2.1 The Morava stabilizer group

We start this section by reviewing more thoroughly the Morava stabilizer group $\mathbb{S}_{n}$ , as well as the extended Morava stabilizer group $\mathbb{G}_{n}$ . We use [Reference FröhlichFrö68, ch. III] as a general reference. Recall from the introduction that $\mathbb{S}_{n}$ is the automorphism group of the height $n$ Honda formal group law $\unicode[STIX]{x1D6E4}_{n}$ over $\mathbb{F}_{p^{n}}$ ; this is the formal group law $F_{\unicode[STIX]{x1D6E4}_{n}}(X,Y)\in \mathbb{F}_{p^{n}}[\![X,Y]\!]$ with $p$ -series $[p]_{\unicode[STIX]{x1D6E4}_{n}}(X)=X^{p^{n}}$ .

We have a homomorphism

(2.1) $$\begin{eqnarray}\mathbb{S}_{n}\rightarrow \mathbb{F}_{p^{n}}^{\times },\quad f(X)\mapsto f^{\prime }(0).\end{eqnarray}$$

More invariantly, this homomorphism sends an automorphism $f$ of the formal group $F$ to its action on the tangent space of $F$ (dual to the Lie algebra). This homomorphism is surjective, and its kernel (which consists of automorphisms that are congruent to the identity modulo  $X^{2}$ ) is a pro- $p$ -group.

The group $\mathbb{S}_{n}$ can be described as the units in the maximal order ${\mathcal{O}}_{n}$ of the central division algebra $\mathbb{D}_{n}$ of Hasse invariant $1/n$ over $\mathbb{Q}_{p}$ . More explicitly, we have that ${\mathcal{O}}_{n}$ is the non-commutative ring

$$\begin{eqnarray}{\mathcal{O}}_{n}=\mathbb{W}(\mathbb{F}_{p^{n}})\langle S\rangle /(S^{n}=p,Sw=w^{\unicode[STIX]{x1D719}}S),\end{eqnarray}$$

where $w\in \mathbb{W}(\mathbb{F}_{p^{n}})$ and $\unicode[STIX]{x1D719}$ is a lift of the Frobenius to $\mathbb{W}(\mathbb{F}_{p^{n}})$ . Then ${\mathcal{O}}_{n}$ is the endomorphism ring of the formal group $\unicode[STIX]{x1D6E4}_{n}$ , and $\mathbb{S}_{n}={\mathcal{O}}_{n}^{\times }$ .

Finally, because $\unicode[STIX]{x1D6E4}_{n}$ is already defined over $\mathbb{F}_{p}$ , the Galois group $\operatorname{Gal}(\mathbb{F}_{p^{n}}/\mathbb{F}_{p})$ acts on $\mathbb{S}_{n}$ , and we define the extended Morava stabilizer group (which we will often just call the Morava stabilizer group) as the semidirect product $\mathbb{S}_{n}\rtimes \operatorname{Gal}(\mathbb{F}_{p^{n}}/\mathbb{F}_{p})$ , i.e. there is a split extension

(2.2) $$\begin{eqnarray}1\rightarrow \mathbb{S}_{n}\rightarrow \mathbb{G}_{n}\rightarrow \operatorname{Gal}(\mathbb{F}_{p^{n}}/\mathbb{F}_{p})\rightarrow 1.\end{eqnarray}$$

2.2 The finite subgroups of $\mathbb{G}_{n}$

Before proceeding with the calculations, let us recall the classification of the maximal finite subgroups of $\mathbb{G}_{n}$ , as well as their action on the coefficients of  $E_{n}$ . Bujard and Hewett [Reference HewettHew95, Reference BujardBuj12] have determined the maximal finite subgroups of $\mathbb{S}_{n}\subset \mathbb{G}_{n}$ . Specifically, let $p>2$ and $n=(p-1)p^{k-1}m$ for $m$ prime to $p$ , and denote by $n_{\unicode[STIX]{x1D6FC}}$ the quotient $n/\unicode[STIX]{x1D719}(p^{\unicode[STIX]{x1D6FC}})$ , where $\unicode[STIX]{x1D719}$ is Euler’s totient function. There are exactly $k+1$ conjugacy classes of maximal finite subgroups of $\mathbb{S}_{n}$ , represented by

$$\begin{eqnarray}G_{0}=C_{p^{n}-1}\quad \text{and}\quad G_{\unicode[STIX]{x1D6FC}}=C_{p^{\unicode[STIX]{x1D6FC}}}\rtimes C_{(p^{n_{\unicode[STIX]{x1D6FC}}}-1)(p-1)}\quad \text{for}~1\leqslant \unicode[STIX]{x1D6FC}\leqslant k.\end{eqnarray}$$

When $p-1$ does not divide $n$ , the only class of maximal finite subgroups is that of  $G_{0}$ , so, in particular, there are no finite subgroups of order divisible by  $p$ .

Consider now the special case when $n=p-1$ . For convenience, we drop the subscripts after $\mathbb{G},\mathbb{S}$ . In this situation, there are two conjugacy classes of maximal finite subgroups of  $\mathbb{S}$ .

1. (i) One of order not divisible by $p$ , namely the cyclic group $C_{p^{n}-1}$ .

2. (ii) One of order divisible by $p$ , namely the semi-direct product $F=C_{p}\rtimes C_{(p-1)^{2}}$ . In the latter case, the action of $C_{(p-1)^{2}}$ on $C_{p}$ is given by a projection

   $$\begin{eqnarray}C_{(p-1)^{2}}\rightarrow C_{p-1}\simeq \text{Aut}(C_{p}).\end{eqnarray}$$

Finally, we recall from (2.2) that the extended Morava stabilizer group $\mathbb{G}$ sits in a split extension

$$\begin{eqnarray}1\rightarrow \mathbb{S}\rightarrow \mathbb{G}\rightarrow \operatorname{Gal}(\mathbb{F}_{p^{n}}/\mathbb{F}_{p})\rightarrow 1,\end{eqnarray}$$

and, therefore, any finite subgroup $G$ of $\mathbb{G}$ sits in a (not necessarily split) extension

(2.3) $$\begin{eqnarray}1\rightarrow G_{\mathbb{S}}\rightarrow G\rightarrow G_{\operatorname{Gal}}\rightarrow 1,\end{eqnarray}$$

where $G_{\mathbb{S}}=G\cap \mathbb{S}$ , and $G_{\operatorname{Gal}}$ is a subgroup of $\operatorname{Gal}(\mathbb{F}_{p^{n}}/\mathbb{F}_{p})\cong C_{n}$ .

In the case when $G_{\mathbb{S}}$ is a maximal abelian subgroup of $\mathbb{S}$ , Bujard [Reference BujardBuj12, Theorem 4.13] has determined that $G_{\operatorname{Gal}}$ can be maximally large, i.e. it is always possible that $G_{\operatorname{Gal}}\cong \operatorname{Gal}(\mathbb{F}_{p^{n}}/\mathbb{F}_{p})$ . This is also known to be true when $G_{\mathbb{S}}$ is a maximal finite subgroup of  $\mathbb{S}$ , cf. [Reference HennHen07, Proposition 20].

The proof of Theorem 1.5 for $G$ will consist of a reduction first from $G$ to its subgroup $G_{\mathbb{S}}$ , and then further to the $p$ -Sylow subgroup of  $G_{\mathbb{S}}$ .

2.3 The action on $E_{\ast }$

The first explicit description of the action of $\mathbb{G}$ on $E_{\ast }$ appears in unpublished work of Hopkins and Miller. The general situation for $\mathbb{G}$ acting on $E_{\ast }$ is determined by Devinatz and Hopkins [Reference Devinatz and HopkinsDH95].Footnote ¹ We focus on the action of  $\mathbb{S}$ . For our purposes, a nice summary of what we need can be found in the work of Nave [Reference NaveNav10, § 2].

Write an element $g\in \mathbb{S}$ as $g=\sum _{j=0}^{n-1}a_{j}S^{j}$ , where $a_{j}\in \mathbb{W}(\mathbb{F}_{p^{n}})$ and $a_{0}$ is a unit. Recall that $u\in \unicode[STIX]{x1D70B}_{-2}E_{n}$ is a generator. The next result follows from [Reference Devinatz and HopkinsDH95, Proposition 3.3 and Theorem 4.4].

Theorem 2.4. Let $\unicode[STIX]{x1D719}$ denote the lift of the Frobenius to $\mathbb{W}(\mathbb{F}_{p^{n}})$ . Then if $g=\sum _{j=0}^{n-1}a_{j}S^{j}$ as above, we have

(2.4) $$\begin{eqnarray}\displaystyle g_{\ast }(u) & \equiv & \displaystyle a_{0}u+a_{n-1}^{\unicode[STIX]{x1D719}}uu_{1}+\cdots +a_{1}^{\unicode[STIX]{x1D719}^{n-1}}uu_{n-1}\quad \text{mod}~(p,\mathfrak{m}^{2})\end{eqnarray}$$

(2.5) $$\begin{eqnarray}\displaystyle g_{\ast }(uu_{i}) & \equiv & \displaystyle a_{0}^{\unicode[STIX]{x1D719}^{i}}uu_{i}+\cdots +a_{i-1}^{\unicode[STIX]{x1D719}}uu_{1}\quad \text{mod}~(p,\mathfrak{m}^{2}).\end{eqnarray}$$

In the case of elements $g$ of finite order, we can be much more explicit. By the classification of finite subgroups of  $\mathbb{S}$ , such a $g$ either has order equal to  $p$ , or prime to  $p$ :

1. (i) if $g$ has order $p$ , it is known [Reference NaveNav10, Lemma 2.2] that $a_{0}\equiv 1~\text{mod}~(p)$ and $a_{1}$ is a unit;

2. (ii) If $g$ has order prime to $p$ , then $g$ is a Teichmuller lift of $a$ , and it follows from [Reference HennHen07] that the induced map of rings $g_{\ast }:E_{\ast }\rightarrow E_{\ast }$ is given by

   (2.6) $$\begin{eqnarray}g_{\ast }(u)=au\quad \text{and}\quad g_{\ast }(u_{i})=a^{p^{i}-1}u_{i}.\end{eqnarray}$$

2.4 The HFPSS

Because the action of finite subgroups on $E_{\ast }$ is well understood in our case of $n=p-1$ (which we implicitly assume in this subsection), so is the computation of the associated group cohomology. In fact, the differentials in the HFPSS are also known. The calculations are due to Hopkins and Miller, although their work has not appeared in print. Published references include the thesis and subsequent paper of Nave [Reference NaveNav99, Reference NaveNav10], or for a detailed computation at $n=2$ , $p=3$ , the work of Goerss et al. [Reference Goerss, Henn and MahowaldGHM04, Reference Goerss, Henn, Mahowald and RezkGHMR05].

