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\title[On Bousfield's conjectures]{On Bousfield's conjectures for the unstable Adams spectral sequence for $SO$ and $U$}
%ế
\author{\lastname{Nguy{\~{\^e}}n} \firstname{Th{\myacetil} C{\myuhorn}{\myacohorn}ng}}
\address{Department of Mathematics, Informatics and Mechanics, VNU University of Science, Vietnam National University, Hanoi}
\curraddr{Hanoi Institute of Mathematics, Vietnam Academy of Science and Technology}
\email{tdntcuong@gmail.com}
\thanks{The author was supported by the Simons foundations}

\subjclass{55T15, 55Q52}
\keywords{\kwd{Injective resolution}
\kwd{Projective resolution}
\kwd{Unstable Adams spectral sequence}
\kwd{Unstable modules}}

\begin{abstract}
The unstable Adams spectral sequence is a spectral sequence that starts from algebraic information about the mod \(2 \) cohomology \(H ^{ * } \left(X \right) \) of a space \(X \) as an unstable algebra over the Steenrod algebra $\mathcal{A}$, and converges, in good cases, to the \(2 \)-localized homotopy groups of \(X \). Bousfield and Don Davis looked at the case when \(X \) was either of the infinite matrix groups \(SO\) or \(U\). Bousfield and Davis created algebraic spectral sequences and conjectured that they agreed with the unstable Adams spectral sequences for \(SO \) and \(U \). To this end the following algebraic decomposition must hold
\[
\mathrm{Ext} _{ \mathcal{U} } ^{ s } \left(\tilde{ H } ^{ * } \left(\mathbb{R} P ^{ \infty }, \Sigma ^{ t } \mathbb{Z} /2 \right) \right) \cong \bigoplus _{ n } \mathrm{Ext}_{ \mathcal{U} } ^{ s } \left(M _{ n } / M _{ n - 1 },\Sigma ^{ t } \mathbb{Z} /2 \right)
\]
where \(M _{ 1 } \subset M _{ 2 } \subset \cdots \) is the well known dyadic filtration of the \(\mathcal{A}\)-module \(\tilde{ H } ^{ * } \left(\mathbb{R} P ^{ \infty }, \mathbb{Z} /2 \right) \cong \mathbb{F} _{ 2 } \left[u \right] \) given by the dyadic expansion of the powers of \(u \). This paper aims at showing that this decomposition holds for numerous values of \(s \) and \(t \).
\end{abstract}

