injection $\operatorname{Lev}_{p}\circ\operatorname{D}\rightarrow\Pi$, by
checking that the product $\star=(X_{2};d,d)$ and the element $d$ do not
satisfy any other relation in $\Pi$ than $\star(d,d)=d(\star)$. This induces
an injection $T_{q}\operatorname{Lev}_{p}\circ
T_{q}\operatorname{D}\hookrightarrow T_{q}\Pi$, where
$T_{q}\Pi=\operatorname{uCom}\circ T_{q}\operatorname{D}$ is spanned by the
partitions $I$ such that $I_{j}=\emptyset$ for all $j>q$. This allows us to
identify the free $\star$-unstable $T_{q}\operatorname{Lev}_{p}\circ
T_{q}\operatorname{D}$-algebra generated by an element of degree one to the
vector space spanned by the sequences $u$ such that $u_{j}=0$ for all $j>q$
and

$\sum_{j\in\mathbb{N}}\frac{u_{j}}{p^{j}}\in\\{1,\frac{1}{p},\dots,\frac{1}{p^{q}}\\}.$

As a vector space, Theorem 4.6.3 identifies
$\bigoplus_{i=0}^{q}J^{\prime}(2p^{i})$ to the vector subspace of $J^{\prime}$
spanned by

$\\{x_{0}^{n_{0}}\dots
x_{q}^{n_{q}}:\sum_{i=0}^{q}n_{i}2p^{i}\in\\{2,2p,\dots,2p^{q}\\}\\}_{0\leq
i\leq q}.$

We define a linear bijection

$K_{T_{q}\operatorname{Lev}_{p}\circ
T_{q}\operatorname{D}}^{\star}(F^{\prime}(2))\rightarrow\bigoplus_{i=0}^{q}J^{\prime}(2p^{i})$

which identifies a partition $I$ such that $I_{j}=\emptyset$ for all $j>q$ and
$\sum_{j\in\mathbb{N}}\frac{|I_{j}|}{p^{j}}\in\\{1,\frac{1}{p},\dots,\frac{1}{p^{q}}\\}$
to the monomial $\prod_{j=0}^{q}x_{j}^{u_{q-j}}$. We readily check that the
actions of $T_{q}\operatorname{Lev}$, $T_{q}\operatorname{D}$ and
$\mathcal{A}^{\prime}$ are compatible with this bijection.

∎

### Morphisms of operads, and links between associated unstable algebras

Recall from Proposition 4.3.9 and Proposition 4.4.7 that any morphism
$f\colon\star_{\mathcal{P}}\rightarrow\star_{\mathcal{Q}}$ in
$(\operatorname{MagCom}_{p}\downarrow\operatorname{Op})$ induces a restriction
functor
$f^{*}\colon\mathcal{K}_{\mathcal{Q}}^{\star_{\mathcal{Q}}}\rightarrow\mathcal{K}_{\mathcal{P}}^{\star_{\mathcal{P}}}$,
and a morphism $f_{*}\colon
K_{\mathcal{P}}^{\star_{\mathcal{P}}}(M)\rightarrow
f^{*}K_{\mathcal{Q}}^{\star_{\mathcal{Q}}}(M)$ in
$\mathcal{K}_{\mathcal{P}}^{\star}$, natural with respect to the unstable
module $M$.

######  Notation 4.6.12.

By abuse of notation, we drop the restriction functor and denote by
$K_{\mathcal{Q}}^{\star_{\mathcal{Q}}}(M)$ the $\star_{\mathcal{P}}$-unstable
algebra $f^{*}K_{\mathcal{Q}}^{\star_{\mathcal{Q}}}(M)$.

######  Example 4.6.13.

The monomorphism
$\operatorname{Lev}_{p}\rightarrow\operatorname{uCom}\circ\operatorname{D^{\pm}}$
mapping $\star\in\operatorname{Lev}_{p}(p)^{\mathfrak{S}_{p}}$ to the
operation
$d^{X_{p}}\in(\operatorname{uCom}\circ\operatorname{D^{\pm}})(p)^{\mathfrak{S}_{p}}$
induces a monomorphism of unstable $p$-level algebras

$K_{\operatorname{Lev}_{p}}^{\star}(F^{\prime}(2))\rightarrow
K_{\operatorname{uCom}\circ\operatorname{D^{\pm}}}^{d^{X_{p}}}(F^{\prime}(2)).$

This morphism corresponds to the inclusion $K^{\prime}(2)\hookrightarrow
K^{\prime}$ of the Carlsson module of weight two into the Carlsson algebra.

######  Proposition 4.6.14.

The Brown–Gitler module of weight $2p^{q}$, $J^{\prime}(2p^{q})$, with the
multiplication given by the internal $p$-ary operation, is the free
$\star$-unstable $T_{q}\operatorname{Lev}_{p}$-algebra generated by
$F^{\prime}(2)$.

###### Proof.

The injective morphism
$T_{q}\operatorname{Lev}_{p}\hookrightarrow\operatorname{uCom}\circ
T_{q}\operatorname{D}$ induces a monomorphism of $\star$-unstable
$T_{q}\operatorname{Lev}_{p}$-algebras

$K_{T_{q}\operatorname{Lev}_{p}}^{\star}(F^{\prime}(2))\rightarrow
K_{\operatorname{uCom}\circ
T_{q}\operatorname{D}}^{d^{X_{p}}}(F^{\prime}(2)).$

This morphism corresponds to the inclusion of the free
$T_{q}\operatorname{Lev}_{p}$-algebra generated by $x_{q}$ as a subalgebra of
$\bigoplus_{i=1}^{2p^{q}}J^{\prime}(i)$. It is easy to show that the image of
this inclusion is $J^{\prime}(2p^{q})$ (for example, by adapting the proof of
Proposition 4.6.10). ∎

######  Proposition 4.6.15.

The truncation tower of $\operatorname{Lev}_{p}$ induces the following
cofiltration of $K^{\prime}(2)$ in the category of unstable level algebras:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
13.41237pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{J^{\prime}(2)}$}}}}}}}{\hbox{\kern 37.41237pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{J^{\prime}(2p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
18.9648pt\raise 6.0361pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-3.0361pt\hbox{$\scriptstyle{\cdot
P^{1}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 13.41238pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
93.26836pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\cdots}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
69.26837pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
130.76836pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{J^{\prime}(2p^{q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
106.76836pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
189.32518pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{J^{\prime}(2p^{q+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces)}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
170.63078pt\raise 5.99445pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.99445pt\hbox{$\scriptstyle{\cdot P^{p^{q}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 165.3252pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
255.03758pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
231.03758pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>
###### Proof.

The operad $\operatorname{Lev}_{p}$ is the limit of the diagram:

$\textstyle{T_{0}\operatorname{Lev}_{p}}$$\textstyle{T_{1}\operatorname{Lev}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{q}\operatorname{Lev}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{q+1}\operatorname{Lev}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where the morphisms $T_{q+1}\operatorname{Lev}_{p}\rightarrow
T_{q}\operatorname{Lev}_{p}$ are the operadic projections. Following
Proposition 4.4.7, the previous diagram induces a diagram:

$\textstyle{K_{T_{0}\operatorname{Lev}_{p}}^{\star}(F^{\prime}(2))}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{T_{q}\operatorname{Lev}_{p}}^{\star}(F^{\prime}(2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{T_{q+1}\operatorname{Lev}_{p}}^{\star}(F^{\prime}(2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

in the category of unstable level algebras. For all $q\in\mathbb{N}$,
Proposition 4.6.14 gives the isomorphism
$K_{T_{q}\operatorname{Lev}_{p}}^{\star}(F^{\prime}(2))\cong
J^{\prime}(2p^{q})$. It is easy to check that the morphism

$K_{T_{q+1}\operatorname{Lev}_{p}}^{\star}(F^{\prime}(2))\rightarrow
K_{T_{q}\operatorname{Lev}_{p}}^{\star}(F^{\prime}(2))$

induced by the operadic projection corresponds to $\cdot P^{p^{q}}\colon
J^{\prime}(2p^{q+1})\rightarrow J^{\prime}(2p^{q})$. Indeed, $\cdot P^{p^{q}}$
is compatible with the internal products on $J(2p^{q+1})$ and $J(2p^{q})$, and
maps $x_{q+1}$ to $x_{q}$. The unstable module $K^{\prime}(2)$ is defined as
the limit of the $J^{\prime}(2p^{q})$’s under the morphisms $\cdot P^{p^{q}}$,
and its internal product is given as the limit of the internal products of the
$J^{\prime}(2p^{q})$’s. Hence, $K^{\prime}(2)$ with its internal product is
the limit of the desired diagram in the category of unstable $p$-level
algebras. ∎

##  5\. Cyclic twists and a Campbell and Selick Theorem

In this section, we review some results obtained by Campbell and Selick in
[6], that we adapt to our study of free unstable algebras over an operad. In
[6], the authors produce a decomposition of the cohomology of (the classifying
space of) simple abelian $p$-groups into injective unstable modules. More
precisely, they show that one can twist the action of the Steenrod algebra on
this cohomology, obtaining an algebra over the Steenrod algebra (unstable as
an $\mathcal{A}$-module but not as an algebra) which then splits into
injective unstable modules.