We start by considering the prime-to- $p$ case.

Example 2.5. If $p$ does not divide the order of $G$ , then the higher cohomology of $G$ with coefficients in $E_{\ast }$ vanishes, giving that

$$\begin{eqnarray}(E^{hG})_{\ast }=(E_{\ast })^{G}.\end{eqnarray}$$

Using (2.6), we can (at least, in principle) determine these invariants. In particular, we see that if $m$ is the order of $G_{\mathbb{S}}$ , then $u^{m}$ is $G_{\mathbb{S}}$ -invariant and is a minimal periodicity element of $(E_{\ast })^{G_{\mathbb{S}}}$ . However, this element is also fixed under the Galois quotient of $G$ , so $E^{hG}$ has periodicity $2m=2|G_{\mathbb{S}}|$ .

The most interesting case for us is when $G=C_{p}$ . In this case, one has the following calculation of the $E_{2}$ -page modulo transfers.

Proposition 2.6 (Hopkins–Miller).

There is an exact sequence

$$\begin{eqnarray}E_{\ast }\xrightarrow[{}]{\text{tr}}H^{\ast }(C_{p},E_{\ast })\rightarrow \mathbb{F}_{p^{n}}[\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FF}^{\pm 1}]/(\unicode[STIX]{x1D6FC}^{2})\rightarrow 0,\end{eqnarray}$$

of graded groups, where the $(s,t)$ -bidegrees are $|\unicode[STIX]{x1D6FC}|=(1,2n)$ , $|\unicode[STIX]{x1D6FD}|=(2,2pn)$ , and $|\unicode[STIX]{x1D6FF}|=(0,2p)$ .

We note that because we have not taken Galois invariants anywhere, we see copies of $\mathbb{F}_{p^{n}}$ (rather than  $\mathbb{F}_{p}$ ) in the above spectral sequence. The passage to $\mathbb{F}_{p}$ can be achieved by adding in the Galois group. For example, we also state the following result of Hopkins–Miller.

Proposition 2.7 (Hopkins–Miller).

When $G\subset \mathbb{G}$ is a maximal finite subgroup containing $p$ -torsion, then there is an exact sequence

$$\begin{eqnarray}E_{\ast }\xrightarrow[{}]{\text{tr}}H^{\ast }(G,E_{\ast })\rightarrow \mathbb{F}_{p}[\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6E5}^{\pm 1}]/(\unicode[STIX]{x1D6FC}^{2}),\end{eqnarray}$$

with $|\unicode[STIX]{x1D6E5}|=(0,2pn^{2})$ and $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}$ as above.

The above determines the $E_{2}$ -page of the HFPSS for $\unicode[STIX]{x1D70B}_{\ast }(E^{hG})$ in two important cases. The differentials in the HFPSS are also well known. For any finite subgroup $G\subset \mathbb{G}$ , there are maps

$$\begin{eqnarray}\operatorname{Ext}_{BP_{\ast }BP}(BP_{\ast },BP_{\ast })\rightarrow H^{\ast }(\mathbb{G},E_{\ast })\rightarrow H^{\ast }(G,E_{\ast }).\end{eqnarray}$$

If $G$ contains an element of order $p$ , then, by [Reference RavenelRav78], the composite sends $\unicode[STIX]{x1D6FC}_{1}$ to $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}_{1}$ to $\unicode[STIX]{x1D6FD}$ , at least up to a non-zero scalar.

One can also show this directly using the functoriality of the Greek letter construction, as in [Reference Goerss, Henn and MahowaldGHM04, Proposition 7]. For this, recall that there is a class $v_{1}\in \operatorname{Ext}_{BP_{\ast }BP}^{0,2(p-1)}(BP_{\ast },BP_{\ast }/(p))$ which defines a permanent cycle in the Adams–Novikov spectral sequence for the mod- $p$ Moore spectrum $V(0)$ . Via the Greek letter construction, $v_{1}$ gives rise to an element $\unicode[STIX]{x1D6FC}_{1}\in \unicode[STIX]{x1D70B}_{2p-3}S_{(p)}^{0}\simeq \mathbb{Z}/p$ detected by a class, which we also denote by $\unicode[STIX]{x1D6FC}_{1}$ , in $\operatorname{Ext}_{BP_{\ast }BP}^{1,2p-2}(BP_{\ast },BP_{\ast })$ . Recalling that $v_{1}$ is sent to $u_{1}u^{1-p}$ under the orientation map $BP\rightarrow E$ , one can check that the image of $v_{1}$ defines a non-trivial element in $H^{0}(G,E_{\ast }/(p))$ . Using the short exact sequence

$$\begin{eqnarray}0\rightarrow E_{\ast }\rightarrow E_{\ast }\rightarrow E_{\ast }/(p)\rightarrow 0,\end{eqnarray}$$

one can see that this class has non-trivial image in $\unicode[STIX]{x1D6FF}^{0}(v_{1})\in H^{1}(G,E_{2p-2})$ (which, if $G$ is maximal, is $\mathbb{F}_{p}$ ). The result follows since this latter group is generated by  $\unicode[STIX]{x1D6FC}$ . One can make a similar argument for  $\unicode[STIX]{x1D6FD}$ , and since $\unicode[STIX]{x1D6FC}_{1}$ and $\unicode[STIX]{x1D6FD}_{1}$ are permanent cycles, so are $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ . We also know via the calculations of Toda [Reference TodaTod67, Reference TodaTod68] that $\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D6FD}_{1}^{p}=0$ and $\unicode[STIX]{x1D6FD}_{1}^{pn+1}=0$ in $\unicode[STIX]{x1D70B}_{\ast }S^{0}$ . Sparsity in the HFPSS, as well as the multiplicative structure, then allow the complete determination of the differentials.

In the following we use the notation $\stackrel{\cdot }{=}$ to denote an equality that holds up to multiplication by a non-zero scalar.

Lemma 2.8. In the spectral sequence for $\unicode[STIX]{x1D70B}_{\ast }E^{hC_{p}}$ , there are differentials

$$\begin{eqnarray}d_{2p-1}(\unicode[STIX]{x1D6FF})\stackrel{\cdot }{=}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}^{p-1}\unicode[STIX]{x1D6FF}^{1-n^{2}}\quad \text{and}\quad d_{2n^{2}+1}(\unicode[STIX]{x1D6FF}^{n^{3}}\unicode[STIX]{x1D6FC})\stackrel{\cdot }{=}\unicode[STIX]{x1D6FD}^{n^{2}+1}.\end{eqnarray}$$

If $G$ is a maximal finite subgroup, then in the spectral sequence for $\unicode[STIX]{x1D70B}_{\ast }E^{hG}$ , there are differentials

$$\begin{eqnarray}d_{2p-1}(\unicode[STIX]{x1D6E5})\stackrel{\cdot }{=}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}^{p-1}\quad \text{and}\quad d_{2n^{2}+1}(\unicode[STIX]{x1D6E5}^{n}\unicode[STIX]{x1D6FC})\stackrel{\cdot }{=}\unicode[STIX]{x1D6FD}^{n^{2}+1}.\end{eqnarray}$$

These differentials determine all others by multiplicativity. More precisely, in the $C_{p}$ -HFPSS, the shorter differential $d_{2p-1}$ is non-zero on all $\unicode[STIX]{x1D6FD}^{b}\unicode[STIX]{x1D6FF}^{d}$ with $d\not \equiv 0~\text{mod}~p$ , and classes of form $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}^{b}\unicode[STIX]{x1D6FF}^{d}$ are hit whenever $b\geqslant p-1$ and $d\not \equiv -1~\text{mod}~p$ . Now consider a monomial $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}^{b}\unicode[STIX]{x1D6FF}^{d}$ that survives to the $E_{2p}=E_{2n^{2}+1}$ -page; in particular, $d\equiv -1~\text{mod}~p$ or $b<p-1$ . If $d\not \equiv -1~\text{mod}~p$ (so we would have that $b<p-1$ ), the potential elements for $d_{2n^{2}+1}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}^{b}\unicode[STIX]{x1D6FF}^{d})$ (which are multiples of $\unicode[STIX]{x1D6FD}^{n^{2}+b+1}\unicode[STIX]{x1D6FF}^{d-n^{3}}$ ) have already been wiped out by $d_{2p-1}$ . When $d\equiv -1~\text{mod}~p$ , the potential targets are still there, and Lemma 2.8 implies that indeed $d_{2n^{2}+1}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}^{b}\unicode[STIX]{x1D6FF}^{d})\stackrel{\cdot }{=}\unicode[STIX]{x1D6FD}^{n^{2}+b+1}\unicode[STIX]{x1D6FF}^{d-n^{3}}$ .

As a specific example, note that $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF}$ is a permanent cycle, but $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF}^{-1}$ is not.

We demonstrate the computation (modulo some elements on the 0-line) of $\unicode[STIX]{x1D70B}_{\ast }E^{hC_{p}}$ for $n=2,p=3$ in Figure 1.

Figure 1. The HFPSS for $\unicode[STIX]{x1D70B}_{\ast }E^{hC_{p}}$ in the case $n=2$ , $p=3$ . Circled dots refer to copies of $\mathbb{F}_{9}$ , and the square represents a copy of $\mathbb{W}(\mathbb{F}_{9})$ . The dotted lines represent multiplication by  $\unicode[STIX]{x1D6FC}$ .

3.1 The Galois property of the Hopkins–Miller spectra

In this subsection, we show that for any finite subgroup $G\subset \mathbb{G}_{n}$ , the extension $E_{n}^{hG}\rightarrow E_{n}$ is a faithful $G$ -Galois extension of ring spectra. This result is essentially due (at least in the $K(n)$ -local case) to Devinatz and Hopkins [Reference Devinatz and HopkinsDH04] and Rognes [Reference RognesRog08]. We note also that, for $n=p-1$ (the case in which we are primarily interested here), the first author in [Reference HeardHea15] has given a computational proof of the vanishing of the Tate construction, which is a key component of the result.

Using the results of [Reference Mathew and MeierMM15], it will suffice to show that the stack $(\operatorname{Spec}\unicode[STIX]{x1D70B}_{0}E_{n})/G$ is affine over the moduli stack $M_{FG}$ of formal groups. This is easy to check at the closed point, and follows in general by a formal argument that we describe here, involving ‘free actions’.

We need to recall the following classical definition.

We refer to [Reference Grothendieck and RaynaudSGA1, Exp. V] for a detailed treatment and to [Reference LurieLur11, § 4] for a discussion with a view towards ring spectra. The following is an important example.

We now specialize to the case of interest in this subsection.