\begin{document}
\maketitle

\section{Introduction}

In this article, we study a conjecture of Bousfield on the unstable Adams spectral sequence for $SO$ and $U$. This is first stated in the original paper~\cite{BD92} of Bousfield and Davis and is studied by Kathryn Lesh in~\cite{Les02}. We begin by recalling the precise statement of the conjecture. We work at the prime $2$ and for a topological space $X$, we write $\chv{X}$ for $\chv{X;\fd{f}_{2}}$. Let $\calA$ be the Steenrod algebra. An $\fd{N}-$graded commutative $\calA-$algebra $K$ is called an unstable algebra if the Steenrod action is unstable, represents the Frobenius twist
\[
Sq^{k}x=
\begin{cases}
0 & \text{if }k>n, \\
x^{2} & \text{if } k=n,
\end{cases}
\]
for all $x\in K^{n}$, and satisfies the Cartan formula
\[
Sq^{k}\moc{ab}=\sum_{i=0}^{k}\moc{Sq^{i}a}\moc{Sq^{k-i}b}
\]
for all $a,b\in K$. In particular, as \(Sq ^{ 0 } \) is the identity element then \(x = x ^{ 2 } \) for all \(x \) homogeneous of degree \(0 \). In other words, the degree \(0 \) elements of an unstable algebra form a Boolean algebra. If we no longer require the product structure, keeping only the instability condition of the Steenrod action, we get the notion of an unstable module. Denote by $\calK$ the category of unstable algebras, and by $\calU$ the category of unstable modules. The Cartan formula on the tensor product $M\otimes N$ of two unstable modules define another unstable one, making $\calU$ a symmetric monoidal category. For an unstable module $M$, we write \(S ^{ * } \left(M \right) \) for the symmetric power of \(M \). This is an algebra but does not satisfy the Cartan formula. Folowing~\cite{Ste62}, we define
\[
U\moc{M}=\frac{S^{*}\moc{M}}{\mocg{Sq^{\moca{x}}x-x^{2}}}.
\]
to get a functor $U:\calU\to\calK$, which is left adjoint to the forgetful functor $\calK\to \calU$. The importance of the categories \(\calU \) and \(\calK \) is justified by the unstable Adams spectral sequence, introduced by Massey and Peterson in~\cite{MP67}, generalized by Bousfield and Curtis in~\cite{BC70}, and generalized further by Bousfield and Kan in~\cite{BK72}. In general, the unstable Adams spectral sequence for a topological space $X$ has the form
\[
E_{2}^{s,t}=\ext{\calK}{s}{\chv{X}}{\chv{S^{t}}}\Longrightarrow \pi_{s-t}\moc{\widehat{X_{2}}}.
\]
However, if the cohomology of the space $X$ is of the form $U\moc{M}$ for some unstable module $M$, then the derived functor $\Ext$ in the nonabelian category $\calK$ can be computed as Ext-groups in the category $\calU$:
\[
\ext{\calK}{s}{\chv{X}}{\chv{S^{t}}}\cong \ext{\calU}{s}{M}{\Sigma^{t}\fh},
\]
where $\Sigma^{t}\fh\cong \tilde{\Hrm}^{*}\moc{S^{t}}$. So, in these cases, we write the unstable Adams spectral sequence for $X$ as:
\[
E_{2}^{s,t}=\ext{\calU}{s}{M}{\Sigma^{t}\fh}\Longrightarrow \pi_{s-t}\moc{\widehat{X_{2}}}.
\]
We will be discussing the unstable Adams spectral sequence for the stable special orthogonal group $SO$ and the stable unitary group $U$. As $\chv{SO}\cong U\moc{\tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}}$ and $\chv{U}\cong U\moc{\tilde{\Hrm}^{*}\moc{\Sigma\moc{\mathbb{C}P^{\infty}_{+}}}}$, the spectral sequences have the following form:
\[
\begin{array}{ccccc}
E_{2}^{s,t}&=&\ext{\calU}{s}{\tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}}{\Sigma^{t}\fh}&\Longrightarrow& \pi_{s-t}\moc{\widehat{SO_{2}}},\\
E_{2}^{s,t}&=&\ext{\calU}{s}{\tilde{\Hrm}^{*}\moc{\Sigma\moc{\mathbb{C}P^{\infty}_{+}}}}{\Sigma^{t}\fh}&\Longrightarrow& \pi_{s-t}\moc{\widehat{U_{2}}}.
\end{array}
\]
Recall that $\chv{\fd{r}P^{\infty}}$ is the polynomial algebra $\fh\mocv{u}$ on one generator of degree~$1$ and
\[
\tilde{\Hrm}^{*}\moc{\Sigma\moc{\mathbb{C}P^{\infty}_{+}}}\cong \Sigma\fh\mocv{v}
\]
where $v$ is of degree~$2$ and the Steenrod action is determined by:
\begin{align*}
Sq^{k}u^{n} &=\binom{n}{k}u^{n+k}, \\
Sq^{2k}v^{n} &=\binom{n}{k}v^{n+k}.
\end{align*}
Let $\alpha\moc{i}$ be the number of $1$'s in the dyadic expansion of $i$ and filter $\tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}$ and $\tilde{\Hrm}^{*}\moc{\Sigma\moc{\mathbb{C}P^{\infty}_{+}}}$by:
\begin{align*}
R_{n} &=\bigl\{u^{d}\bigm|\alpha(d)\leq n\bigr\}, \\
C_{n} &=\bigl\{\Sigma v^{d} \bigm|\alpha(d)\leq n\bigr\}.
\end{align*}
In~\cite{BD92}, the authors examine the Postnikov tower of \(SO \) and get a spectral sequence:
\[
E _{ 2 } ^{ s,t } = \bigoplus_{n\geq 1}\ext{\calU}{s}{\frac{C_{n}}{C_{n-1}}}{\Sigma^{t}\fh} \Longrightarrow \pi _{ s - t } \moc{\widehat{U_{2}}}
\]
A similar reasoning is followed for \(SO \):
\[
E _{ 2 } ^{ s,t } = \bigoplus_{n\geq 1}\ext{\calU}{s}{\frac{R_{n}}{R_{n-1}}}{\Sigma^{t}\fh} \Longrightarrow \pi _{ s - t } \moc{\widehat{SO_{2}}}
\]
Moreover, the authors show that these new spectral sequences can be constructed algebraically and they hope to identify these spectral sequences to the unstable Adams spectral sequences for \(U \) and \(SO \) respectively. The first step of the identification is the comparison of the \(E _{ 2 } \) terms.
\begin{conj}[Bousfield's splitting conjecture]\label{MainConj}
There are isomorphisms:
\begin{align*}
\bigoplus_{n\geq 1}\ext{\calU}{s}{\frac{R_{n}}{R_{n-1}}}{\Sigma^{t}\fh} &\cong \ext{\calU}{s}{\tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}}{\Sigma^{t}\fh},\\
\displaystyle\bigoplus_{n\geq 1}\ext{\calU}{s}{\frac{C_{n}}{C_{n-1}}}{\Sigma^{t}\fh} &\cong \ext{\calU}{s}{\tilde{\Hrm}^{*}\moc{\Sigma\moc{\mathbb{C}P^{\infty}_{+}}}}{\Sigma^{t}\fh}.
\end{align*}
\end{conj}

