_E$ and an isogeny $q_A\colon(G_E)_a\to(G_E)_b$ with kernel $A$. This creates a relationship between power operations and a particular subclass of isogenies. Stable operations give an action of all isomorphisms, and one can combine this with the action of special isogenies to get an action of all isogenies, but this is slightly artificial.

Going back to the case of $HP$, the stories with stable operations, unstable operations and power operations for $\Sigma_{p^d}$ give coaction maps \begin{align*} HP^0(X) &\to HP^0(X)[\zeta_0,\zeta_1,\zeta_2,\dotsc][\zeta_0^{-1}] \\ HP^0(X) &\to HP^0(X)[\zeta_0,\zeta_1,\zeta_2,\dotsc] \\ HP^0(X) &\to HP^0(X)[\zeta_0,\zeta_1,\zeta_2,\dotsc,\zeta_d]/(1-\zeta_d) \end{align*} These are all compatible in an obvious way. Indeed, because $G_E=\text{spf}(E^0BS^1)$ the claim is true by construction when $X=BS^1$. The class $\mathcal{X}$ of spaces where it is true is closed under products and coproducts. Moreover, if $f:X\to Y$ induces a surjection in cohomology, and $Y\in\mathcal{X}$, then $X\in\mathcal{X}$. In the case $E=HP$ we can now use the relationship between $BS^1$, $BC_p=K(\mathbb{Z}/p,1)$ and $K(\mathbb{Z}/p,n)$ to deduce that all spaces lie in $\mathcal{X}$. If $E$ is Landweber exact then one can make a similar argument using the spaces in the $MU$ spectrum instead of $K(\mathbb{Z},n)$.