We will consider group actions in this category, or equivalently morphisms of algebraic stacks of the form $(\operatorname{Spec}B)/G\rightarrow M_{FG}$ . In particular, giving a $G$ -action on a pair $(B,\unicode[STIX]{x1D6E4})$ yields for each $\unicode[STIX]{x1D70E}\in B$ the data of an automorphism $\unicode[STIX]{x1D70E}:B\simeq B$ and an isomorphism of formal groups $f_{\unicode[STIX]{x1D70E}}:\unicode[STIX]{x1D70E}_{\ast }\unicode[STIX]{x1D6E4}\simeq \unicode[STIX]{x1D6E4}$ which satisfy a homomorphism and cocycle condition, respectively. We will represent $\unicode[STIX]{x1D6E4}$ by a formal group law over $B$ given by a power series $F(X,Y)\in B[\![X,Y]\!]$ . In this case, $f_{\unicode[STIX]{x1D70E}}=f_{\unicode[STIX]{x1D70E}}(X)$ is a power series in $B[\![X]\!]$ such that

$$\begin{eqnarray}F(f_{\unicode[STIX]{x1D70E}}(X),f_{\unicode[STIX]{x1D70E}}(Y))=f_{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D70E}_{\ast }F(X,Y)).\end{eqnarray}$$

We will need a general criterion to show that a map of the form $(\operatorname{Spec}B)/G\rightarrow M_{FG}$ is affine. For this, it will suffice to consider (for various  $n$ ) the faithfully flat cover $M_{FG}^{(n)}\rightarrow M_{FG}$ where $M_{FG}^{(n)}$ is the moduli stack of formal groups together with a coordinate modulo degrees at least $n+1$ .

Proof. Let $B^{\prime }$ be the $B$ -algebra which classifies coordinates modulo degrees at least $n+1$ on the formal group  $\unicode[STIX]{x1D6E4}$ . Abstractly, $B^{\prime }\simeq B[x_{1}^{\pm 1},x_{2},\ldots ,x_{n}]$ . The $G$ -action on $(B,\unicode[STIX]{x1D6E4})$ induces a $G$ -action on $(B^{\prime },\unicode[STIX]{x1D6E4}^{\prime })$ , where $\unicode[STIX]{x1D6E4}^{\prime }$ is the extension of scalars of $\unicode[STIX]{x1D6E4}$ to  $B^{\prime }$ .

We need to show that $G$ acts freely on $\operatorname{Hom}(B^{\prime },k)$ for any field  $k$ . Let $\unicode[STIX]{x1D70E}\in G\setminus \{1\}$ fix an element of $\operatorname{Hom}(B^{\prime },k)$ given by a map $\unicode[STIX]{x1D719}:B^{\prime }\rightarrow k$ . We get that $\unicode[STIX]{x1D70E}$ fixes the restriction $\unicode[STIX]{x1D719}|_{B}:B\rightarrow k$ , which means that the prime ideal $\mathfrak{p}=\text{ker}(\unicode[STIX]{x1D719}|_{B})$ is fixed by $\unicode[STIX]{x1D70E}$ , and that $\unicode[STIX]{x1D70E}$ acts trivially on the residue field $k(\mathfrak{p})$ , so that $\unicode[STIX]{x1D70E}\in I_{\mathfrak{p}}$ . Note that the fiber above $\unicode[STIX]{x1D719}|_{B}$ of the map (which is preserved by  $\unicode[STIX]{x1D70E}$ )

(3.1) $$\begin{eqnarray}\operatorname{Hom}(B^{\prime },k)\rightarrow \operatorname{Hom}(B,k)\end{eqnarray}$$

is given by the set of coordinates to degree $n$ on the formal group $(\unicode[STIX]{x1D719}|_{B})_{\ast }(\unicode[STIX]{x1D6E4})$ , and the action of $\unicode[STIX]{x1D70E}$ on this set of coordinates is given by composing with the image of  $f_{\unicode[STIX]{x1D70E}}$ . However, since the image of $f_{\unicode[STIX]{x1D70E}}\in k[\![X]\!]$ is not the identity, it acts freely on the above set of coordinates up to degree  $n$ . In particular, the action of $\unicode[STIX]{x1D70E}$ on the fiber of (3.1) above $\unicode[STIX]{x1D719}|_{B}$ has no fixed points, which is a contradiction and which completes the proof.◻

It follows, in particular, that there is a descent spectral sequence for computing $\operatorname{Pic}(E_{n}^{hG})$ (for any finite subgroup $G\subset \mathbb{G}_{n}$ ) based on the equivalence of spectra

$$\begin{eqnarray}\mathfrak{pic}(E_{n}^{hG})\simeq \unicode[STIX]{x1D70F}_{{\geqslant}0}\,\mathfrak{pic}(E_{n})^{hG}.\end{eqnarray}$$

We know that $\operatorname{Pic}(E_{n})$ is cyclic (by regularity, cf. [Reference Baker and RichterBR05]), and we want to prove that $\operatorname{Pic}(E_{n}^{hG})$ is also cyclic. In the rest of this section, we will reduce the analysis of this spectral sequence to the case where $G$ is a $p$ -subgroup of  $\mathbb{S}_{n}$ .

3.2 Elimination of the Galois piece

In this subsection, we make a basic reduction that enables us to eliminate the Galois piece. In other words, we reduce to proving cyclicity of the Picard group of $E_{n}^{hG}$ when $G\subset \mathbb{S}_{n}$ .

We will need a few basic properties of $\mathbb{W}(k)$ . Most of them can be encapsulated appropriately in the notion of ‘formal étaleness’, but we will spell out some details for the convenience of the reader.

Let $f:A_{0}\rightarrow A_{0}^{\prime }$ be a morphism of $p$ -local commutative rings. We say that $f$ is a uniform ${\mathcal{F}}_{p}$ -isomorphism if there exists $N>0$ such that:

1. (i) $\text{ker}(f)^{N}=0$ ;

2. (ii) given $x\in A_{0}^{\prime }$ , $x^{p^{N}}\in \text{im}(f)$ .

Proof. Let $\unicode[STIX]{x1D6E4}=G/\tilde{G}$ be the quotient group, so that by the Galois correspondence, the morphism of $\mathbf{E}_{\infty }$ -rings $A=R^{hG}\rightarrow B=R^{h\tilde{G}}$ is a faithful $\unicode[STIX]{x1D6E4}$ -Galois extension. We claim first that it is actually a Galois extension on homotopy groups, i.e.  $A\rightarrow B$ is étale, in the sense that $\unicode[STIX]{x1D70B}_{0}(A)\rightarrow \unicode[STIX]{x1D70B}_{0}(B)$ is étale and the natural map $\unicode[STIX]{x1D70B}_{\ast }A\otimes _{\unicode[STIX]{x1D70B}_{0}A}\unicode[STIX]{x1D70B}_{0}B\rightarrow \unicode[STIX]{x1D70B}_{\ast }B$ is an isomorphism; this will enable us to make the desired conclusion. We refer to [Reference Baker and RichterBR07] for the relationship between Galois extensions on homotopy and Galois extensions in the sense of Rognes [Reference RognesRog08], and to [Reference LurieLur11, § 4] as a convenient reference for some of the facts we need.

In particular, by [Reference LurieLur11, Corollary 4.15], we need to show that $\unicode[STIX]{x1D6E4}$ acts freely on $\unicode[STIX]{x1D70B}_{0}(B)$ . We will do this by functoriality. By the universal property of the ( $p$ -typical) Witt vectors, we obtain a $G$ -equivariant map

$$\begin{eqnarray}\mathbb{W}(k)\rightarrow \unicode[STIX]{x1D70B}_{0}(R),\end{eqnarray}$$

where $G$ acts on $\mathbb{W}(k)$ via the induced action on $k$ . Of course, the $G$ -action on $\mathbb{W}(k)$ factors through $\unicode[STIX]{x1D6E4}=G/\tilde{G}$ , and we claim that this lifts to a $\unicode[STIX]{x1D6E4}$ -equivariant map

$$\begin{eqnarray}\mathbb{W}(k)\rightarrow \unicode[STIX]{x1D70B}_{0}(B)=\unicode[STIX]{x1D70B}_{0}(R^{h\tilde{G}}).\end{eqnarray}$$

The equivariance alone gives the $\unicode[STIX]{x1D6E4}$ -equivariant lift $\mathbb{W}(k)\rightarrow \unicode[STIX]{x1D70B}_{0}(R)^{\tilde{G}}$ , which of course receives a map from $\unicode[STIX]{x1D70B}_{0}(R^{h\tilde{G}})$ . Using Lemma 3.9, we can upgrade it to the claimed $\unicode[STIX]{x1D6E4}$ -equivariant map $\mathbb{W}(k)\rightarrow \unicode[STIX]{x1D70B}_{0}(B)$ .

The $\unicode[STIX]{x1D6E4}$ -action on $W(k)$ is free (Example 3.2). It follows that the $\unicode[STIX]{x1D6E4}$ -action on $\unicode[STIX]{x1D70B}_{0}(B)$ is free. Applying [Reference LurieLur11, Corollary 4.15], we conclude that $A\rightarrow B$ is an algebraic $\unicode[STIX]{x1D6E4}$ -Galois extension in that $\unicode[STIX]{x1D70B}_{0}(A)\rightarrow \unicode[STIX]{x1D70B}_{0}(B)$ is one, and $\unicode[STIX]{x1D70B}_{\ast }(A)\otimes _{\unicode[STIX]{x1D70B}_{0}(A)}\unicode[STIX]{x1D70B}_{0}(B)\rightarrow \unicode[STIX]{x1D70B}_{\ast }(B)$ is an isomorphism.

Finally, we prove the result about the Picard group. Let $M$ be an invertible $A$ -module. By assumption, $M\otimes _{A}B$ is a suspension of a free $B$ -module of rank one; by desuspending $M$ appropriately, we may assume that $M\otimes _{A}B\simeq B$ . We would like to see that $M\simeq A$ . Since $B$ is faithfully flat over  $A$ , we can conclude that $\unicode[STIX]{x1D70B}_{0}(M)$ is an invertible $\unicode[STIX]{x1D70B}_{0}(A)$ -module and that the map $\unicode[STIX]{x1D70B}_{0}(M)\otimes _{\unicode[STIX]{x1D70B}_{0}(A)}\unicode[STIX]{x1D70B}_{\ast }(A)\rightarrow \unicode[STIX]{x1D70B}_{\ast }(M)$ is an isomorphism (because this happens after base-change to $B$ ). In particular, $M$  simply arises from an invertible module over  $\unicode[STIX]{x1D70B}_{0}(A)$ .