Mahowald suggests that the spectral sequences constructed in~\cite{BD92} can be demystified by taking the destabilization of the stable Adams resolutions of the connective \(so \) and \(u \) spectra. Following Mahowald's suggestions, in~\cite{Les02} Kathryn Lesh constructs other spectral sequences:
\begin{align*}
E _{ 2 } ^{ s,t } = \bigoplus_{n\geq 1}\ext{\calU}{s}{\frac{R_{n}}{R_{n-1}}}{\Sigma^{t}\fh} & \Longrightarrow \pi _{ s - t } \moc{\widehat{SO_{2}}},\\
E _{ 2 } ^{ s,t } = \bigoplus_{n\geq 1}\ext{\calU}{s}{\frac{C_{n}}{C_{n-1}}}{\Sigma^{t}\fh} &\Longrightarrow \pi _{ s - t } \moc{\widehat{U_{2}}}
\end{align*}
Lesh also conjectures that these are other models for the unstable Adams spectral sequences of \(SO \) and \(U \).

In this paper, we focus on the Bousfield splitting conjecture and get the following result.

\begin{theo}[Corollaries~\ref{mainThm1} and~\ref{mainThm2}]
Conjecture~\ref{MainConj} holds if either $s\leq 1$ or $s\geq \mocv{t/2}$.
\end{theo}

We will verify Conjecture~\ref{MainConj} by computing both sides of the isomorphims. The key observation is that the Ext-groups appearing in this conjecture are $\fh-$vector spaces of finite dimension. Therefore, in order to verify these isomorphisms, we only need to count the dimension of these vector spaces.


\section{Unstable Modules}

This section provides some basis facts of the category $\calU$ of unstable modules used throughout the present paper. We refer to~\cite{Sch94} for a thorough treatment of unstable modules.

\subsection{Projective unstable modules}

Denote by $\calV_{\fh}$ the category of $\fh-$vector spaces. The functor $F_{n}:\calU\to \calV_{\fh},M\mapsto M^{n}$ is representable, and we denote by $F(n)$ the representing object. As the functor $F_{n}$ is exact, $F(n)$ is a projective unstable module. For all unstable modules $M$, there is an epimorphism
\[
\bigoplus_{\substack{x\in M^{n},\\n\geq 0}}F(n)\twoheadrightarrow M.
\]
Therefore, the $F(n)$'s$,n\geq 0$ form a system of projective generators for the category $\calU$ of unstable modules.