Here, we see the polynomial part of the cohomology of
$B\left(\mathbb{Z}/p\mathbb{Z}^{\times s}\right)$ as a free unstable
$\operatorname{uCom}$-algebra generated by the free object
$F^{\prime}(2)^{\oplus s}$ of $\mathcal{U}^{\prime}$ on $s$ generators of
degree 2. The main result of this section is Theorem 5.2.1, which generalises
Theorem 1 of [6], replacing $\operatorname{uCom}$ by any operad $\mathcal{P}$
equipped with a central operation $\star$, and replacing the module
$F^{\prime}(2)$ by any object $N$ of $\mathcal{U}^{\prime}$, with mild
conditions on the dimensions of $N$ and $\mathcal{P}$. We also study the
decompositions obtained in the same way as those of [6], and we investigate
the occurrences in which the Carlsson module $K^{\prime}(2)$ appears.

The main part of this section, including the proof of Theorem 5.2.1, is
inspired by the techniques used by Duflot, Kuhn and Winstead in [14].

From now on, $p>0$ can be any prime number, including the case $p=2$. In the
case $p=2$, all degrees are divided by $2$. We use the notation introduced in
Section 4.4 for free unstable algebras over an operad.

###  5.1. Scalar extension of cyclic twists

Recall that $Q_{s}\operatorname{D}$ denotes the polynomial algebra on one
indeterminate $d$ satisfying $d^{s}=1$. A $Q_{s}\operatorname{D}$-algebra in
$\mathcal{U}^{\prime}$ is an unstable module endowed with a cyclic
automorphism of order $s$ compatible with the action of
$\mathcal{A}^{\prime}$. The aim of this section is to show that any
$Q_{s}\operatorname{D}$-algebra in $\mathcal{U}^{\prime}$ is isomorphic to the
$Q_{s}\operatorname{D}$-algebra with same underlying object of
$\mathcal{U}^{\prime}$ endowed with a trivial action of
$Q_{s}\operatorname{D}$. Moreover, if the considered
$Q_{s}\operatorname{D}$-algebra is also endowed with an action of an operad
$\mathcal{P}$, making it into a $\mathcal{P}\circ
Q_{s}\operatorname{D}$-algebra, then this isomorphism is compatible with the
action of $\mathcal{P}$. Note that we only obtain these results after
extending the scalars to $\mathbb{F}_{p^{s}}$, endowing $\mathbb{F}_{p^{s}}$
with a cyclic action of $Q_{s}\operatorname{D}$ induced by the Frobenius map.

For this section, we follow the first half of Section 4 of [14]. The algebra
$Q_{s}\operatorname{D}$ is identified with the group ring of the Gallois group
$\operatorname{Gal}(\mathbb{F}_{p^{s}}/\mathbb{F}_{p})$.

Denote by $\phi:\mathbb{F}_{p^{s}}\rightarrow\mathbb{F}_{p^{s}}$ the Frobenius
automorphism. We fix a primitive element $\omega\in\mathbb{F}_{p^{s}}$. We
obtain an $\mathbb{F}$-basis $\\{\omega,\phi\omega,\dots,\phi^{s-1}\omega\\}$
of $\mathbb{F}_{p^{s}}$. We endow $\mathbb{F}_{p^{s}}$ with a
$Q_{s}\operatorname{D}$-action such that $d$ acts on $\mathbb{F}_{p^{s}}$
through $\phi^{-1}$. As such, $\mathbb{F}_{p^{s}}$ is the free
$Q_{s}\operatorname{D}$-algebra generated by $\omega$. We consider
$\mathbb{F}_{p^{s}}$ as an object of $\mathcal{U}^{\prime}$ concentrated in
degree zero.

######  Notation 5.1.1.

In this section, we denote by $N$ a $Q_{s}\operatorname{D}$-algebra in
$\mathcal{U}^{\prime}$. Denote by $N_{\operatorname{triv}}$ the
$Q_{s}\operatorname{D}$-algebra in $\mathcal{U}^{\prime}$ obtained by
adjoining a trivial action of $Q_{s}\operatorname{D}$ to the underlying
unstable module of $N$.

######  Remark 5.1.2.

The action of $\mathcal{A}^{\prime}$ on $\mathbb{F}_{p^{s}}\otimes N$ is given
by $\theta(\lambda\otimes x)=\lambda\otimes\theta x$ for all
$\theta\in\mathcal{A}^{\prime}$. The action of $Q_{s}\operatorname{D}$ on
$\mathbb{F}_{p^{s}}\otimes N$ is the diagonal action, $d(\lambda\otimes
x)=\phi^{-1}\lambda\otimes dx$.

Define a linear map $\beta:\mathbb{F}_{p^{s}}\otimes
N_{\operatorname{triv}}\rightarrow\mathbb{F}_{p^{s}}\otimes N$ by setting

$\beta(\phi^{i}\omega\otimes x)=\phi^{i}\omega\otimes d^{-i}x.$

######  Lemma 5.1.3.

The map $\beta:\mathbb{F}_{p^{s}}\otimes
N_{\operatorname{triv}}\rightarrow\mathbb{F}_{p^{s}}\otimes N$ is an
isomorphism of $Q_{s}\operatorname{D}$-algebras in $\mathcal{U}^{\prime}$.

###### Proof.

The linear map $\beta$ is clearly compatible with the action of
$\mathcal{A}^{\prime}$ and with $d$. The inverse of $\beta$ is given by:

$\phi^{i}\omega\otimes x\mapsto\phi^{i}\omega\otimes d^{i}x.$

∎

We can consider the submodules of fixed points under the action of $d$. Taking
the fixed points map yields an isomorphism in $\mathcal{U}^{\prime}$:

$\beta^{Q_{s}\operatorname{D}}:N_{\operatorname{triv}}\rightarrow(\mathbb{F}_{p^{s}}\otimes
N)^{Q_{s}\operatorname{D}}.$

Explicitely, there is an isomorphism
$N_{\operatorname{triv}}\rightarrow(\mathbb{F}_{p^{s}}\otimes
N_{\operatorname{triv}})^{Q_{s}\operatorname{D}}$ sending $x$ to
$\sum_{i=0}^{s-1}\phi^{i}\omega\otimes x$, and:

$\beta^{Q_{s}\operatorname{D}}(x)=\sum_{i=0}^{s-1}\phi^{i}\omega\otimes
d^{-i}x$

######  Notation 5.1.4.

We denote by $N_{\operatorname{inv}}=(\mathbb{F}_{p^{s}}\otimes
N)^{Q_{s}\operatorname{D}}$.