Therefore, it suffices to show that the Picard group of the (ordinary) commutative ring $\unicode[STIX]{x1D70B}_{0}(A)$ is trivial. This will follow if we can show that $\unicode[STIX]{x1D70B}_{0}(A)$ is a (possibly non-noetherian) local ring. Since $\unicode[STIX]{x1D70B}_{0}(A)\rightarrow \unicode[STIX]{x1D70B}_{0}(B)$ is finite étale, it suffices to show that $\unicode[STIX]{x1D70B}_{0}(B)$ is a local ring, i.e. that its spectrum has a unique closed point. However, $\unicode[STIX]{x1D70B}_{0}(B)\rightarrow \unicode[STIX]{x1D70B}_{0}(R)^{\widetilde{G}}$ is an ${\mathcal{F}}_{p}$ -isomorphism as above, so that it induces an isomorphism on Zariski spectra. Therefore, the fact that $\unicode[STIX]{x1D70B}_{0}(R)^{\widetilde{G}}$ is a local ring forces $\unicode[STIX]{x1D70B}_{0}(B)$ to be one, and completes the proof.◻

In our example of interest, $R$ will be $E_{n}$ , $G$ will be a finite subgroup of the extended Morava stabilizer group, and $\tilde{G}=G_{\mathbb{S}}$ , as in (2.3).

3.3 Reduction to the $p$ -Sylow

Here, we will use basic facts about the structure of $\mathbb{S}_{n}$ , for which we refer the reader to § 2.1.

Proof. Observe first that $\operatorname{Pic}(E_{n}^{hG})=\unicode[STIX]{x1D70B}_{0}(\mathfrak{pic}(E_{n})^{hG})$ is a torsion group (cf. [Reference Mathew and StojanoskaMS16, § 5.3]), so it suffices to show that the suspension generates the $q$ -part $\operatorname{Pic}(E_{n}^{hG})_{q}$ of $\operatorname{Pic}(E_{n}^{hG})$ for any prime number  $q$ . We consider two cases.

(i) $q=p$ . In this case, the map

$$\begin{eqnarray}\unicode[STIX]{x1D719}_{p}:\operatorname{Pic}(E_{n}^{hG})_{p}\rightarrow \operatorname{Pic}(E_{n}^{hG_{p}})_{p}\end{eqnarray}$$

(induced by base-change along $E_{n}^{hG}\rightarrow E_{n}^{hG_{p}}$ ) is injective thanks to the transfer map

$$\begin{eqnarray}\operatorname{Pic}(E_{n}^{hG_{p}})_{p}\simeq \unicode[STIX]{x1D70B}_{0}\,\mathfrak{pic}(E_{n})^{hG_{p}}\rightarrow \operatorname{Pic}(E_{n}^{hG})_{p}\simeq \unicode[STIX]{x1D70B}_{0}\,\mathfrak{pic}(E_{n})^{hG}.\end{eqnarray}$$

Namely, recall that if $X$ is a $p$ -local spectrum with a $G$ -action, the composite map $X^{hG}\rightarrow X^{hG_{p}}\stackrel{\text{tr}}{\rightarrow }X^{hG}$ is an equivalence. Since the map $\unicode[STIX]{x1D719}_{p}$ carries the suspension of the unit to itself, the fact that this class generates the target implies that it must generate the source.

(ii) $q\neq p$ . Let $G_{q}\subseteq G$ be a $q$ -Sylow subgroup. As above, we observe that the map $\unicode[STIX]{x1D719}_{q}$ defined via

$$\begin{eqnarray}\unicode[STIX]{x1D719}_{q}:\operatorname{Pic}(E_{n}^{hG})_{q}\rightarrow \operatorname{Pic}(E_{n}^{hG_{q}})_{q}\end{eqnarray}$$

is injective. So, it suffices to consider the case where $G$ is itself a $q$ -group.

We consider the HFPSS converging to $\operatorname{Pic}(E_{n}^{hG})=\unicode[STIX]{x1D70B}_{0}\,\mathfrak{pic}(E_{n})^{hG}$ . In this case, since $G$ is a $q$ -group, we observe that the only contributions come from $H^{0}(G,\mathbb{Z}/2)$ (which is $\mathbb{Z}/2$ , generated by the suspension) and $H^{1}(G,(E_{n})_{0}^{\times })\simeq H^{1}(G,(\mathbb{F}_{p^{n}})^{\times })$ via the Teichmüller lift. In other words, we need to show (to prove cyclicity of the Picard group in this case) that the group $H^{1}(G,(\mathbb{F}_{p^{n}})^{\times })=\operatorname{Hom}(G,\mathbb{F}_{p^{n}}^{\times })$ is generated by the class $\unicode[STIX]{x1D713}$ corresponding to the $G$ -action on $\unicode[STIX]{x1D70B}_{-2}(E_{n})\otimes _{(E_{n})_{0}}\mathbb{F}_{p^{n}}$ . (This will give the order of the Picard group as well as solve the extension problem.) However, we know (by general facts about even-periodic ring spectra) that this is simply the action on the Lie algebra, i.e. the homomorphism of (2.1). However, as we remarked before, the kernel of (2.1) is pro- $p$ , so its restriction to $G$ must be injective. Hence, $G$ injects into $\mathbb{F}_{p^{n}}^{\times }$ , which forces $H^{1}(G,\mathbb{F}_{p^{n}}^{\times })$ to be generated by the class of the injection.◻

4.1 The general case

Our goal in this subsection is to prove the main result of this paper, namely the following theorem.

By combining Propositions 3.10, and 3.11, it is enough to prove the result when $G=C_{p}$ . To handle this case, we start by recalling from Proposition 2.6 that, modulo the image of the transfer,

(4.1) $$\begin{eqnarray}H^{\ast }(C_{p},E_{\ast })/\text{transfers}\simeq \mathbb{F}_{p^{n}}[\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FF}^{\pm 1}]/(\unicode[STIX]{x1D6FC}^{2}),\end{eqnarray}$$

where $|\unicode[STIX]{x1D6FC}|=(1,2n)$ , $|\unicode[STIX]{x1D6FD}|=(2,2pn),$ and $|\unicode[STIX]{x1D6FF}|=(0,2p)$ . Recall also that the homotopy groups $\unicode[STIX]{x1D70B}_{\ast }(E^{hC_{p}})$ are $2p^{2}$ -periodic; this sets $\mathbb{Z}/(2p^{2})$ as a lower bound on the order of the Picard group. We will use the Picard spectral sequence to determine an upper bound. Since we will be comparing the Picard spectral sequence with the additive one (for computing $\unicode[STIX]{x1D70B}_{\ast }E^{hC_{p}}$ ), we will write $d_{k,\times }$ for the Picard differentials and $d_{k}$ for the additive ones.

Our first task is to work out the $E_{2}$ -term $H^{s}(C_{p},\unicode[STIX]{x1D70B}_{t}\,\mathfrak{pic}(E))$ , and we are interested in the abutment for $t-s=0$ . In the range where $t\geqslant 2$ , this follows immediately from (4.1). Namely, we have

$$\begin{eqnarray}H^{s}(C_{p},\unicode[STIX]{x1D70B}_{t}\,\mathfrak{pic}(E))=(\mathbb{F}_{p^{n}}[\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FF}^{\pm 1}]/(\unicode[STIX]{x1D6FC}^{2}))^{s,t-1}\quad \text{when}~t\geqslant 2,s\geqslant 1\end{eqnarray}$$

since in this case $\unicode[STIX]{x1D70B}_{t}\,\mathfrak{pic}(E)=\unicode[STIX]{x1D70B}_{t-1}E$ , equivariantly. (The superscript notation on the right-hand side designates the bidegree.)

Let us single out the classes in the $t-s=0$ column, i.e. those with potential contribution to the Picard group of $E^{hC_{p}}$ , that come from the additive spectral sequence (so, $s>1$ ). They are

$$\begin{eqnarray}e_{c}=\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}^{pc-1}\unicode[STIX]{x1D6FF}^{n-1-c(pn-1)},\end{eqnarray}$$

indexed by integers $c\geqslant 1$ , and have cohomological degree $s(e_{c})=2pc-1$ .

Proof. In the additive spectral sequence, by Lemma 2.8, the classes $e_{c}$ are never the source of a $d_{2p-1}$ differential, but can be the target. In particular, there are additive differentials

$$\begin{eqnarray}d_{2p-1}(\unicode[STIX]{x1D6FD}^{a}\unicode[STIX]{x1D6FF}^{b})\stackrel{\cdot }{=}b\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}^{a+p-1}\unicode[STIX]{x1D6FF}^{b-n^{2}}.\end{eqnarray}$$

For this image to be (a multiple of) $e_{c}$ , we must have $b=n^{2}+n-1-c(pn-1)\equiv c-1~\text{mod}~p$ .

1. – If $c\not \equiv 1~\text{mod}~p$ , then $e_{c}$ is the target of $d_{2p-1}$ on an element with $t=2p(c-1)$ . Since such $c\geqslant 2$ , we have that $t\geqslant 2p>2p-1$ , so that Theorem 1.4 implies that this differential can be imported in the Picard spectral sequence. Hence, $e_{c}$ itself is killed.

2. – If $c\equiv 1~\text{mod}~p$ , this differential is zero (as then $b\equiv 0~\text{mod}~p$ ), so $e_{c}$ survives the $d_{2p-1}$ .

The next and only possible additive differential is a $d_{2n^{2}+1}$ . We have

$$\begin{eqnarray}d_{2n^{2}+1}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF}^{n^{3}})\stackrel{\cdot }{=}\unicode[STIX]{x1D6FD}^{n^{2}+1},\end{eqnarray}$$

which determines the rest. Namely, we have, for all $a,b\in \mathbb{Z}$ , $a\geqslant 0$ ,

$$\begin{eqnarray}d_{2n^{2}+1}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}^{a}\unicode[STIX]{x1D6FF}^{bp-1})\stackrel{\cdot }{=}\unicode[STIX]{x1D6FD}^{a+n^{2}+1}\unicode[STIX]{x1D6FF}^{pb-n^{3}-1}.\end{eqnarray}$$

When $c\equiv 1~\text{mod}~(p)$ , we have that $n-1-c(pn-1)\equiv -1~\text{mod}~p$ , hence

$$\begin{eqnarray}d_{2n^{2}+1}(e_{c})\stackrel{\cdot }{=}\unicode[STIX]{x1D6FD}^{pc+n^{2}}\unicode[STIX]{x1D6FF}^{c(1-pn)+n-n^{3}-1}.\end{eqnarray}$$

To import this differential to the Picard spectral sequence, we need to have that

$$\begin{eqnarray}2n^{2}+1\leqslant t(e_{c})=2pc-2.\end{eqnarray}$$

This holds whenever

$$\begin{eqnarray}p-2+\frac{5}{2p}\leqslant c\quad \Leftrightarrow \quad c\geqslant p-1.\end{eqnarray}$$

Since $c\equiv 1~\text{mod}~(p)$ , we see we can import all differentials except that involving $e_{1}$ , and the result follows.◻

We can now prove our main result.