\begin{defi}
For a set $X$ of nonnegative integers, denote by $F\moc{X}$ the direct sum of $F(n),n\in X$.
\end{defi}

\begin{rema}
As a morphism $F\moc{a}\to F\moc{b}$ is determined by a Steenrod operation of degree $a-b$, if $X,Y$ are two sets of nonnegative integers of cardinal $m$ and $n$, respectively, then we represent a morphism $\varphi:F\moc{X}\to F\moc{Y}$ by a matrix $M\in\Mat_{n,m}\moc{\calA}$.
\end{rema}

\subsection{The module \texorpdfstring{$F\moc{1}$}{F moc1} and the dyadic filtration}

It is straightforward from the definition that $F(n)$ has the following description:
\[
F(n)\cong \frac{\Sigma^{n}\calA}{\mocg{Sq^{I}\Sigma^{n}1 \,\middle|\, I\text{ is admissible and }e\moc{I}>n}}.
\]
Hereby, we denote the generator $\Sigma^{n}1$ of $F(n)$ by $\imath_{n}$.

From this description of $F(n)$, we can identify $F\moc{1}$ as the submodule of $\tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}$ generated by the variable $u$, that is $F\moc{1}=\fh\mocg{u,u^{2},u^{4},\ldots}$ and
\[
Sq^{n}u^{2^{k}}=
\begin{cases}
u^{2^{k+1}} &\text{if }n=2^{k} \\
u^{2^{k}} &\text{if }n=0,\\
0 &\text{otherwise.}
\end{cases}
\]
We can use the module $F\moc{1}$ to describe the composition series of the dyadic filtration on $\tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}$ and $\tilde{\Hrm}^{*}\moc{\Sigma\moc{\mathbb{C}P^{\infty}_{+}}}$. Recall from the introduction that for all $n\geq 1$, we have
\[
R_{n}:=\bigl\{u^{d} \bigm|\alpha(d)\leq n\bigr\}.
\]
Note that $R_{1}=F\moc{1}$. Moreover, the morphism
\[
R_{m}\otimes R_{n}\to \tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}\otimes \tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}\xrightarrow{\mathrm{mult}} \tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}
\]
is surjective onto $R_{m+n}$, for all $m,n\geq 1$. Denote by \(\Lambda ^{ n } \) the \(n \)-th exterior power functor, then the following result is straightforward.

\begin{lemm}\label{dyafil}
The epimorphism $F\moc{1}^{\otimes n}\to R_{n}$ induces an isomorphism of unstable modules:
\[
\Lambda^{n}\moc{F\moc{1}}\xrightarrow{\cong}\frac{R_{n}}{R_{n-1}}
\]
for all $n\geq 1$.
\end{lemm}

There is a clean way to describe the dyadic filtration of $\tilde{\Hrm}^{*}\moc{\Sigma\moc{\mathbb{C}P^{\infty}_{+}}}$ using that of $\tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}$. To this end, we will need some endofunctors of the category $\calU$: the suspension functor $\Sigma$ and the doubling functor $\Phi$.

The suspension functor $\Sigma:\calU\to \calU$ is easy to describe. What $\Sigma M$ is to the unstable module $M$ is what $\tilde{\Hrm}^{*}\moc{\Sigma X}$ is to $\tilde{\Hrm}^{*}\moc{X}$, that is
\[
\moc{\Sigma M}^{n}=
\begin{cases}
M^{n-1} &\text{if }n\geq 1, \\
\mocn{0} &\text{if }n=0,
\end{cases}
\]
and the Steenrod action is defined by
\[
Sq^{n}\Sigma x=\Sigma Sq^{n}x.
\]
The functor $\Sigma$ is exact and admits a left adjoint $\Omega$, called the loop functor of unstable modules. The loop functor $\Omega$ is right exact, and we denote by $\Omega_{s}$ the $s-$th left derived functor of $\Omega$.