Suppose now that $N$ is endowed with an action of $\mathcal{P}$ that makes it
into a $\mathcal{P}\circ Q_{s}\operatorname{D}$-algebra in
$\mathcal{U}^{\prime}$. We endow $\mathbb{F}_{p^{s}}\otimes N$ with a
structure of $\mathcal{P}\circ Q_{s}\operatorname{D}$-algebra in
$\mathcal{U}^{\prime}$ by extension of scalars, that is, for all
$\mu\in\mathcal{P}(k)$, $\lambda_{1},\dots,\lambda_{k}\in\mathbb{F}_{p^{s}}$,
$x_{1},\dots,x_{k}\in N^{k}$,

$\mu(\lambda_{1}\otimes x_{1},\dots,\lambda_{k}\otimes
x_{k})=\lambda_{1}\cdots\lambda_{k}\otimes\mu(x_{1},\dots,x_{k}).$

This induces a structure of $\mathcal{P}\circ Q_{s}\operatorname{D}$-algebra
in $\mathcal{U}^{\prime}$ on $N_{\operatorname{inv}}$. Note that
$\beta^{Q_{s}\operatorname{D}}$ is not a $\mathcal{P}$-algebra map. However,
extending the inclusion
$N_{\operatorname{inv}}\hookrightarrow\mathbb{F}_{p^{s}}\otimes N$ yields a
$\mathcal{P}$-algebra map $\tilde{\iota}\colon\mathbb{F}_{p^{s}}\otimes
N_{\operatorname{inv}}\rightarrow\mathbb{F}_{p^{s}}\otimes N$ and we get

######  Lemma 5.1.5.

The map $\tilde{\iota}\colon\mathbb{F}_{p^{s}}\otimes
N_{\operatorname{inv}}\rightarrow\mathbb{F}_{p^{s}}\otimes N$ is an
isomorphism of $\mathcal{P}\circ Q_{s}\operatorname{D}$-algebras in
$\mathcal{U}^{\prime}$.

###### Proof.

This is a direct consequence of [32], Lemma 3.4. ∎

###  5.2. A Campbell–Selick Theorem

In the preceding section, we have shown how a cyclic twist of a
$\mathcal{P}$-algebra in $\mathcal{U}^{\prime}$ could be ‘straightened’
without perturbing the structure of $\mathcal{P}$-algebra in
$\mathcal{U}^{\prime}$. However, this result only stood after extending the
scalars to $\mathbb{F}_{p^{s}}$. In this section we prove a generalisation of
Theorem 1 of [6], showing that after applying the preceding result, we can
restrict again the scalars to $\mathbb{F}$, in the case where we started with
the cyclic twist of a free $d^{\star p}$-unstable $\mathcal{P}\circ
Q_{s}\operatorname{D}$-algebra. Note that this yields an isomorphism in
$\mathcal{U}^{\prime}$ which is not compatible with the $\mathcal{P}$-algebra
structure.

To prove this result, we use the techniques presented in the preceding section
and inspired by [14], Section 4.

######  Theorem 5.2.1.

Let $\mathcal{P}$ be an operad such that for all $n\in\mathbb{N}$,
$\mathcal{P}(n)$ has finite dimension. Let
$\star\in\mathcal{P}(p)^{\mathfrak{S}_{p}}$ be a $\mathcal{P}$-central
operation, and $M$ a gradually finite reduced and connected object of
$\mathcal{U}^{\prime}$. There is an isomorphism in $\mathcal{U}^{\prime}$:

$K_{\mathcal{P}}^{\star}(M^{\oplus s})\cong K_{\mathcal{P}\circ
Q_{s}\operatorname{D}}^{d^{\star p}}(M).$

######  Remark 5.2.2.

In the case $p=2$, we set $\mathcal{A}^{\prime}=\mathcal{A}$ and
$\mathcal{U}^{\prime}=\mathcal{U}$, and Theorem 4.4.9 must be replaced with
the result of [17]. Under those conventions, our theorem is valid for all
prime $p>0$.

###### Proof.

Since $M$ is connected and reduced, and since $\star$ is
$\mathcal{P}$-central, Theorem 4.4.9 gives a graded $\mathcal{P}$-algebra
isomorphism $\varphi:K_{\mathcal{P}}^{\star}(M^{\oplus s})\rightarrow
S(\mathcal{P},(\Sigma\Omega M)^{\oplus s})$ and a graded $\mathcal{P}\circ
Q_{s}\operatorname{D}$ algebra isomorphism

$\varphi^{\prime}\colon K_{\mathcal{P}\circ Q_{s}\operatorname{D}}^{d^{\star
p}}(M)\rightarrow S(\mathcal{P}\circ Q_{s}\operatorname{D},\Sigma\Omega M).$

Note that the $\mathcal{P}\circ Q_{s}\operatorname{D}$-algebra
$S(\mathcal{P}\circ Q_{s}\operatorname{D},\Sigma\Omega M)$ is isomorphic to
$S(\mathcal{P},\mathbb{F}_{p^{s}}\otimes\Sigma\Omega M)$ with
$Q_{s}\operatorname{D}$ acting on $\mathbb{F}_{p^{s}}$ by means of the
Frobenius automorphism $\phi$. Endowing $S(\mathcal{P},(\Sigma\Omega
M)^{\oplus s})$ with a trivial action of $Q_{s}\operatorname{D}$ yields a
$\mathcal{P}\circ Q_{s}\operatorname{D}$-algebra isomorphic to
$S(\mathcal{P},\mathbb{F}_{p^{s}}\otimes\Sigma\Omega
M)_{\operatorname{triv}}$, where the notation $N_{\operatorname{triv}}$ for a
$Q_{s}\operatorname{D}$-algebra $N$ was introduced in 5.1.1. So, as a
$Q_{s}\operatorname{D}$-algebra in $\mathcal{U}^{\prime}$, we can identify
$K_{\mathcal{P}}^{\star}(M^{\oplus s})$ to $K_{\mathcal{P}\circ
Q_{s}\operatorname{D}}^{d^{\star p}}(M)_{\operatorname{triv}}$.

Lemmas 5.1.3 and 5.1.5 respectively give an isomorphism in
$\mathcal{U}^{\prime}$

$\beta^{Q_{s}\operatorname{D}}\colon K_{\mathcal{P}}^{\star}(M^{\oplus
s})\rightarrow(K_{\mathcal{P}\circ Q_{s}\operatorname{D}}^{d^{\star
p}}(M))_{\operatorname{inv}},$

where the notation $N_{\operatorname{inv}}$ is given in 5.1.4, and an
isomorphism of $\mathcal{P}\circ Q_{s}\operatorname{D}$ algebras in
$\mathcal{U}^{\prime}$

$\tilde{\iota}\colon\mathbb{F}_{p^{s}}\otimes(K_{\mathcal{P}\circ
Q_{s}\operatorname{D}}^{d^{\star
p}}(M))_{\operatorname{inv}}\rightarrow\mathbb{F}_{p^{s}}\otimes
K_{\mathcal{P}\circ Q_{s}\operatorname{D}}^{d^{\star p}}(M).$

Now, since we assumed that $\mathcal{P}(n)$ is of finite dimension for all
$n\in\mathbb{N}$, and since $M$ is gradually finite, the graded vector space
$K_{\mathcal{P}\circ Q_{s}\operatorname{D}}^{d^{\star p}}(M)\cong
S(\mathcal{P},\mathbb{F}_{p^{s}}\otimes M)$ is itself gradually finite, so the
isomorphism $\mathbb{F}_{p^{s}}\otimes\left(K_{\mathcal{P}\circ
Q_{s}\operatorname{D}}^{d^{\star
p}}(M)\right)_{\operatorname{inv}}\cong\mathbb{F}_{p^{s}}\otimes
K_{\mathcal{P}\circ Q_{s}\operatorname{D}}^{d^{\star p}}(M)$ implies that
there is an isomorphism of $\mathcal{P}\circ Q_{s}\operatorname{D}$-algebras
in $\mathcal{U}^{\prime}$ between $\left(K_{\mathcal{P}\circ
Q_{s}\operatorname{D}}^{d^{\star p}}(M)\right)_{\operatorname{inv}}$ and
$K_{\mathcal{P}\circ Q_{s}\operatorname{D}}^{d^{\star p}}(M)$.