Proof of Theorem 4.1.

We have seen that in the range $t\geqslant 2$ of the HFPSS for $\unicode[STIX]{x1D70B}_{t-s}\,\mathfrak{pic}(E^{hC_{p}})$ , we only have a single possible contribution to the 0-stem, namely a group of order at most  $p^{n}$ , arising from the class  $e_{1}$ . This class has bidegree $(2p-1,2p-1)$ , and it follows from Theorem 7.1 that there is a differential

(4.2) $$\begin{eqnarray}d_{2n^{2}+1,\times }(e_{1})\stackrel{\cdot }{=}d_{2n^{2}+1}(e_{1})+\unicode[STIX]{x1D701}\unicode[STIX]{x1D6FD}{\mathcal{P}}^{n/2}(e_{1}),\end{eqnarray}$$

for $\unicode[STIX]{x1D701}\in \mathbb{F}_{p}^{\times }$ . Let $f_{t}=\unicode[STIX]{x1D6FD}^{n^{2}+n+1}\unicode[STIX]{x1D6FF}^{-n^{2}(n+1)}$ so that we have (in the additive spectral sequence)

$$\begin{eqnarray}d_{2n^{2}+1}(e_{1})\stackrel{\cdot }{=}f_{t}.\end{eqnarray}$$

While it is possible to work out exactly what $\unicode[STIX]{x1D6FD}{\mathcal{P}}^{n/2}(e_{1})$ is, we will not need to be so precise. Instead we claim that the group $\mathbb{F}_{p^{n}}e_{1}$ on the $E_{2}$ -page can contribute at most an element of order $p$ to Picard group.

Choose $\unicode[STIX]{x1D709}^{\prime }\in \mathbb{F}_{p^{n}}$ such that $\unicode[STIX]{x1D6FD}{\mathcal{P}}^{n/2}(e_{1})=\unicode[STIX]{x1D709}^{\prime }f_{t}$ . Then we have

$$\begin{eqnarray}d_{2n^{2}+1}(e_{1})=\unicode[STIX]{x1D709}f_{t}\quad \text{and}\quad \unicode[STIX]{x1D6FD}{\mathcal{P}}^{n/2}(e_{1})=\unicode[STIX]{x1D709}^{\prime }f_{t},\end{eqnarray}$$

for $\unicode[STIX]{x1D709},\unicode[STIX]{x1D709}^{\prime }\in \mathbb{F}_{p^{n}}$ and $\unicode[STIX]{x1D709}\neq 0$ . It follows from (4.2) that, for any $a\in \mathbb{F}_{p^{n}}$ , we have

$$\begin{eqnarray}\displaystyle d_{2n^{2}+1,\times }(ae_{1}) & = & \displaystyle d_{2n^{2}+1}(ae_{1})+\unicode[STIX]{x1D701}\unicode[STIX]{x1D6FD}{\mathcal{P}}^{n/2}(ae_{1})\nonumber\\ \displaystyle & = & \displaystyle a\unicode[STIX]{x1D709}f_{t}+a^{p}\unicode[STIX]{x1D701}\unicode[STIX]{x1D6FD}{\mathcal{P}}^{n/2}(e_{1})\nonumber\\ \displaystyle & = & \displaystyle a\unicode[STIX]{x1D709}f_{t}+a^{p}\unicode[STIX]{x1D701}\unicode[STIX]{x1D709}^{\prime }f_{t}\nonumber\\ \displaystyle & = & \displaystyle (a\unicode[STIX]{x1D709}+a^{p}\unicode[STIX]{x1D701}\unicode[STIX]{x1D709}^{\prime })f_{t}.\nonumber\end{eqnarray}$$

Since $\unicode[STIX]{x1D709}\neq 0$ , there are at most $p$ choices of $a$ that make the above vanish, and so we do indeed have an upper bound of $\mathbb{Z}/p$ for the contribution of $e_{1}$ in $\operatorname{Pic}(E^{hC_{p}})$ .

Finally, we know from Proposition 6.5 that $H^{1}(C_{p},\unicode[STIX]{x1D70B}_{1}\,\mathfrak{pic}(E))\simeq \mathbb{Z}/p$ , while as usual it follows from [Reference Baker and RichterBR05] that the invariants $H^{0}(C_{p},\unicode[STIX]{x1D70B}_{0}\,\mathfrak{pic}(E))$ are $\mathbb{Z}/2$ . Thus, we have the following in the 0-stem of the Picard spectral sequence:

1. – a group of order at most $2$ in $t=s=0$ ;

2. – a group of order at most $p$ in $t=s=1$ ; and

3. – a group of order at most $p$ in $t=s=2p-1$ .

This gives an upper bound of $2p^{2}$ for the order of the Picard group, but as noted previously this is also a lower bound, and the result follows. A picture of the Picard spectral sequence for $n=2$ , $p=3$ and $G=C_{3}$ is shown in Figure 2.◻

Figure 2. The HFPSS for $\unicode[STIX]{x1D70B}_{\ast }\,\mathfrak{pic}\,E_{p-1}^{hC_{p}}$ in the case $n=2$ , $p=3$ . The differential out of $(0,5)$ has kernel $\mathbb{Z}/3$ . (Here $\bullet$ denotes $\mathbb{Z}/3$ , 

 denotes $\mathbb{F}_{9}$ , and $\times$ denotes $\mathbb{Z}/2$ .)

4.2 The maximal finite subgroup

In this short section we sketch a direct proof in the case that $G$ is a maximal finite subgroup of $\mathbb{G}_{n}$ with $p$ -torsion. We include this to show that after taking Galois invariants the calculation is slightly easier; in particular, we do not need to use Theorem 7.1. The calculation of $\operatorname{Pic}(E_{n}^{hG})$ in this case was previously known to Hopkins.

We start by recalling that, modulo the image of the transfer, we have

(4.3) $$\begin{eqnarray}H^{\ast }(G,E_{\ast })/\text{transfers}\simeq \mathbb{F}_{p}[\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6E5}^{\pm 1}]/(\unicode[STIX]{x1D6FC}^{2}),\end{eqnarray}$$

where $|\unicode[STIX]{x1D6E5}|=(0,2pn^{2})$ , and that the homotopy groups $\unicode[STIX]{x1D70B}_{\ast }(E^{hG})$ are $2p^{2}n^{2}$ periodic; this sets $\mathbb{Z}/(2p^{2}n^{2})$ as a lower bound on the order of the Picard group. As usual, we will use the Picard spectral sequence to determine an upper bound.

In the range $t\geqslant 2$ we have classes

$$\begin{eqnarray}e_{c}=\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}^{pc-1}\unicode[STIX]{x1D6E5}^{(c(1-pn)+n-1)/n^{2}},\end{eqnarray}$$

in the 0-stem, indexed by integers $c\geqslant 1$ , which have cohomological degree $s(e_{c})=2pc-1$ . Once again, one can check that the classes $e_{c}$ for $c\neq 1$ do not survive the Picard spectral sequence. One key difference to the case of $C_{p}$ is that we see immediately that $e_{1}$ can contribute a group of at most order $p$ to $\operatorname{Pic}(E^{hG})$ , thus avoiding the use of Theorem 7.1.

We are then left with calculating the contributions from $t=0$ and $t=1$ . It follows from [Reference Baker and RichterBR05] that $H^{0}(G,\unicode[STIX]{x1D70B}_{0}\,\mathfrak{pic}(E))\simeq H^{0}(G,\mathbb{Z}/2)\simeq \mathbb{Z}/2$ , and so we are left to determine $H^{1}(G,\unicode[STIX]{x1D70B}_{1}\,\mathfrak{pic}(E))\simeq H^{1}(G,E_{0}^{\times })$ . This can be done in at least a couple of different ways; for example, as in Proposition 6.5, or using an exponential map and several Bockstein spectral sequences as in [Reference KaramanovKar10]. In any case, the result is a group of order  $pn^{2}$ .

Figure 3. The HFPSS for $\unicode[STIX]{x1D70B}_{\ast }\,\mathfrak{pic}\,E^{hG}$ in the case $n=2$ , $p=3$ for $G\subset \mathbb{G}$ a maximal finite subgroup whose order is divisible by  $p$ . (Here $\Box$ denotes $\mathbb{Z}/4$ , $\bullet$ denotes $\mathbb{Z}/3$ , and $\times$ denotes $\mathbb{Z}/2$ .)

Putting all of this together, we obtain an upper bound $2\cdot pn^{2}\cdot p$ for the abutment, i.e. for the Picard group of $E^{hG}$ . However, by Remark 2.9 we know the Picard group contains a cyclic group of order $2p^{2}n^{2}$ , and so it follows that $\operatorname{Pic}(E^{hG})$ is indeed cyclic of order $2p^{2}n^{2}$ . We illustrate this for $n=2$ , $p=3$ in Figure 3.

5.1 Overview

In this section, we describe a tool for working with the following general situation. Let $R$ be a complete noetherian local ring with maximal ideal $\mathfrak{m}\subset R$ such that the residue field $R/\mathfrak{m}$ has characteristic  $p$ . Suppose $G$ is a finite group acting on  $R$ . Our goal is to calculate the cohomology groups $H^{i}(G,R^{\times })$ . We assume that the additive analog, the cohomology groups $H^{i}(G,R)$ , are understood by other means.

One approach towards $H^{i}(G,R^{\times })$ is to consider the $\mathfrak{m}$ -adic filtration on  $R$ . It induces a $G$ -stable filtration on  $R^{\times }$ ,

(5.1) $$\begin{eqnarray}R^{\times }\supset 1+\mathfrak{m}\supset 1+\mathfrak{m}^{2}\supset \cdots ,\end{eqnarray}$$

which leads to a spectral sequence

(5.2) $$\begin{eqnarray}E_{1}^{i,j}=H^{i}(G,(1+\mathfrak{m}^{j})/(1+\mathfrak{m}^{j+1}))\;\Longrightarrow \;H^{i}(G,R^{\times }).\end{eqnarray}$$

Here the differentials run $d_{k}:E_{k}^{i,j}\rightarrow E_{k}^{i+1,j+k}$ . For $j>0$ , there is a $G$ -equivariant isomorphism (sending $1+x\mapsto x$ )

$$\begin{eqnarray}(1+\mathfrak{m}^{j})/(1+\mathfrak{m}^{j+1})\simeq \mathfrak{m}^{j}/\mathfrak{m}^{j+1}\end{eqnarray}$$

between the associated graded module of the filtration (5.1) and the $\mathfrak{m}$ -adic filtration on $R$ itself. The latter, of course, leads to a spectral sequence (indexed similarly)

(5.3) $$\begin{eqnarray}E_{1}^{i,j}=H^{i}(G,\mathfrak{m}^{j}/\mathfrak{m}^{j+1})\;\Longrightarrow \;H^{i}(G,R).\end{eqnarray}$$

In practice, one may (and is likely to) understand the additive spectral sequence (5.3) concretely in specific instances, for instance, because of the additional tools provided by multiplicativity. The additive and the multiplicative spectral sequences have almost the same $E_{1}$ pages, and in this section we will give a tool for comparing differentials in a range of dimensions.