To describe the doubling functor $\Phi:\calU\to\calU$, recall that the Steenrod operations represent the Frobenius twist of an unstable algebra. Therefore, we can use the Steenrod action to define the Frobenius twist of an unstable module as follows. For an unstable module $M$, let $Sq_{0}$ be the operator that associates an element $x$ with $Sq^{\moca{x}}x$. The operator $Sq_{0}$ defines a map $M\to M$. However, this map is not of degree~$0$. For this reason, we introduce the functor $\Phi:\calU\to \calU$, associating an unstable module $M$ with $\Phi M$, such that:
\[
\moc{\Phi M}^{n}=
\begin{cases}
M^{k} &\text{if }n=2k, \\
\mocn{0} &\text{otherwise,}
\end{cases}
\]
and the Steenrod action is defined by
\[
Sq^{n}\Phi x=
\begin{cases}
\Phi Sq^{k}x &\text{if }n=2k, \\
\mocn{0} &\text{otherwise.}
\end{cases}
\]
Now, we can represent the operator $Sq_{0}$ as the natural transformation $\lambda:\Phi\to \Id$, sending each unstable module $M$ to the morphism
\begin{align*}
\lambda_{M}:\Phi M&\longrightarrow M, \\
\Phi x &\longmapsto Sq_{0}x.
\end{align*}

It turns out that the kernel, as well as the cokernel of $\lambda_{M}$ can be determined using $\Sigma$ and $\Omega$.

\begin{prop}[{\cite[Proposition~2.4]{NC19},\cite[Proposition~1.7.3]{Sch94}}]\label{exactseq}
Let $M$ be an unstable module. Then $\Omega_{s}M=\mocn{0}$ for all $s>1$. Moreover, $\Omega_{1}M$ and $\Omega M$ fit in the following natural exact sequence:
\[
0\to \Sigma\Omega_{1}M\to \Phi M\xrightarrow{\lambda_{M}}M\xrightarrow{\sigma_{M}}\Sigma\Omega M\to 0,
\]
where $\sigma_{M}$ is the unit of the adjunction $\moc{\Omega\dashv\Sigma}$.
\end{prop}

With the help of $\Sigma$ and $\Phi$, we can write:
\[
\tilde{\Hrm}^{*}\moc{\Sigma\moc{\mathbb{C}P^{\infty}_{+}}}=\Sigma\Phi\chv{\fd{r}P^{\infty}}
\]
and
\[
C_{n}=\bigl\{\Sigma v^{d}\bigm|\alpha(d)\leq n\bigr\}=\Sigma\Phi \moc{R_{n} \oplus \fh}.
\]
It follows from Lemma~\ref{dyafil} that we have:

\begin{lemm}\label{omegaCn}
There are isomorphisms of unstable modules:
\[
\Sigma\Phi\Lambda^{n}\moc{F\moc{1}}\xrightarrow{\cong}\frac{C_{n}}{C_{n-1}}
\]
for all $n\geq 1$.
\end{lemm}

\subsection{The loop functor of unstable modules} Now, we compute the loop functor $\Omega$ on $\tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}$ and $\Lambda^{n}\moc{F\moc{1}}$ for all $n\geq 1$.

\begin{prop}
We have an isomorphism of unstable modules:
\[
\Omega \chv{\fd{r}P^{\infty}} \cong \Phi \chv{\fd{r}P^{\infty}}.
\]
\end{prop}
\begin{proof}
Recall that $\chv{\fd{r}P^{\infty}}$ is the polynomial algebra $\fh\mocv{u}$ on one generator $u$ of degree~$1$. Define an $\fh-$linear transformation $\alpha:\fh\mocv{u}\to \Sigma\Phi \fh\mocv{u}$ by:
\[
\alpha\moc{u^{n}}=
\begin{cases}
\Sigma\Phi u^{k} &\text{if }n=2k+1, \\
0 & \text{otherwise.}
\end{cases}
\]
Because of the formulae
\begin{align*}
Sq^{k}u^{2n}&=
\begin{cases}
\binom{n}{m}u^{2m+2n} &\text{if }k=2m, \\
0 &\text{otherwise},
\end{cases} \\
Sq^{k}\Sigma\Phi u^{n} &=
\begin{cases}
\Sigma\Phi\binom{n}{m} u^{m+n} &\text{if }k=2m, \\
0 &\text{otherwise},
\end{cases}
\end{align*}
then $\alpha$ is also a morphism over the Steenrod algebra. It is clear that $\alpha$ is surjective and the sequence
\[
0\to \Phi \fh\mocv{u}\xrightarrow{ \lambda _{ \fh\mocv{u} } }\fh\mocv{u}\xrightarrow{\alpha}\Sigma\Phi \fh\mocv{u}\to 0
\]
is exact. Therefore, the conclusion follows from Proposition~\ref{exactseq}.
\end{proof}

\begin{coro}\label{hq:27}
We have an isomorphism of unstable modules:
\[
\Omega \tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}} \cong \Phi \chv{\fd{r}P^{\infty}}.
\]
\end{coro}

\begin{proof}
It follows from the fact that $\chv{\fd{r}P^{\infty}}\cong \tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}\oplus\fh$ and $\Omega\fh\cong\mocn{0}$.
\end{proof}