Hence, in $\mathcal{U}^{\prime}$,

$K_{\mathcal{P}}^{\star}(M^{\oplus s})\cong K_{\mathcal{P}\circ
Q_{s}\operatorname{D}}^{d^{\star p}}(M).\qed$

###  5.3. Splitting of Campbell and Selick modules

In this section we study a decomposition of the underlying unstable module of
$K_{\mathcal{P}}^{\star}(M^{\oplus s})$ obtained by using 5.2.1 and defining a
grading on $K_{\mathcal{P}\circ Q_{s}\operatorname{D}}^{d^{\star p}}(M)$. We
recall how this technique allowed Campbell and Selick [6] to obtain a
splitting of the cohomology of $B\left(\mathbb{Z}/p\mathbb{Z}^{\times
s}\right)$ into injective modules. We show that the underlying unstable module
of the free unstable $p$-level algebra generated by $s$ elements of degree 2
(of degree 1 in characteristic 2) admits a further decomposition involving
$s$-copies of the Carlsson module of weight two (weight one in characteristic
2).

Here $\mathcal{P}$ designates an operad such that for all $n\in\mathbb{N}$,
$\mathcal{P}(n)$ is finite-dimensional,
$\star\in\mathcal{P}(p)^{\mathfrak{S}_{p}}$ is a $\mathcal{P}$-central
operation, $\omega\in\mathbb{F}_{p^{s}}$ is a primitive element and
$\phi:\mathbb{F}_{p^{s}}\rightarrow\mathbb{F}_{p^{s}}$ is the Frobenius map.

######  Definition 5.3.1.

  * •

We define a $\mathbb{Z}/(p^{s}-1)\mathbb{Z}$-graduation on
$\mathbb{F}_{p^{s}}$, called the weight, by declaring that
$w(\phi^{i}\omega)=p^{i}$. Note that the action of $d=\phi^{-1}$ divides the
weight by $p$.

  * •

Let $M$ be a connected reduced object of $\mathcal{U}^{\prime}$ (where, by
convention, we set $\mathcal{U}^{\prime}=\mathcal{U}$ in the case $p=2$).
Following Theorem 4.4.9, the free $d^{\star p}$-unstable $\mathcal{P}\circ
Q_{s}\operatorname{D}$-algebra $K_{\mathcal{P}\circ
Q_{s}\operatorname{D}}^{d^{\star p}}(M)$ is isomorphic, as a graded
$\mathcal{P}\circ Q_{s}\operatorname{D}$-algebra, to $S(\mathcal{P}\circ
Q_{s}\operatorname{D},\Sigma\Omega M)$. This graded $\mathcal{P}\circ
Q_{s}\operatorname{D}$-algebra is itself isomorphic to
$S(\mathcal{P},\mathbb{F}_{p^{s}}\otimes\Sigma\Omega M)$. Considering
$\mathcal{P}$ and $\Sigma\Omega M$ as concentrated in weight zero, this yields
a $d^{\star p}$-unstable
$\mathcal{P}\circ\mathcal{Q}_{s}\operatorname{D}$-algebra,
$\mathbb{Z}/(p^{s}-1)\mathbb{Z}$-graded as an object of
$\mathcal{U}^{\prime}$, that we denote by:

$Q_{s}M_{\mathcal{P}}^{\star}.$

Indeed, the action of $\mathcal{A}^{\prime}$ is compatible with the weight.

  * •

The sub-object in $\mathcal{U}^{\prime}$ of elements of weight $[j]_{p^{s}-1}$
for $j\in\mathbb{Z}$ is denoted by $Q_{s}M_{\mathcal{P}}^{\star}(j)$.

  * •

For all $s>0$, consider $\mathbb{F}=\mathbb{F}_{p}$ as a $Q_{s}D$-algebra with
trivial $d$. Denote by
$\gamma_{s}\colon\mathbb{F}_{p^{s}}\rightarrow\mathbb{F}$ the unique non-
trivial morphism of $Q_{s}\operatorname{D}$-algebras, sending $\phi^{j}\omega$
to $1$ for all $j\in\\{0,\dots,s-1\\}$.

  * •

For all $s>0$, we extend the map $\gamma_{s}\otimes id_{\Sigma\Omega
M}\colon\mathbb{F}_{p^{s}}\otimes\Sigma\Omega M\rightarrow\Sigma\Omega
M\cong\mathbb{F}\otimes\Sigma\Omega M$ into a $\mathcal{P}\circ
Q_{s}\operatorname{D}$-algebra morphism

$\gamma_{s}\colon Q_{s}M_{\mathcal{P}}^{\star}\rightarrow
K_{\mathcal{P}}^{\star}(M).$

  * •

For all $s>0$ and $w\in\mathbb{Z}$, we denote by

$\gamma_{s;w}\colon Q_{s}M_{\mathcal{P}}^{\star}(w)\rightarrow
K_{\mathcal{P}}^{\star}(M)$

the composite

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
23.1697pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Q_{s}M_{\mathcal{P}}^{\star}(w)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
47.1697pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{Q_{s}M_{\mathcal{P}}^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
86.16736pt\raise 5.1875pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\gamma_{s}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 102.30302pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
102.30302pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{K_{\mathcal{P}}^{\star}(M)}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

######  Lemma 5.3.2.

For all gradually finite, connected and reduced objects $M$ of
$\mathcal{U}^{\prime}$, and all integers $s>0$, there is a decomposition in
$\mathcal{U}^{\prime}$:

$K_{\mathcal{P}}^{\star}(M^{\oplus
s})\cong\bigoplus_{j\in\mathbb{Z}/(p^{s}-1)\mathbb{Z}}Q_{s}M_{\mathcal{P}}^{\star}(j),$

with isomorphisms $Q_{s}M_{\mathcal{P}}^{\star}(j)\cong
Q_{s}M_{\mathcal{P}}^{\star}(pj)$.

###### Proof.

The first affirmation is a direct corollary of Theorem 5.2.1, with the
decomposition described above. The isomorphisms
$Q_{s}M_{\mathcal{P}}^{\star}(j)\cong Q_{s}M_{\mathcal{P}}^{\star}(pj)$ are
induced by the (invertible) action of $d\in Q_{s}\operatorname{D}$. ∎

######  Lemma 5.3.3.

For all $i\in\\{0,\dots,s-1\\}$,
$Q_{s}M_{\operatorname{Lev}_{p}}^{\star}(p^{i})$ admits a direct factor of the
type $K_{\operatorname{Lev}_{p}}^{\star}(M)$.

###### Proof.

There is a unique operad morphism
$f\colon\operatorname{Lev}_{p}\rightarrow\operatorname{Lev}_{p}\circ
Q_{s}\operatorname{D}$ induced by $\star\mapsto d^{\star p}$. Since the action
of $f(\star)$ is compatible with the weight of
$Q_{s}M_{\operatorname{Lev}_{p}}^{\star}$, and since
$Q_{s}M_{\operatorname{Lev}_{p}}^{\star}$ is $f(\star)$-unstable, the object
$Q_{s}M_{\operatorname{Lev}_{p}}^{\star}(p^{i})$ of $\mathcal{U}^{\prime}$ can
be seen as an unstable level algebra. Thus, the map $g\colon M\rightarrow
Q_{s}M_{\operatorname{Lev}_{p}}^{\star}(p^{i})$ in $\mathcal{U}^{\prime}$
sending $x\in M$ to $(1_{\operatorname{Lev}_{p}};y_{i}\otimes x)\in
Q_{s}M_{\operatorname{Lev}_{p}}^{\star}(N)(p^{i})$ can be extended into an
unstable $p$-level algebra map $g\colon
K_{\operatorname{Lev}_{p}}^{\star}(M)\rightarrow
Q_{s}M_{\operatorname{Lev}_{p}}^{\star}(p^{i})$.

The map $\gamma_{s;p^{i}}\colon
Q_{s}M_{\operatorname{Lev}_{p}}^{\star}(p^{i})\rightarrow
K_{\operatorname{Lev}_{p}}^{\star}(M)$ described above is compatible with the
action of $\operatorname{Lev}_{p}$ defined by $d^{\star p}$ on the source.

Note that $\gamma_{s;p^{i}}$ sends $g(x)$ back to $x$ for all $x\in M$. Thus,
$\gamma_{s;p^{i}}\circ g$ is an endomorphism of the unstable $p$-level algebra
$K_{\operatorname{Lev}_{p}}^{\star}(M)$ which fixes $M$. Hence, it is the
identity of $K_{\operatorname{Lev}_{p}}^{\star}(M)$. ∎

######  Lemma 5.3.4.

Suppose that $p=2$. For all $i\in\\{0,\dots,s-1\\}$,
$Q_{s}F(1)_{\operatorname{Com}}^{X_{2}}(2^{i})$ admits a direct factor of the
type $\tilde{H}(B\mathbb{Z}/2\mathbb{Z},\mathbb{F}_{2})$.