5.2 The $p$ -truncated exponential

Let $R$ be a commutative ring and let $I\subset R$ be an ideal such that $I^{2}=0$ . Then there is an isomorphism of groups

$$\begin{eqnarray}I\simeq 1+I\subset R^{\times },\quad x\mapsto 1+x.\end{eqnarray}$$

We now review a crucial and elementary piece of algebra that will enable generalizations of this fact when certain primes are inverted in  $R$ .

Definition 5.1. Let $R$ be a $\mathbb{Z}[1/(p-1)!]$ -algebra. We define a function

(5.4) $$\begin{eqnarray}\exp _{p}:R\rightarrow R,\quad \exp _{p}(x)=1+x+\frac{x^{2}}{2!}+\cdots +\frac{x^{p-1}}{(p-1)!},\end{eqnarray}$$

called the $p$ -truncated exponential, and a function

(5.5) $$\begin{eqnarray}\log _{p}:R\rightarrow R,\quad \log _{p}(x)=\mathop{\sum }_{n=1}^{p-1}(-1)^{n-1}\frac{(x-1)^{n}}{n},\end{eqnarray}$$

called the $p$ -truncated logarithm.

The significance arises from the following well-known application.

Lemma 5.2. If $I\subset R$ is an ideal with $I^{p}=0$ and $R$ is a $\mathbb{Z}[1/(p-1)!]$ -algebra, then the map

$$\begin{eqnarray}\exp _{p}:I\rightarrow 1+I\subset R^{\times }\end{eqnarray}$$

is a group isomorphism (whose inverse is given by the truncated logarithm).

We note that the inverse isomorphisms $\exp _{p},\log _{p}$ between $I$ and $1+I$ are compatible with the $I$ -adic filtration, and with the isomorphisms on associated graded modules $I^{k}/I^{k+1}\simeq (1+I^{k})/(1+I^{k+1})$ given by $1+x\mapsto x$ .

Example 5.3. Suppose $(A,\mathfrak{m})$ is a local ring whose residue field is of characteristic  $p$ , a prime larger than  $p$ , or zero. Then for each $k>0$ , we have a functorial isomorphism of groups

$$\begin{eqnarray}(1+\mathfrak{m}^{k})/(1+\mathfrak{m}^{pk})\simeq \mathfrak{m}^{k}/\mathfrak{m}^{pk},\end{eqnarray}$$

given by the truncated logarithm. This leads to an identification of the spectral sequences (5.3) and (5.2) in a ‘stable’ range of dimensions (which we will make more precise below).

We will actually need a slight generalization of this fact (which measures the first-order failure of $\exp _{p}$ to be a homomorphism in general).

Definition 5.4. Given $x_{1},\ldots ,x_{k}\in R$ , we define

(5.6) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{p}(x_{1},\ldots ,x_{k})\stackrel{\text{def}}{=}\frac{(\mathop{\sum }_{i}x_{i})^{p}-\mathop{\sum }_{i}x_{i}^{p}}{p!}=\mathop{\sum }_{\substack{ j_{1},\ldots ,j_{k}\in [0,p-1] \\ j_{1}+\cdots +j_{k}=p}}^{p-1}\frac{x_{1}^{j_{1}}x_{2}^{j_{2}}\ldots x_{k}^{j_{k}}}{j_{1}!\ldots j_{k}!},\end{eqnarray}$$

so that we obtain a function $\unicode[STIX]{x1D70E}_{p}(\cdot ,\ldots ,\cdot ):R^{k}\rightarrow R$ . If $M$ is a $\mathbb{Z}[1/(p-1)!]$ -module, we will also use $\unicode[STIX]{x1D70E}_{p}$ to denote the induced function

$$\begin{eqnarray}\unicode[STIX]{x1D70E}_{p}:M^{k}\rightarrow \operatorname{Sym}^{p}M,\end{eqnarray}$$

defined by the same formula.

Proposition 5.5. Let $R$ be a $\mathbb{Z}[1/(p-1)!]$ -algebra. Suppose that $x_{1},\ldots ,x_{k}\in R$ and let $I=(x_{1},\ldots ,x_{k})\subset R$ . Then

(5.7) $$\begin{eqnarray}\prod \exp _{p}(x_{i})-\exp _{p}\Bigl(\sum x_{i}\Bigr)\equiv \unicode[STIX]{x1D70E}_{p}(x_{1},\ldots ,x_{k})\quad \text{mod}~I^{p+1}.\end{eqnarray}$$

With notation as above, observe that $\unicode[STIX]{x1D70E}_{p}(x_{1},\ldots ,x_{k})\in I^{p}$ and that $\exp _{p}(x)\in 1+I$ for $x\in I$ . As a result, we also find that, if $I$ is contained in the Jacobson radical of  $R$ , then we have the formula

(5.8) $$\begin{eqnarray}\frac{\exp _{p}(\sum x_{i})}{\prod \exp _{p}(x_{i})}\equiv 1-\unicode[STIX]{x1D70E}_{p}(x_{1},\ldots ,x_{k})\quad \text{mod}~I^{p+1}.\end{eqnarray}$$

For example, taking $k=2$ and taking $x_{1}=y,x_{2}=-y$ , we find

(5.9) $$\begin{eqnarray}\exp _{p}(y)\exp _{p}(-y)\equiv 1+\unicode[STIX]{x1D70E}_{p}(y,-y)\quad \text{mod}~I^{p+1}.\end{eqnarray}$$

Note, in addition, that $\unicode[STIX]{x1D70E}_{p}(y,-y)=y^{p}\unicode[STIX]{x1D70E}_{p}(1,-1)$ .

Corollary 5.6. If $y_{0},\ldots ,y_{n}\in I$ and $I$ is contained in the radical of $R$ , then

(5.10) $$\begin{eqnarray}\frac{\exp _{p}(\mathop{\sum }_{i=0}^{n}(-1)^{i}y_{i})}{\mathop{\prod }_{i=0}^{n}\exp _{p}(y_{i})^{(-1)^{i}}}\equiv 1-\unicode[STIX]{x1D70E}_{p}(y_{0},-y_{1},\ldots ,(-1)^{n}y_{n})+\mathop{\sum }_{i~\text{odd}}\unicode[STIX]{x1D70E}_{p}(y_{i},-y_{i})\quad \text{mod}~I^{p+1}.\end{eqnarray}$$

We give a name to the discrepancy term; given $y_{0},\ldots ,y_{n}$ , we set

(5.11) $$\begin{eqnarray}\unicode[STIX]{x1D707}_{p}(y_{0},\ldots ,y_{n})\stackrel{\text{def}}{=}\unicode[STIX]{x1D70E}_{p}(y_{0},-y_{1},\ldots ,(-1)^{n}y_{n})-\mathop{\sum }_{i~\text{odd}}\unicode[STIX]{x1D70E}_{p}(y_{i},-y_{i}).\end{eqnarray}$$

5.3 The stable range and first differential

We begin by describing a version of the general setup of comparison of spectral sequences from the introduction to this section.

Proposition 5.7. Let $G$ be a finite group acting on a local noetherian ring $(R,\mathfrak{n})$ with $(p-1)!$ invertible. Consider the two $\text{truncated}$ spectral sequences

$$\begin{eqnarray}\displaystyle E_{1}^{s,t} & = & \displaystyle H^{s}(G,\mathfrak{n}^{t}/\mathfrak{n}^{t+1})\;\Longrightarrow \;H^{i}(G,\mathfrak{n}^{k}/\mathfrak{n}^{pk}),\nonumber\\ \displaystyle E_{1}^{s,t} & = & \displaystyle H^{i}(G,\mathfrak{n}^{t}/\mathfrak{n}^{t+1})\;\Longrightarrow \;H^{i}(G,(1+\mathfrak{n}^{k})/(1+\mathfrak{n}^{pk})),\nonumber\end{eqnarray}$$

where $t\in [k,pk-1]$ . Then the natural identification of $E_{1}$ -terms extends to an isomorphism of spectral sequences given by the $p$ -truncated exponential.

We now describe the setup that will enable us to ultimately give a formula for the first unequal differential. For convenience, we will work with nonunital commutative rings here. If $A$ is a nonunital commutative ring, we define an abelian monoid $1+A$ consisting of formal elements $1+x,x\in A$ with the obvious multiplication rule.

Construction 5.8. Suppose $\mathfrak{m}^{\bullet }$ is a nonunital cosimplicial commutative $\mathbb{Z}[1/(p-1)!]$ -algebra such that $\mathfrak{m}^{\bullet ,p+1}=0$ .Footnote ² We have a descending multiplicative filtration by powers

$$\begin{eqnarray}\mathfrak{m}^{\bullet }\supset \mathfrak{m}^{\bullet ,2}\supset \cdots .\end{eqnarray}$$

Consider also the cosimplicial abelian group $1+\mathfrak{m}^{\bullet }$ .Footnote ³ We also have a filtration of cosimplicial abelian groups

$$\begin{eqnarray}1+\mathfrak{m}^{\bullet }\supset 1+\mathfrak{m}^{\bullet ,2}\supset \cdots ,\end{eqnarray}$$

and the successive quotients are isomorphic to those of the additive one.

Both lead to spectral sequences starting from the associated graded modules $\mathfrak{m}^{\bullet ,j}/\mathfrak{m}^{\bullet ,j+1}$ (for $1\leqslant j\leqslant p$ ) converging to the cohomology of $\mathfrak{m}^{\bullet }$ (respectively $1+\mathfrak{m}^{\bullet }$ ), i.e. the $E_{1}$ -pages are identified. We have spectral sequences

(5.12) $$\begin{eqnarray}\displaystyle E_{1}^{i,j} & = & \displaystyle H^{i}(\mathfrak{m}^{\bullet ,j}/\mathfrak{m}^{\bullet ,j+1})\;\Longrightarrow \;H^{i}(\mathfrak{m}^{\bullet })\end{eqnarray}$$

(5.13) $$\begin{eqnarray}\displaystyle E_{1}^{i,j,\times } & = & \displaystyle H^{i}(\mathfrak{m}^{\bullet ,j}/\mathfrak{m}^{\bullet ,j+1})\;\Longrightarrow \;H^{i}(1+\mathfrak{m}^{\bullet }).\end{eqnarray}$$

The $E_{1}$ -terms are identified. We will denote the differentials at the $k$ th page by $d_{k}$ and $d_{k}^{\times }$ , respectively.