Similarly, we have:

\begin{prop}\label{md:28}
We have an isomorphism of unstable modules $
\Omega R_{n}\cong \Phi R_{n-1} $ for all $n\geq 1$.
\end{prop}

\begin{coro}
We have an isomorphism of unstable modules $
\Omega \Lambda^{n}\moc{F\moc{1}}\cong \Phi \Lambda^{n-1}\moc{F\moc{1}} $ for all $n\geq 1$.
\end{coro}

\subsection{Brown--Gitler modules}

For an $\fh-$vector space $M$, denote by $M^{\sharp}$ the linear dual $\ho{\fh}{M}{\fh}$. The functor $J^{n}:\calU\to \calV_{\fh},M\mapsto M^{n,\sharp}$ from the category $\calU$ of unstable modules to the category of $\fh-$vector spaces is representable for all natural numbers $n$ (see~\cite[Chapter~2]{Sch94}). Denote by $J(n)$ the representing object. As the functor $J^{n}$ is exact, $J(n)$ in an injective unstable module for all integers $n\geq 0$. It is cogenerated in an unstable way by an element $\imath_{n}$ of degree $n$, that is,
\[
J(n)^{k}=
\begin{cases}
\mocn{0} &\text{if }k>n, \\
\fh\mocg{\imath_{n}} &\text{if }k=n,\\
\fh\mocg{\theta_{I}\nmoct{Sq^{I} \text{ is admissible of degree }n-k, e\moc{Sq^{I}}\leq k}}&\text{if }k<n,
\end{cases}
\]
where $Sq^{I}\theta_{I}=\imath_{n}$. As
\[
\ho{\calU}{J(n)}{J\moc{m}}\cong J(n)^{\sharp},
\]
a morphism from $J(n)$ to $J\moc{m}$ is determined by a Steenrod operation $\theta$. Follows~\cite{Sch94}, we denote such morphism by $\bullet\theta$. The composition law is
\[
\moc{\bullet\omega}\circ\moc{\bullet\sigma}=\bullet\moc{\sigma\omega}.
\]
For all unstable modules $M$, the morphism
\[
M\hookrightarrow \prod_{\substack{x\in M^{n,\sharp},\\n\geq 0}}J(n)
\]
is injective. It follows that the $J(n)$'s form a system of injective cogenerators for the category $\calU$.


\section{Injective resolutions and the conjecture}

In~\cite{NC16}, the author develops a new technique to construct the minimal injective resolution of $\Sigma^{t}\fh$. In this section, we will use these constructions to verify Conjecture~\ref{MainConj} for $s\geq \mocv{t/2}$.

Recall that the squaring of the Bockstein operation $Sq^{1}$ is trivial. Therefore, the Bockstein operation $Sq^{1}$ defines the following complex:

\begin{defi}
The sequence
\[
\cdots\xrightarrow{\bullet Sq^{1}}J\moc{4k}\xrightarrow{\binom{\bullet Sq^{1}}{\bullet Sq^{2k}}}
\begin{smallmatrix}
J\moc{4k-1}\\ \oplus\\J\moc{2k}
\end{smallmatrix}\xrightarrow{\moc{\bullet Sq^{1},\bullet Sq^{2k}}}J\moc{4k-2}\xrightarrow{\bullet Sq^{1}}J\moc{4k-3}\xrightarrow{\bullet Sq^{1}}\cdots
\]
is called the Bockstein complex of Brown--Gitler modules.
\end{defi}