###### Proof.

G. Carlsson showed in [7] that there is a morphism of unstable modules

$f\colon\tilde{H}(B\mathbb{Z}/2\mathbb{Z},\mathbb{F}_{2})\rightarrow K(1).$

There are isomorphisms $\tilde{H}(B\mathbb{Z}/2\mathbb{Z},\mathbb{F}_{2})\cong
K_{\operatorname{Com}}^{X_{2}}(F(1))$ and $K(1)\cong
K_{\operatorname{Lev}}^{\star}(F(1))$ (see [17]). There is a unique morphism
of operads $\operatorname{Lev}\rightarrow\operatorname{Com}\circ
Q_{s}\operatorname{D}$ sending $\star$ to $(X_{2};d,d)$, and this allows us to
extend the morphism in $\mathcal{U}$: $F(1)\rightarrow
Q_{s}F(1)_{\operatorname{Com}}^{X_{2}}$ sending $\iota_{1}$ to
$(X_{1};\phi^{i}\omega\otimes\iota_{1})$ into an unstable level algebra
morphism $g\colon K(1)\rightarrow Q_{s}F(1)_{\operatorname{Com}}^{X_{2}}$.

It is easy to check that $\gamma_{s;2^{i}}\circ g\circ f$ is the identity of
$\tilde{H}(B\mathbb{Z}/2\mathbb{Z},\mathbb{F}_{2})$. ∎

######  Example 5.3.5.

At $p=2$, in the cases where
$\mathcal{P}=\operatorname{Com},\operatorname{uCom}$ and $M=F(1)$, we recover
the splittings considered by Campbell and Selick in [6]. Indeed,
$K_{\operatorname{uCom}}^{\star}(F(1)^{\oplus s})$ is isomorphic, as an
unstable algebra, to the mod 2 cohomology of the classifying space
$B\mathbb{Z}/2\mathbb{Z}^{\times s}$ of the simple abelian group
$(\mathbb{Z}/2\mathbb{Z})^{\times s}$. On the other side, the commutative
algebra $Q_{s}F(1)_{\operatorname{uCom}}^{X_{2}}$, with its
$\mathbb{Z}/(2^{s}-1)\mathbb{Z}$-weighting, corresponds the polynomial algebra
with $\mathcal{A}$-action denoted by $M_{s}$ in [6]. Then, Lemma 5.3.4 shows
that, for all $i\in\\{0,\dots,s-1\\}$,
$Q_{s}F(1)_{\operatorname{Com}}^{X_{2}}(2^{i})$ contains a direct factor of
the type $\tilde{H}(B\mathbb{Z}/2\mathbb{Z},\mathbb{F}_{2})$.

For $s=2$ we get a splitting in $\mathcal{U}$:

$H^{*}(B\mathbb{Z}/2\mathbb{Z}^{\times 2},\mathbb{F}_{2})\cong M_{2}(0)\oplus
M_{2}(1)\oplus M_{2}(2),$

with $M_{2}(0)=\mathbb{F}\oplus Q_{2}F(1)_{\operatorname{Com}}^{X_{2}}(0)$,
and $M_{2}(1)\cong M_{2}(2)$ contains a direct factor isomorphic to the
unstable module $\tilde{H}^{*}(B\mathbb{Z}/2\mathbb{Z},\mathbb{F})$.

For $s=3$ we get a splitting in $\mathcal{U}$:

$H^{*}(B\mathbb{Z}/2\mathbb{Z}^{\times 3},\mathbb{F}_{2})\cong M_{3}(0)\oplus
M_{3}(1)\oplus M_{3}(2)\oplus M_{3}(3)\oplus M_{3}(4)\oplus M_{3}(5)\oplus
M_{3}(6),$

with $M_{3}(0)=\mathbb{F}\oplus Q_{3}F(1)_{\operatorname{Com}}^{X_{2}}(0)$,
$M_{3}(1)\cong M_{3}(2)\cong M_{3}(4)$ each contains a factor of the type
$\tilde{H}^{*}(B\mathbb{Z}/2\mathbb{Z},\mathbb{F}_{2})$, and $M_{3}(3)\cong
M_{3}(6)\cong M_{3}(5)$.

######  Example 5.3.6.

Denote by $F=F(1)$ if $p=2$ and $F=F^{\prime}(2)$ if $p>2$ is an odd prime. If
$p=2$, we also set $K^{\prime}(2)=K(1)$. Lemmas 5.3.2 and 5.3.3 affirm that
that for all $s>0$, $K_{\operatorname{Lev}_{p}}^{\star}(F^{\oplus s})$ admits
a decomposition in $\mathcal{U}^{\prime}$ of the form

$K_{\operatorname{Lev}_{p}}^{\star}(F^{\oplus
s})\cong\bigoplus_{j\in\mathbb{Z}/(p^{s}-1)\mathbb{Z}}Q_{s}F_{\operatorname{Lev}_{p}}^{\star}(j),$

with isomorphisms in $\mathcal{U}^{\prime}$:

$Q_{s}F_{\operatorname{Lev}_{p}}^{\star}(j)\cong
Q_{s}F_{\operatorname{Lev}_{p}}^{\star}(pj)$

for all $j\in\mathbb{Z}/(p^{s}-1)\mathbb{Z}$, and that each
$Q_{s}F_{\operatorname{Lev}_{p}}^{\star}(p^{i})$ for $i\in\\{0,\dots,s-1\\}$
contains a copy of $K^{\prime}(2)$ as a direct factor.

For instance, when fixing $p=2$, and $s=2$, we can decompose
$K_{\operatorname{Lev}}^{\star}(F^{\oplus 2})$ as an unstable module:

$K_{\operatorname{Lev}}^{\star}(F^{\oplus
2})=Q_{2}F_{\operatorname{Lev}}^{\star}(0)\oplus
Q_{2}F_{\operatorname{Lev}}^{\star}(1)\oplus
Q_{2}F_{\operatorname{Lev}}^{\star}(2),$

with $Q_{2}F_{\operatorname{Lev}}^{\star}(1)\cong
Q_{2}F_{\operatorname{Lev}}^{\star}(2)$, and
$Q_{2}F_{\operatorname{Lev}}^{\star}(1)$ can again be decomposed as
$K(1)\oplus L$ for a certain unstable module $L$.

##  6\. Twisted actions on unstable algebras and their classification

The aim of this section is to review and generalise the results obtained by
Duflot, Kuhn and Winstead in [14]. The first main result of this section is
Theorem 6.2.3, which shows that, given any operad $\mathcal{P}$ with a
completely symmetric operation $\star\in\mathcal{P}(p)^{\mathfrak{S}_{p}}$,
any vector space $V$, and any endomorphism $M$ of $V$, the equation

$P_{0}v=Mv^{\star p}$

extends to an action of $\mathcal{A}^{\prime}$ on the free
$\mathcal{P}$-algebra $S(\mathcal{P},V)$ with $V$ concentrated in degree two.
We denote by $S_{M}(\mathcal{P},V)$ the resulting $\mathcal{P}$-algebra in
$\mathcal{U}^{\prime}$.

The second main result of this section is Theorem 6.3.2, which gives necessary
and sufficient conditions on the endomorphisms $M_{1}$ and $M_{2}$ of a finite
dimension $V$ for the underlying unstable modules of
$S_{M_{1}}(\mathcal{P},V)$ and $S_{M_{2}}(\mathcal{P},V)$ to be isomorphic.

To prove the first result, we follow [14], Sections 2 and 3, which rely on the
elegant approach to the Adem relations by Bullett and McDonald [2]. We
introduce the notion of higher derivation for an algebra over an operad, which
generalises both the notion of higher derivation of [1] and the notion of
operadic derivation [25], and a notion of formal power series with
coefficients in an algebra over an operad.

Like in the preceding section, $p$ is any prime number, and the degrees are
divided by two if $p=2$.

