# Galois extensions of ring spectra and subextensions

In Galois extensions of structured ring spectra, Rognes introduces the notion of a faithful $$G$$-Galois extension of ring spectra. Let me recall what this means:

We have a commutative ring spectrum $$R$$ with an action of $$G$$, and a $$G$$-equivariant morphism of commutative ring spectra $$S\to R$$ where $$S$$ has the trivial $$G$$-action. This is a $$G$$-Galois extension if:

i) The canonical map $$S\to R^{hG}$$ is an equivalence

ii) The canonical map $$R\otimes_S R\to F(G_+,R)$$ is an equivalence

iii) $$R$$ is a faithful $$S$$-module

This makes sense for stably dualizable topological groups and in fact for my question, I'm interested in topological groups that aren't finite discrete. I am, however, happy to restrict to groups that have the homotopy type of a finite CW-complex.

He proves the following result, of which I will sketch a proof at the end of the post for convenience, and to perhaps understand my question better (proposition 7.2.2.in [Rog08]):

If $$K$$ is an allowable subgroup of $$G$$, then $$R^{hK}\to R$$ is a faithful $$K$$-Galois extension.

Here, an allowable subgroup is a subgroup $$K$$ which is stably dualizable, such that $$G_{hK}\simeq^{stably} G/K$$, and such that there exists a continuous section of $$G\to G/K$$.

My question is the following :

Is this last condition about the existence of a continuous section necessary to get this result ?

In my sketch of the proof below I will point out precisely where we use it - I don't follow the exact same proof as Rognes, but we use the hypothesis for essentially the same thing.

An obvious remark is that when $$G$$ is finite discrete, this continuous section always exists, so there is no problem there (and in fact every subgroup is allowable). When $$G$$ splits as a product (topologically, but in particular if it splits as a product of topological groups) $$K\times G/K$$ then the section obviously exists, which is what allows for instance Mathew to induct on $$n$$ when looking at $$\mathbb T^n$$-Galois extensions in Torus actions on stable module categories, Picard groups and localizing subcategories.

Of course, the best answer would be a proof that doesn't use this condition, or an example where the result fails. Most of the examples I know are finite groups, so I'm not really sure what to look for.

An example where the condition fails is $$S^1$$ with any finite subgroup, but one would have to find $$S^1$$-Galois extensions.

Sketch of proof: I will use Mathew's language of descendability, which he uses in [Mat15] to give an alternative characterization of $$G$$-Galois extensions, cf. proposition 3.7. in that paper.

$$R$$ is dualizable over $$S$$, by 6.2.1 in [Rog08] (of course this doesn't refer to subgroups, so I won't expand on the proof of that fact), and faithful by definition. Therefore, by Thm 3.38 in The Galois group of a stable homotopy theory, $$S\to R$$ admits descent.

It follows that $$R^{hH}\to R\otimes_S R^{hH}$$ also admits descent. Let's call that new algebra $$\tilde R$$. We can now check $$H$$-Galois-ness of $$R^{hH}\to R$$ by tensoring with $$\tilde R$$ : $$\tilde R\to \tilde R\otimes_{R^{hH}}R\simeq R\otimes_S R\simeq F(G_+,R)$$. Now $$\tilde R = R\otimes_S R^{hH}\simeq F(G/H_+,R)$$ because $$R$$ is dualizable and $$G_{hH}\simeq G/H$$, so this map is $$F(G/H_+,R)\to F(G_+,R)$$.

If we have a section, we can identify this map with the canonical map $$F(G/H_+,R)\to F(H_+,F(G/H_+,R))$$ and so by proposition 3.7. in [Mat15], we get that $$R^{hH}\to R$$ is indeed $$H$$-Galois.

Of course this is just a sketch, one needs to identify the maps a bit more precisely.

I guess this reduces the question to looking at $$G$$-Galois extensions of the form $$R\to F(G_+,R)$$ i.e. the "trivial ones", in other words,

what do we need for $$F(G/H_+,R)\to F(G_+,R)$$ to be an $$H$$-Galois extension ? Are there cases with no section where this fails ?

But for instance whenever $$G$$ is connected, $$G\to G/H$$ is a principal $$H$$-bundle and $$\pi_1(G/H)$$ acts nilpotently on $$H_*(G;\mathbb F_p)$$, proposition 5.6.3. of [Rog08] will yield a positive answer for $$R= \mathbb F_p$$, so not "any" counterexample will work. For instance, these conditions are satisfied by $$S^1\to S^1$$ which is indeed a $$\mathbb Z/p$$-principal bundle.

References:

[Rog08] : Galois extensions of structured ring spectra, arxiv:math/0502183

[Mat15]: Torus actions on stable module categories, Picard groups and localizing subcategories, arXiv:1512.01716

[Mat16]: The Galois group of a stable homotopy theory, arXiv:1404.2156

• Note that the condition on $H$-fixed points for $F(G_+,R)$ is automatic if we assume $G_{hH}\simeq G/H$, which I want to assume, so the main part is about the non-ramification : $F(G_+,R)\otimes_{F(G/H_+,R)} F(G_+,R) \simeq F(H_+,F(G_+,R))$. Because $G\times_{G/H} G \simeq G\times H$, this is related in some sense to the Eilenberg-Moore spectral sequence, but I don't know enough about the latter Oct 27 '20 at 9:37

It turns out that I was misapplying Rognes' 5.6.3. : when $$p\neq n$$, the action of $$\pi_1(S^1)$$ on $$H_*(C_n; \mathbb F_p)$$ need not be nilpotent.
In particular, the following is a counterexample : take any odd prime $$p$$, then the projection $$S^1\to \mathbb RP^1$$ is a mod $$p$$-equivalence, so that $$\mathbb F_p^{S^1}\otimes_{\mathbb F_p^{\mathbb RP^1}} \mathbb F_p^{S^1} \simeq \mathbb F_p^{S^1}$$ which is not equivalent to $$\mathbb F_p^{S^1\times C_2}$$, even abstractly.
This gives a counterexample where all of $$G,H,G/H$$ are stably dualizable, so it's better than the other counterexamples I stumbled upon, which had $$G= EC_n, H =C_n$$ (and so $$G/H = BC_n$$ which is not stably dualizable)