Using the truncated exponential, we obtain isomorphisms of filtered cosimplicial abelian groups

$$\begin{eqnarray}\mathfrak{m}^{\bullet }/\mathfrak{m}^{\bullet ,p}\simeq (1+\mathfrak{m}^{\bullet })/(1+\mathfrak{m}^{\bullet ,p}),\quad \mathfrak{m}^{\bullet ,2}\simeq 1+\mathfrak{m}^{\bullet ,2}\end{eqnarray}$$

inducing the above isomorphism on the associated graded modules.

The above construction yields the following result.

The most important goal is to obtain a description of the first ‘unstable’ differential $d_{p-1}$ (in the present setup of nonunital algebras with $(p+1)$ -fold products zero, it is also the last differential in the spectral sequence).

Let $P^{\bullet }$ be a cosimplicial $\mathbb{Z}[1/(p-1)!]$ -module. Then we can consider the cosimplicial $\mathbb{Z}[1/(p-1)!]$ -module $\operatorname{Sym}^{p}P^{\bullet }$ obtained by applying the $p$ th symmetric power levelwise. We construct an operation

$$\begin{eqnarray}H^{i}(P^{\bullet })\rightarrow H^{i+1}(\operatorname{Sym}^{p}P^{\bullet }).\end{eqnarray}$$

Construction 5.10. Let $x\in P^{i}$ . We define

$$\begin{eqnarray}\unicode[STIX]{x1D719}(x)\stackrel{\text{def}}{=}\unicode[STIX]{x1D707}_{p}(\unicode[STIX]{x2202}^{0}x,\unicode[STIX]{x2202}^{1}x,\unicode[STIX]{x2202}^{2}x,\ldots ,\unicode[STIX]{x2202}^{i}x)\in \operatorname{Sym}^{p}P^{i+1}.\end{eqnarray}$$

Here the $\unicode[STIX]{x2202}^{k}$ are the cosimplicial operators. We claim that:

1. (i) if $x$ is a cocycle in $P^{i}$ , then $\unicode[STIX]{x1D719}(x)$ is a cocycle in $\operatorname{Sym}^{p}P^{i+1}$ ;

2. (ii) if $x,y\in P^{i}$ are cohomologous cocycles, then $\unicode[STIX]{x1D719}(x)$ and $\unicode[STIX]{x1D719}(y)$ are cohomologous in $\operatorname{Sym}^{p}P^{i+1}$ .

The proofs of both these claims will be contained in Proposition 5.11 below. As a result, we will also abuse notation and use $\unicode[STIX]{x1D719}$ to denote the induced map in cohomology.

Return to the previous spectral sequences for computing the cohomology of the cosimplicial abelian groups $\mathfrak{m}^{\bullet }$ and $1+\mathfrak{m}^{\bullet }$ . The $(p-1)$ th differential in each spectral sequence, namely $d_{p-1}$ or $d_{p-1,\times }$ , runs from a sub-object of $H^{i}(\mathfrak{m}^{\bullet }/\mathfrak{m}^{\bullet ,2})$ to a quotient of $H^{i+1}(\mathfrak{m}^{\bullet ,p})$ .

Let us explain this formula. We have a natural (multiplication) morphism of cosimplicial abelian groups $\operatorname{Sym}^{p}(\mathfrak{m}^{\bullet }/\mathfrak{m}^{\bullet ,2})\rightarrow \mathfrak{m}^{\bullet ,p}$ . The operator $\unicode[STIX]{x1D719}$ gives us a natural map from $H^{i}(\mathfrak{m}^{\bullet }/\mathfrak{m}^{\bullet ,2})$ to $H^{i+1}(\operatorname{Sym}^{p}(\mathfrak{m}^{\bullet }/\mathfrak{m}^{\bullet ,2}))$ and we compose that with the multiplication map.

Proof. We unwind the definitions. A class $x\in H^{i}(\mathfrak{m}^{\bullet }/\mathfrak{m}^{\bullet ,2})$ that has survived to the $E_{p-1}$ stage of the spectral sequence, by definition, lifts to a class in $H^{i}(\mathfrak{m}^{\bullet }/\mathfrak{m}^{\bullet ,p})$ , which we represent by a degree $i$ cycle  $\widetilde{x}$ . The short exact sequence of cosimplicial abelian groups

$$\begin{eqnarray}0\rightarrow \mathfrak{m}^{\bullet ,p}\rightarrow \mathfrak{m}^{\bullet }\rightarrow \mathfrak{m}^{\bullet }/\mathfrak{m}^{\bullet ,p}\rightarrow 0,\end{eqnarray}$$

leads to a connecting homomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D6FF}:H^{i}(\mathfrak{m}^{\bullet }/\mathfrak{m}^{\bullet ,p})\rightarrow H^{i+1}(\mathfrak{m}^{\bullet ,p}).\end{eqnarray}$$

The differential $d_{p-1}(x)$ is given by the image of $\unicode[STIX]{x1D6FF}(\widetilde{x})$ .

In the multiplicative setting, we know that the cycle $\exp _{p}(\widetilde{x})\in (1+\mathfrak{m}^{\bullet })/(1+\mathfrak{m}^{\bullet ,p})$ lifts the image of $x$ (or rather, $1+x$ ) in the $E_{1}$ -page of the multiplicative spectral sequence. Now $d_{p-1,\times }(x)$ is given as follows: one considers the short exact sequence of cosimplicial abelian groups

$$\begin{eqnarray}0\rightarrow (1+\mathfrak{m}^{\bullet ,p})\rightarrow (1+\mathfrak{m}^{\bullet })\rightarrow (1+\mathfrak{m}^{\bullet })/(1+\mathfrak{m}^{\bullet ,p})\rightarrow 0,\end{eqnarray}$$

and the induced connecting homomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{\times }:H^{i}((1+\mathfrak{m}^{\bullet })/(1+\mathfrak{m}^{\bullet ,p}))\rightarrow H^{i+1}(1+\mathfrak{m}^{\bullet ,p}).\end{eqnarray}$$

Unwinding the definitions, the multiplicative differential $d_{p-1,\times }(x)$ is given by

$$\begin{eqnarray}d_{p-1,\times }(x)=\unicode[STIX]{x1D6FF}_{\times }(\exp _{p}(\widetilde{x}))\in H^{i+1}(1+\mathfrak{m}^{\bullet ,p})\simeq H^{i+1}(\mathfrak{m}^{\bullet ,p}),\end{eqnarray}$$

or rather the image in the quotient of this at the $E_{p-1}$ -page.

It follows now that we need to compare the additive and the multiplicative boundary maps $\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FF}^{\times }$ . Given a cycle $\widetilde{x}\in \mathfrak{m}^{\bullet }/\mathfrak{m}^{\bullet ,p}$ in degree  $i$ , projecting to a cycle and induced cohomology class $x\in \mathfrak{m}^{\bullet }/\mathfrak{m}^{\bullet ,2}$ , we claim that

(5.14) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{\times }(\exp _{p}(\widetilde{x}))=1+\unicode[STIX]{x1D6FF}(\widetilde{x})+\unicode[STIX]{x1D719}(\widetilde{x})\in H^{i+1}(1+\mathfrak{m}^{\bullet ,p})\simeq H^{i+1}(\mathfrak{m}^{\bullet ,p}).\end{eqnarray}$$

To prove this, we unwind the definitions again. Lift $\widetilde{x}$ to a chain $x^{\prime }\in \mathfrak{m}^{\bullet }$ (in degree  $i$ ). The additive connecting homomorphism $\unicode[STIX]{x1D6FF}(\widetilde{x})$ is obtained by considering the image in cohomology of the alternating sum $y^{\prime }=\sum _{k}(-1)^{k}\unicode[STIX]{x2202}^{k}x^{\prime }$ , which is an $(i+1)$ -cocycle in $\mathfrak{m}^{\bullet ,p}$ . For the multiplicative version, we have to consider the cocycle

$$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{\times }(\exp _{p}(\widetilde{x}))=\mathop{\prod }_{k}(\unicode[STIX]{x2202}^{k}\exp _{p}(x^{\prime }))^{(-1)^{k}}=\mathop{\prod }_{k}\exp _{p}(\unicode[STIX]{x2202}^{k}x^{\prime })^{(-1)^{k}}\in 1+\mathfrak{m}^{\bullet ,p+1}.\end{eqnarray}$$

Now we use (5.10), which gives us

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}_{\times }(\exp _{p}(\widetilde{x})) & = & \displaystyle \exp _{p}\Bigl(\sum (-1)^{k}\unicode[STIX]{x2202}^{k}x^{\prime }\Bigr)+\unicode[STIX]{x1D707}_{p}(\unicode[STIX]{x2202}^{0}x^{\prime },\unicode[STIX]{x2202}^{1}x^{\prime },\ldots ,\unicode[STIX]{x2202}^{i}x^{\prime })\nonumber\\ \displaystyle & = & \displaystyle \exp _{p}(y^{\prime })+\unicode[STIX]{x1D707}_{p}(\unicode[STIX]{x2202}^{0}x^{\prime },\unicode[STIX]{x2202}^{1}x^{\prime },\ldots ,\unicode[STIX]{x2202}^{i}x^{\prime })\nonumber\\ \displaystyle & = & \displaystyle 1+y^{\prime }+\unicode[STIX]{x1D707}_{p}(\unicode[STIX]{x2202}^{0}x^{\prime },\unicode[STIX]{x2202}^{1}x^{\prime },\ldots ,\unicode[STIX]{x2202}^{i}x^{\prime }).\nonumber\end{eqnarray}$$

In cohomology, this is the desired claim, provided we can justify the claims about $\unicode[STIX]{x1D719}$ made in Construction 5.10.