###  6.1. Higher operadic derivations and actions of the Steenrod algebra

In this section, we recall the notion of derivation of an algebra over an
operad [25, 13], we introduce the notion of higher derivation of an algebra
over an operad inspired by [1] (see also [26], and we show how to produce,
from a $\mathcal{P}$-algebra $R$, a $\mathcal{P}$-algebra of formal power
series with coefficients in $R$. Following [2] and [14], we give conditions
for a higher derivation of a $\mathcal{P}$-algebra $R$ to induce an action of
$\mathcal{A}^{\prime}$ on $R$, yielding a $\mathcal{P}$-algebra in
$\mathcal{U}^{\prime}$.

We fix the operad $\mathcal{P}$ and the graded $\mathcal{P}$-algebra $R$.

######  Definition 6.1.1.

A derivation of $R$ of degree $m$ is a linear map $D_{1}:R\rightarrow R^{+m}$
of degree $m$ satisfying, for all $\mu\in\mathcal{P}(k)$,
$x_{1},\dots,x_{k}\in R$,

(6.eq.2)
$D_{1}(\mu(x_{1},\dots,x_{k}))=\sum_{i=1}^{k}(-1)^{m|\mu|+m(|x_{1}|+\dots+|x_{i-1}|)}\mu(x_{1},\dots,x_{i-1},D_{1}x_{i},x_{i+1},\dots,x_{k}).$

A higher derivation of $R$ of degree $m$ is a sequence

$D_{*}=\\{D_{0},D_{1},\dots,D_{n},\dots\\}$

such that for all $i\in\mathbb{N}$, $D_{i}:R\rightarrow R^{+mi}$ is a linear
map of degree $mi$, and for all $\mu\in\mathcal{P}(k)$, $x_{1},\dots,x_{k}\in
R$,

(6.eq.3)
$D_{i}(\mu(x_{1},\dots,x_{k}))=\sum_{j_{1}+\dots+j_{k}=i}\operatorname{sign}\mu(D_{j_{1}}x_{1},\dots,D_{j_{k}}x_{k}),$

with
$\operatorname{sign}=(-1)^{m\left(|\mu|+\sum_{l=1}^{k}j_{l}(|x_{1}|+\dots+|x_{l-1}|)\right)}$.

######  Remark 6.1.2.

With these definitions, if $D_{*}$ is a higher derivation of $R$ of degree $m$
such that $D_{0}=\operatorname{id}_{R}$, then $D_{1}$ is a derivation of $R$
of degree $m$.

Our sign convention is the classical Koszul sign convention [18]. In what
follows, all of our higher derivations are of even degree, on
$\mathcal{P}$-algebras concentrated in even degrees, and the operad is always
concentrated in degree zero, so we can drop the sign convention. Without the
sign convention, when $\mathcal{P}=\operatorname{uCom}$, we recover the notion
of higher derivation of [14] inspired by [1, 26].

######  Definition 6.1.3.

Define $\overline{\operatorname{D}}=\mathbb{F}\left[[d]\right]$. Denote by
$R[[d]]=\overline{\operatorname{D}}\circ R=\mathbb{F}\left[[d]\right]\otimes
R$, endowed with the $\mathcal{P}$-algebra structure induced by that of $R$,
that is,

$\mu\left((d^{j_{1}};x_{1,j_{1}})_{j_{1}\in\mathbb{N}},\dots,(d^{j_{n}};x_{n,j_{n}})_{j_{n}\in\mathbb{N}}\right)=(d^{i};\sum_{j_{1}+\dots+j_{n}=i}\mu(x_{1,j_{1}},\dots,x_{n,j_{n}}))_{i\in\mathbb{N}}.$

We also denote $R\left[[d,f]\right]=(R\left[[d]\right])\left[[f]\right]$. A
higher derivation $D_{*}$ on $R$ induces a $\mathcal{P}$-algebra morphism

$\displaystyle D_{*}[[d]]:$ $\displaystyle R$ $\displaystyle\rightarrow
R\left[[d]\right]$ $\displaystyle x$ $\displaystyle\mapsto(d^{i};D_{i}(x)),$

and a $\mathcal{P}$-algebra morphism:

$\displaystyle D_{*}[[d,f]]:$ $\displaystyle R[[d]]$ $\displaystyle\rightarrow
R\left[[d,f]\right]$ $\displaystyle(d^{i},x_{i})_{i}$
$\displaystyle\mapsto(f^{i};D_{*}[[d]](x_{i}))_{i}.$

Note that:

$\displaystyle D_{*}[[d,f]]((d^{i},x_{i})_{i})$
$\displaystyle=\left(f^{i};\left(d^{j};D_{j}(x_{i})\right)_{j}\right)_{i}$
$\displaystyle=\left(\sum_{k=0}^{n}((f^{n-k};d^{k});D_{k}(x_{n-k}))\right)_{n}\in(\overline{\operatorname{D}}\circ\overline{\operatorname{D}})\circ
R$

######  Remark 6.1.4.

Note that this is nothing but an extension of scalars of $R$ to the ring of
formal series $\mathbb{F}\left[[d]\right]$ with coefficients in $\mathbb{F}$.
In the case where $\mathcal{P}=\operatorname{uCom}$ is the operad of
associative, commutative, unital algebras, we recover the classical notion of
formal power series with coefficient in the commutative ring $R$.

######  Definition 6.1.5.

Following Bullett and MacDonald [2], we say that $D_{*}$ follows the Adem
relations if the following diagram commutes:

(6.eq.4)
$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D_{*}\left[[s^{p}]\right]}$$\scriptstyle{D_{*}\left[[t^{p}]\right]}$$\textstyle{R\left[[s]\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D_{*}\left[[s,t^{p}+st^{p-1}+\dots+s^{p-1}t]\right]}$$\textstyle{R\left[[t]\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D_{*}\left[[t,s^{p}+ts^{p-1}+\dots+t^{p-1}s]\right]}$$\textstyle{R\left[[t,s]\right],}$

where $t,s$ are two indeterminates. A higher derivation is said to be unstable
if $D_{i}(x)=0$ for all $i>\frac{|x|}{2}$.

######  Proposition 6.1.6.

Let $R$ be a graded $\mathcal{P}$-algebra that is generated by elements of
degree $2$ as a $\mathcal{P}$-algebra. Suppose that $D_{*}$ is a higher
derivation of $R$ such that:

  1. i)

$D_{*}$ is unstable,

  2. ii)

$D_{0}D_{1}=D_{1}D_{0}$, and

  3. iii)

$D_{i}D_{1}=(i+1)D_{i+1}$ for all $i\in\\{1,\dots,p-1\\}$.

Then $D_{*}$ satisfies the Adem relations on $R$.

###### Proof.

Since $R$ is generated by $R^{2}$ as a $\mathcal{P}$-algebra, and since
$D_{*}\left[[t,s^{p}+ts^{p-1}+\dots+t^{p-1}s]\right]\circ
D_{*}\left[[t^{p}]\right]$,
$D_{*}\left[[s,t^{p}+st^{p-1}+\dots+s^{p-1}t]\right]\circ
D_{*}\left[[s^{p}]\right]$ are two $\mathcal{P}$-algebra morphisms, we need
only verify that the diagram 6.eq.4 commutes on degree $2$ elements. Let $x\in
R^{2}$. One has

$D_{*}[[d,f]]\circ
D_{*}\left[[d]\right](x)=\left(\sum_{k=0}^{n}((f^{n-k};d^{k});D_{k}D_{n-k}(x))\right)_{n}.$

Since $D_{*}$ is unstable, $D_{k}D_{n-k}(x)=0$ as soon as $n-k>1$ or
$k>(n-k)(p-1)+1$, and $D_{i}D_{1}(x)=(i+1)D_{i+1}(x)=0$. One has:

$\displaystyle\left(D_{*}[[d,f]]\circ D_{*}\left[[d]\right](x)\right)_{0}$
$\displaystyle=\left(1_{\operatorname{D}};D_{0}D_{0}(x)\right),$
$\displaystyle\left(D_{*}[[d,f]]\circ D_{*}\left[[d]\right](x)\right)_{1}$
$\displaystyle=(d,D_{1}D_{0}(x))+(f,D_{0}D_{1}(x)),$
$\displaystyle\left(D_{*}[[d,f]]\circ D_{*}\left[[d]\right](x)\right)_{p+1}$
$\displaystyle=(d^{p}f;D_{p}D_{1}(x)),$

and $\left(D_{*}[[d,f]]\circ D_{*}\left[[d]\right](x)\right)_{i}=0$ for all
$i\notin\\{0,1,p+1\\}$. Setting

$S_{1}=\left(D_{*}[[s^{p}+ts^{p-1}+\dots+t^{p-1}s,t^{p}]]\circ
D_{*}\left[[s^{p}+ts^{p-1}+\dots+t^{p-1}s]\right](x)\right),$

and

$S_{2}=\left(D_{*}[[t^{p}+st^{p-1}+\dots+s^{p-1}t,s^{p}]]\circ
D_{*}\left[[t^{p}+st^{p-1}+\dots+s^{p-1}t]\right](x)\right),$

since we assumed that $D_{0}D_{1}=D_{1}D_{0}$, it is clear that
$(S_{1})_{i}=(S_{2})_{i}$ for all $i\neq p+1$. Now, since

$(s^{p}+ts^{p-1}+\dots+t^{p-1}s)^{p}t^{p}=(s^{p}t+s^{p-1}t^{2}+\dots+t^{p}s),$

which is symmetric in $s$ and $t$, we deduce that $S_{1}=S_{2}$. ∎

Following Bullett and MacDonald [2], we deduce the following:

######  Proposition 6.1.7.

Suppose that $R$ and $D_{*}$ are as in Proposition 6.1.6, that $D_{*}$ is of
degree $2(p-1)$, and that $D_{0}=\operatorname{id}_{R}$. Then the equation

$P^{i}x=D_{i}x$

endows $R$ with the structure of a $\mathcal{P}$-algebra in
$\mathcal{U}^{\prime}$.

###  6.2. Twisted actions on unstable algebras

In this section, we shall prove Theorem 6.2.3, which shows that for all
endomorphism $M$ of the vector space $V$ concentrated in degree two, there is
always a (unique) action of $\mathcal{A}^{\prime}$ on $S(\mathcal{P},V)$,
compatible with the action of $\mathcal{P}$, such that $P_{0}v=Mv^{\star p}$
for all $v\in V$. Once again, our reasoning follows a very similar path to
that of [14], Section 3, and we show how this result produces a large family
of examples of unstable $\operatorname{Lev}_{p}$-algebras.

######  Lemma 6.2.1.

If $\partial:V\rightarrow\mathcal{P}(p)\otimes_{\mathfrak{S}_{p}}V^{\otimes
p}$ is a linear map, then there exists a unique derivation $D_{1}$ of
$S(\mathcal{P},V)$ of degree $2(p-1)$ such that $D_{1}=\partial$ on $V$, and
$D_{1}=0$ on $\mathcal{P}(0)$.

###### Proof.

This is derived from equation 6.eq.2. ∎

######  Lemma 6.2.2.

Suppose $D_{1}$ is any derivation of $S(\mathcal{P},V)$ of degree $m$ such
that $D_{1}=0$ on $\mathcal{P}(0)$. Then there exists a unique unstable higher
derivation $D_{*}$ of $S(\mathcal{P},V)$ of degree $m$ such that
$D_{*}=\\{\operatorname{id}_{S(\mathcal{P},V)},D_{1},\dots\\}$.

###### Proof.

As a vector space, $S(\mathcal{P},V)$ is spanned by the monomials

$\mathfrak{t}=\mu(x_{1},\dots,x_{k}),$

with $\mu\in\mathcal{P}(k)$, and $x_{1},\dots,x_{k}\in V$. For all
$i\in\mathbb{N}$, one deduces from the definition of higher derivations that

$D_{i}\mathfrak{t}=\sum_{j_{1}+\dots+j_{k}=i}\operatorname{sign}\mu(D_{j_{1}}x_{1},\dots,D_{j_{k}}x_{k}).$

Now, if $j_{l}>1$, $D_{j_{l}}x_{l}=0$ because of the instability condition,
and $D_{0},D_{1}$ are respectively defined on $V$ by the identity and by
$\partial$. Finally, we obtain a general expression:

$D_{i}\mathfrak{t}=\sum\operatorname{sign}\mu(\partial^{A}_{1}x_{1},\dots,\partial^{A}_{k}x_{k}),$

where the sum is the set $E^{i,k}$ of subsets $A\subset[k]$ such that $|A|=i$,
and we denote by $\partial^{A}_{l}=\partial$ if $l\in A$ and
$\partial^{A}_{l}=\operatorname{id}_{V}$ if $l\notin A$. ∎

######  Theorem 6.2.3.

Let $\mathcal{P}$ be an operad endowed with a completely symmetric $p$-ary
operation $\star$. Given a vector space $V$ concentrated in degree two and
$M\in\operatorname{End}(V)$, the equation $P^{1}v=Mv^{\star p})$ (or
$Sq^{1}v=Mv^{\star 2}$ in the case $p=2$) extends to an unstable action of
$\mathcal{A}^{\prime}$ on $S(\mathcal{P},V)$ satisfying the Cartan formula.

###### Proof.

Let $M$ be a linear endomorphism of $V$. Using Lemmas 6.2.1 and 6.2.2, define
a higher derivation $M_{*}$ of degree $2(p-1)$ on $S(\mathcal{P},V)$ such that
$M_{1}v=Mv^{\star p})$ for all $v\in V$. Since the characteristic is $p$,
$M_{1}$ is linear. To apply Proposition 6.1.7, we just have to check that
$M_{i}M_{1}=(i+1)M_{i+1}$ for all $i\in\\{1,\dots,p-1\\}$. For all $j\in[k]$,
$i\in\\{1,\dots,p-1\\}$, $l\in\\{0,\dots,i\\}$, Denote by $E^{i,k}_{l,j}$ the
set of subsets of $[k]$ of cardinal $i-l$ not containing $j$. Note that:

$\displaystyle M_{i}M_{1}\mathfrak{t}$
$\displaystyle=M_{i}\left(\sum_{j=1}^{n}\mu(x_{1},\dots,x_{j-1},M_{1}x_{j},x_{j+1},\dots,x_{k})\right)$
$\displaystyle=\sum_{j=1}^{k}\sum_{l=0}^{i}\sum_{A\in
E^{i,k}_{l,j}}\mu(\partial^{A}_{1}x_{1},\dots,\partial^{A}_{j-1}x_{j-1},M_{l}M_{1}x_{j},\partial^{A}_{j+1}x_{j+1},\dots,\partial^{A}_{k}x_{k}).$

Suppose that $l\in[p-1]$. Since $\star$ is completely symmetric,

$M_{l}M_{1}(x_{j})=M_{l}\left(Mx_{j}^{\star
p}\right)=\binom{p}{l}(\star;M_{1}(Mx_{j})^{\times l},Mx_{j}^{\times p-k})=0.$

Thus, only the terms with $l=0$ remain. We get

$\displaystyle M_{i}M_{1}\mathfrak{t}$ $\displaystyle=\sum_{j=1}^{n}\sum_{A\in
E^{i,k}_{0,j}}\mu(\partial^{A}_{1}x_{1},\dots,\partial^{A}_{j-1}x_{j-1},M_{1}x_{j},\partial^{A}_{j+1}x_{j+1},\dots,\partial^{A}_{k}x_{k}).$
$\displaystyle=\sum_{A\in E^{k,i+1}}\sum_{j\in
A}\mu(\partial^{A}_{1}x_{1},\dots,\partial^{A}_{j-1}x_{j-1},M_{1}x_{j},\partial^{A}_{j+1}x_{j+1},\dots,\partial^{A}_{k}x_{k})$
$\displaystyle=(i+1)M_{i+1}\mathfrak{t}.$

We have checked that $M_{*}$ satisfies the conditions of Proposition 6.1.7. ∎

######  Notation 6.2.4.

From now on we refer to this particular example of a $\mathcal{P}$-algebra in
$\mathcal{U}^{\prime}$ as $S_{M}(\mathcal{P},V)$.