The justification of Construction 5.10 now follows from reversing the above argument. Keep the notation from there. We consider the commutative nonunital cosimplicial algebra

$$\begin{eqnarray}A^{\bullet }=\bigoplus _{0<k\leqslant p}\operatorname{Sym}^{k}P^{\bullet }\end{eqnarray}$$

and its quotient $\overline{A}^{\bullet }=\bigoplus _{0<k<p}\operatorname{Sym}^{k}P^{\bullet }$ . Choose a cycle $x\in P^{\bullet }\subset \overline{A}^{\bullet }$ , which clearly extends to a cycle in $A^{\bullet }$ . Then $\exp _{p}(x)$ is a cycle in $1+\overline{A}^{\bullet }$ . The coboundary of this class in $\operatorname{Sym}^{p}P^{\bullet }\simeq 1+\operatorname{Sym}^{p}P^{\bullet }\subset 1+A^{\bullet }$ is given by $\unicode[STIX]{x1D719}(x)$ by the above analysis. This shows that $\unicode[STIX]{x1D719}(x)$ is a cycle and that its cohomology class depends only on that of  $x$ .◻

5.4 Identification with $\unicode[STIX]{x1D6FD}{\mathcal{P}}^{0}$

Let $P^{\bullet }$ be a cosimplicial $\mathbb{Z}[1/(p-1)!]$ -module. In the previous subsection, we described a natural operation (based on a formula)

(5.15) $$\begin{eqnarray}\unicode[STIX]{x1D719}:H^{i}(P^{\bullet })\rightarrow H^{i+1}(\operatorname{Sym}^{p}P^{\bullet }).\end{eqnarray}$$

Suppose $i>0$ .Footnote ⁴ In the setting of cosimplicial $\mathbb{F}_{p}$ -vector spaces, the work of Priddy [Reference PriddyPri73] has classified all such operations, which amount to determining the cohomology of the symmetric powers of a cosimplicial vector space (one reduces to an Eilenberg–MacLane complex). In particular, for cosimplicial $\mathbb{Z}[1/(p-1)!]$ -modules, the only such natural operations are the scalar multiples of a single operation called $\unicode[STIX]{x1D6FD}{\mathcal{P}}^{0}$ , as one sees from Priddy’s work (and the Bockstein spectral sequence). In other words, in the ‘universal’ case when $H^{\ast }(P^{\bullet })$ is a $\mathbb{Z}[1/(p-1)!]$ (generated by  $\unicode[STIX]{x1D704}$ ) concentrated in degree  $i$ , we have $H^{i+1}(\operatorname{Sym}^{p}P^{\bullet })\simeq \mathbb{Z}/p$ , generated by the class $\unicode[STIX]{x1D6FD}{\mathcal{P}}^{0}\unicode[STIX]{x1D704}$ .

Given a cosimplicial $\mathbb{F}_{p}$ -vector space $P^{\bullet }$ , the cosimplicial commutative ring $\operatorname{Sym}^{\ast }P^{\bullet }$ yields (upon taking the totalization) an $\mathbf{E}_{\infty }$ -algebra over $\mathbb{F}_{p}$ . Then $\unicode[STIX]{x1D6FD}{\mathcal{P}}^{0}$ is identified with the power operation on $\mathbf{E}_{\infty }$ -algebras over $\mathbb{F}_{p}$ with the same name.

In this section, we will prove the following proposition.

Proposition 5.12. For cosimplicial $\mathbb{Z}[1/(p-1)!]$ -modules, we have $\unicode[STIX]{x1D719}=-\unicode[STIX]{x1D6FD}{\mathcal{P}}^{0}$ . As a result, in the situation of Proposition 5.11, we have

(5.16) $$\begin{eqnarray}d_{p-1,\times \!}=d_{p-1}-\unicode[STIX]{x1D6FD}{\mathcal{P}}^{0}.\end{eqnarray}$$

While one can compare explicit formulas, we will approach the determination of the operator $\unicode[STIX]{x1D719}$ by considering examples. We note that this formula will not be used in the following. Rather, it will suffice to know that $\unicode[STIX]{x1D719}$ is Frobenius-semilinear.

First of all, we give an example to show that $\unicode[STIX]{x1D719}$ is not identically zero, i.e. that the additive and multiplicative versions of $d_{p-1}$ can, in fact, differ.

Example 5.13. Consider the commutative algebra $\mathbb{F}_{p}[x]/x^{p+1}$ and the ideal $\mathfrak{m}=(x)$ , so that $\mathfrak{m}^{p+1}=0$ . We then have an isomorphism

$$\begin{eqnarray}(1+\mathfrak{m})/(1+\mathfrak{m}^{p})\simeq (\mathbb{Z}/p)^{p-1},\end{eqnarray}$$

as one sees via the truncated exponential and logarithm. However, we also have

$$\begin{eqnarray}1+\mathfrak{m}\simeq \mathbb{Z}/p^{2}\oplus (\mathbb{Z}/p)^{p-2}.\end{eqnarray}$$

In fact, the unit $1+x\in 1+\mathfrak{m}$ has order exactly  $p^{2}$ . Meanwhile, $\exp _{p}(x^{2}),\ldots ,\exp _{p}(x^{p-1})$ each have order  $p$ (one could also take $1+x^{2},\ldots ,1+x^{p-1}$ ). One sees that the map

$$\begin{eqnarray}1+\mathfrak{m}\rightarrow (1+\mathfrak{m})/(1+\mathfrak{m}^{p})\end{eqnarray}$$

can be identified with the direct sum of the reduction map $\mathbb{Z}/p^{2}\rightarrow \mathbb{Z}/p$ and the identity $(\mathbb{Z}/p)^{p-2}\rightarrow (\mathbb{Z}/p)^{p-2}$ .

Now let $X_{\bullet }$ be a simplicial set and suppose that there exists a class in $H^{n}(X_{\bullet },\mathbb{Z}/p)$ which does not lift to $H^{n}(X_{\bullet },\mathbb{Z}/p^{2})$ . Form the nonunital cosimplicial commutative algebra $F(X_{\bullet },\mathfrak{m})=\mathfrak{m}^{X_{\bullet }}$ , whose group of units is $F(X_{\bullet },1+\mathfrak{m})$ . We can apply the machinery of the previous subsection to this example, where the filtration in question is the $\mathfrak{m}$ -adic filtration.

The additive spectral sequence converges to the cohomology of $X_{\bullet }$ with coefficients in $(\mathbb{Z}/p)^{p}$ and the multiplicative spectral sequence converges to the cohomology with coefficients in $\mathbb{Z}/p^{2}\oplus (\mathbb{Z}/p)^{p-2}$ . We see in particular that, while there are no differentials additively (the filtration splits), there is necessarily a $d_{p-1,\times }$ since there are cohomology classes in $H^{n}(X_{\bullet },\mathbb{Z}/p)$ which do not lift to $H^{n}(X_{\bullet },\mathbb{Z}/p^{2})$ .

We now describe an example where the additive spectral sequence supports a differential but the multiplicative spectral sequence does not.

Proof of Proposition 5.12.

We will analyze the last example more carefully to determine the precise multiple.Footnote ⁵ Note that $\overline{\mathfrak{m}}/\overline{\mathfrak{m}}^{2}\simeq \mathbb{F}_{p}$ (generated by the image of $1-\unicode[STIX]{x1D701}_{p}$ ) and $\overline{\mathfrak{m}}^{p}\simeq \mathbb{F}_{p}$ (generated by the image of $(1-\unicode[STIX]{x1D701}_{p})^{p}$ ). The (additive) differential $d_{p-1}$ comes from the short exact sequence

(5.17) $$\begin{eqnarray}0\rightarrow \overline{\mathfrak{m}}^{p}\rightarrow \overline{\mathfrak{m}}\rightarrow \overline{\mathfrak{m}}/\overline{\mathfrak{m}}^{p}\rightarrow 0,\end{eqnarray}$$

and we have the following commutative diagram of short exact sequences.

(5.18)

Here the map $\mathbb{Z}/p^{2}\rightarrow \overline{\mathfrak{m}}$ carries $1\mapsto 1-\unicode[STIX]{x1D701}_{p}$ . Note that $p(1-\unicode[STIX]{x1D701}_{p})\equiv (1-\unicode[STIX]{x1D701}_{p})^{p}$ in $\overline{\mathfrak{m}}$ as one sees from the expression $\prod _{i=1}^{p-1}(T-\unicode[STIX]{x1D701}_{p}^{i})=T^{p-1}+T^{p-2}+\cdots +1$ with $T=1$ .

Take $X_{\bullet }=K(\mathbb{Z}/p,n)$ , so that $H^{n}(X_{\bullet },\mathbb{F}_{p})=H^{n+1}(X_{\bullet },\mathbb{F}_{p})=\mathbb{F}_{p}$ . We let $\unicode[STIX]{x1D704}$ be a generator of the former, so that $\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D704}$ is a generator of the latter for $\unicode[STIX]{x1D6FD}$ the Bockstein. Again, this analysis is based on the cosimplicial commutative ring $\overline{\mathfrak{m}}^{X_{\bullet }}$ .

1. (i) Note that $H^{n}(X_{\bullet },\overline{\mathfrak{m}}/\overline{\mathfrak{m}}^{p})\simeq H^{n}(X_{\bullet },\mathbb{F}_{p})\otimes _{\mathbb{F}_{p}}\overline{\mathfrak{m}}/\overline{\mathfrak{m}}^{p}\simeq \overline{\mathfrak{m}}/\overline{\mathfrak{m}}^{p}\unicode[STIX]{x1D704}$ and similarly for $H^{n+1}(X_{\bullet },\overline{\mathfrak{m}}^{p})$ . The differential in question comes from the connecting homomorphism in the short exact sequence (5.17), so that by naturality of (5.18), it carries $(1-\unicode[STIX]{x1D701}_{p})\unicode[STIX]{x1D704}$ to $(1-\unicode[STIX]{x1D701}_{p})^{p}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D704}\in H^{n+1}(X_{\bullet },\overline{\mathfrak{m}^{p}})$ .

2. (ii) The associated graded module of $\overline{\mathfrak{m}}$ comes from the reduction of a $\mathbb{Z}$ -algebra, so $\unicode[STIX]{x1D6FD}{\mathcal{P}}^{0}$ is the composite of the Bockstein and ${\mathcal{P}}^{0}$ . Now ${\mathcal{P}}^{0}((1-\unicode[STIX]{x1D701}_{p})\unicode[STIX]{x1D704})=(1-\unicode[STIX]{x1D701}_{p})^{p}\unicode[STIX]{x1D704}\in H^{n}(X,\overline{\mathfrak{m}}^{p})$ since ${\mathcal{P}}^{0}$ is the Frobenius on underlying rings. Applying $\unicode[STIX]{x1D6FD}$ to that, we obtain $(1-\unicode[STIX]{x1D701}_{p})^{p}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D704}$ .

Putting these together, it follows that if the class corresponding to $(1-\unicode[STIX]{x1D701}_{p})\unicode[STIX]{x1D704}$ is to survive in the multiplicative spectral sequence (as it must), the formula must be as desired.◻

We refer to [Reference MumfordMum66, Lecture 27] for an instance of the appearance of these Bockstein operators in the deformation theory of the Picard scheme in characteristic  $p$ . In that case, there are no algebraic obstructions, but the multiplicative operations are controlled by the Bocksteins.