######  Remark 6.2.5.

Define an action of $\operatorname{D}$ on $V$ by $d(v)=Mv$. This extends to an
action of $\mathcal{P}\circ\operatorname{D}$ on $S_{M}(\mathcal{P},V)$ by
setting, for all $\mathfrak{t}=\mu(x_{1},\dots,x_{k})$,

$M\mathfrak{t}=\mu(Mx_{1},\dots,Mx_{k}).$

######  Lemma 6.2.6.

If $\star\in\mathcal{P}(p)^{\mathfrak{S}_{p}}$ is $\mathcal{P}$-central, then
$S_{M}(\mathcal{P},V)$ is a $d^{\star p}$-unstable
$\mathcal{P}\circ\operatorname{D}$-algebra.

###### Proof.

We just have to prove that $S_{M}(\mathcal{P},V)$ is $d^{\star p}$-unstable.
For all $\mathfrak{t}=\mu(x_{1},\dots,x_{k})$, one has

$\displaystyle P_{0}\mathfrak{t}$
$\displaystyle=\mu(P_{0}x_{1},\dots,P_{0}x_{k})=\mu(Mx_{1}^{\star
p},\dots,Mx_{k}^{\star p})$ $\displaystyle=M\mu(x_{1}^{\star
p},\dots,x_{k}^{\star p}),$

and since $\star$ is $\mathcal{P}$-central,

$P_{0}\mathfrak{t}=\left(M\mu(x_{1},\dots,x_{k})\right)^{\star p}).$

∎

######  Example 6.2.7.

The preceding remark and lemma implies that for all
$M\in\operatorname{End}(V)$, $S_{M}(V)=S_{M}(\operatorname{Com},V)$ is endowed
with the structure of a $\operatorname{Com}\circ\operatorname{D}$-algebra, and
following Proposition 4.3.9 applied to the inclusion of operads
$\operatorname{Lev}_{p}\rightarrow\operatorname{Com}\circ\operatorname{D}$
(see Propositions 2.2.8, 2.2.6), this induce on $S_{M}(V)$ a structure of an
unstable $\operatorname{Lev}_{p}$-algebra.

###  6.3. Classification of twisted actions

This section is devoted to the proof of Theorem 6.3.2, which gives conditions
on $M_{1},M_{2}\in\operatorname{End}(V)$ to obtain an isomorphism in
$\mathcal{U}^{\prime}$ between the resulting $\mathcal{P}$-algebras with
twisted actions, $S_{M_{1}}(\mathcal{P},V)$ and $S_{M_{2}}(\mathcal{P},V)$.

######  Proposition 6.3.1.

Suppose that for all $k\in\mathbb{N}$, $\mathcal{P}(k)$ is finite dimensional,
and that $V$ is finite dimensional. Let $M\in\operatorname{End}(V)$, and
$A\in\operatorname{Aut}(V)$ be such that $A^{s}=\operatorname{id}_{V}$ and
$AM=MA$. There is an isomorphism of $\mathcal{P}$-algebras in
$\mathcal{U}^{\prime}$

$\left(S_{M}(\mathcal{P},V)\right)_{\operatorname{inv}}\cong
S_{AM}(\mathcal{P},V),$

and so, there is an isomorphism in $\mathcal{U}^{\prime}$:

$S_{M}(\mathcal{P},V)\cong S_{AM}(\mathcal{P},V).$

###### Proof.

The action of $A$ on $S_{M}(\mathcal{P},V)$ induces a structure of
$\mathcal{P}\circ Q_{s}\operatorname{D}$-algebra on $S_{M}(\mathcal{P},V)$.
Applying Lemma 5.1.3 to $N=S_{M}(\mathcal{P},V)$ give an isomorphism of
$Q_{s}\operatorname{D}$-algebras in $\mathcal{U}^{\prime}$

$\beta^{Q_{s}\operatorname{D}}\colon
S_{M}(\mathcal{P},V)\rightarrow\left(S_{M}(\mathcal{P},V)\right)_{\operatorname{inv}}.$

Note that with our notations, Lemma 5.1.5 gives an isomorphism of
$\mathcal{P}\circ Q_{s}\operatorname{D}$-algebras

$\tilde{\iota}\colon\mathbb{F}_{p^{s}}\otimes\left(S_{M}(\mathcal{P},V)\right)_{\operatorname{inv}}\rightarrow\mathbb{F}_{p^{s}}\otimes
S_{AM}(\mathcal{P},V).$

Since $V$ is finite dimensional, and since for all $k\in\mathbb{N}$,
$\mathcal{P}(k)$ is finite dimensional, $\tilde{\iota}$ restricts to an
isomorphism of $\mathcal{P}$-algebras in $\mathcal{U}^{\prime}$:

$\left(S_{M}(\mathcal{P},V)\right)_{\operatorname{inv}}\cong
S_{AM}(\mathcal{P},V).\qed$

######  Theorem 6.3.2.

Suppose that $\star\in\mathcal{P}(p)^{\mathfrak{S}_{p}}$ is
$\mathcal{P}$-central and non-nilpotent (see Definition 2.3.7, and that $V$ is
finite dimensional. Given two elements $M_{1}$ and $M_{2}$ of
$\operatorname{End}(V)$, there is an isomorphism
$S_{M_{1}}(\mathcal{P},V)\cong S_{M_{2}}(\mathcal{P},V)$ in
$\mathcal{U}^{\prime}$ if and only if

$\dim\left(\operatorname{Ker}\left(M^{i}_{1}\right)\right)=\dim\left(\operatorname{Ker}\left(M^{i}_{2}\right)\right)$

for all $i\geq 1$. In particular, if $M$ is invertible and $\star$ is
$\mathcal{P}$-central, $S_{M}(\mathcal{P},V)$ is isomorphic to
$K_{\mathcal{P}}^{\star}(F^{\oplus\dim(V)})$, where $F=F(1)$ in the case
$p=2$, and $F=F^{\prime}(2)$ in the case $p>2$.

###### Proof.

We first establish that

(6.eq.5) $\operatorname{Ker}(M^{j}:V\rightarrow
V)=\operatorname{Ker}(P^{p^{j-1}}\cdots P^{p}P^{1}\colon V\rightarrow
S_{M}(\mathcal{P},V)^{p^{j}}),$

for all $j\geq 1$. Since we have assumed that
$\star\in\mathcal{P}(p)^{\mathfrak{S}_{p}}$ is non-nilpotent and
$\mathcal{P}$-central, it is a direct consequence of Lemma 6.2.6.

Now, assuming $S_{M_{1}}(\mathcal{P},V)$ is isomorphic to
$S_{M_{2}}(\mathcal{P},V)$ as objects of $\mathcal{U}^{\prime}$, equation
6.eq.5 implies that, for all $i\geq 1$,

$\dim\left(\operatorname{Ker}\left(M_{1}^{i}\right)\right)=\dim\left(\operatorname{Ker}\left(M_{2}^{i}\right)\right).$

We now have to prove the converse statement. Suppose that for all $i\geq 1$,

$\dim\left(\operatorname{Ker}\left(M_{1}^{i}\right)\right)=\dim\left(\operatorname{Ker}\left(M_{2}^{i}\right)\right).$

Proposition 6.3.1 shows that in $\mathcal{U}^{\prime}$,
$S_{M}(\mathcal{P},V)\cong S_{AM}(\mathcal{P},V)$, for all automorphisms $A$
of $V$, cyclic of order $s$, such that $MA=AM$. It is also clear that
$S_{M}(\mathcal{P},V)\cong S_{CMC^{-1}}(\mathcal{P},V)$ for all automorphisms
$C$ of $V$. Proposition 4.7 of [14] finishes the proof of Theorem 6.3.2. ∎

######  Remark 6.3.3.

Denote by $F=F^{\prime}(2)$, ($F=F(1)$ if $p=2$). If
$M\in\operatorname{Aut}(V)$, then Theorem 6.3.2 gives an isomorphism
$S_{M}(\mathcal{P},V)\cong K_{\mathcal{P}}^{\star}(V^{0}\otimes F)$ where
$V^{0}=\Sigma^{-2}V$ is the vector space $V$ concentrated in degree zero.

If $M$ acts on a basis of $V$ as an $s$-cycle, then this can be seen as a
particular case of Theorem 5.2.1, with $M=F$.

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Sacha Ikonicoff, pims–cnrs postdoctoral associate, University of Calgary
e-mail<EMAIL_ADDRESS>