Here the same matrix notation as for direct sums of free modules is used for morphisms between direct sum of Brown--Gitler modules. We write the Bockstein complex cohomologically as $\moc{\calB^{-k},\beta^{-k}:\calB^{-k}\to \calB^{1-k}}_{k\geq 1}$, where
\[
\calB^{-k}=
\begin{cases}
J\moc{4m-1}\oplus J\moc{2m} &\text{if }k=4m-1, \\
J\moc{k} &\text{otherwise}.
\end{cases}
\]

\begin{lemm}[{\cite[Proposition~7.8]{NC16}}]
The Bockstein complex is exact.
\end{lemm}

It turns out that the Bockstein complex plays an important role in the minimal injective resolution of $\Sigma^{t}\fh$.
\begin{theo}[{\cite[Proposition~7.8]{NC16}}]\label{MinRes}
For all integers $t\geq 1$, the injective dimension of $\Sigma^{t}\fh$ is $t-1$. Moreover, if $\moc{I^{k},\partial^{k}}_{t-1\geq k\geq 0}$ is the minimal injective resolution of $\Sigma^{t}\fh$, then each term $I^{k}$ is a finite direct sum of Brown--Gitler modules and for $s\geq \mocv{t/2}$, we have:
\[
I^{s}\cong \calB^{s-t}.
\]
\end{theo}

\begin{coro}
For all integers $t\geq 1$ and $t-1\geq s\geq \mocv{t/2}$, we have:
\begin{align*}
\ext{\calU}{s}{\tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}}{\Sigma^{t}\fh} &=
\begin{cases}
\fh & \text{if } t-s\equiv 3(4), \\
\mocn{0} & \text{otherwise,}
\end{cases} \\
\ext{\calU}{s}{\frac{R_{n}}{R_{n-1}}}{\Sigma^{t}\fh} & =
\begin{cases}
\fh & \text{if } t-s\equiv 3(4) \text{ and }\alpha\moc{t-s}=n, \\
\mocn{0} & \text{otherwise.}
\end{cases}
\end{align*}
\end{coro}

\begin{proof}
Denote $\tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}$ by $\tilde{H}$. It follows from Theorem~\ref{MinRes} that we have
\begin{align*}
\ho{\calU}{\tilde{H}}{I^{s}} &\cong \ho{\calU}{\tilde{H}}{\calB^{s-t}}\\
&\cong
\begin{cases}
\tilde{H}^{t-s,\sharp}\oplus \tilde{H}^{\moc{\frac{t-s+1}{2}},\sharp}& \text{if } t-s\equiv 3(4), \\
\tilde{H}^{t-s,\sharp} & \text{otherwise,}
\end{cases}\\
&\cong
\begin{cases}
\fh^{\oplus 2} & \text{if } t-s\equiv 3(4), \\
\fh & \text{otherwise,}
\end{cases}
\end{align*}
and
\begin{align*}
\ho{\calU}{\tilde{H}}{\partial^{s}} &\cong \ho{\calU}{\tilde{H}}{\beta^{s-t}}\\
&\cong \nmocn{
\begin{array}{cclrll}
\binom{\Id}{\Id}&:&\fh&\to& \fh^{\oplus 2}& \text{ if } t-s\equiv 0(4), \\
0&:&\fh&\to&\fh &\text{ if }t-s\equiv 1(4),\\
\Id&:&\fh&\to&\fh &\text{ if }t-s\equiv 2(4),\\
\moc{0,0}&:&\fh^{\oplus 2}&\to& \fh & \text{ if } t-s\equiv 3(4).
\end{array}}
\end{align*}
It follows that we have
\[
\ext{\calU}{s}{\tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}}{\Sigma^{t}\fh} =
\begin{cases}
\fh & \text{if } t-s\equiv 3(4), \\
\mocn{0} & \text{otherwise.}
\end{cases}
\]
Now, as $R_{n}/R_{n-1}\cong\Lambda^{n}\moc{F\moc{1}}$, we have:
\begin{align*}
\ho{\calU}{\frac{R_{n}}{R_{n-1}}}{I^{s}} &\cong \ho{\calU}{\Lambda^{n}\moc{F\moc{1}}}{\calB^{s-t}}\\
&\cong
\begin{cases}
\Lambda^{n}\moc{F\moc{1}}^{t-s,\sharp}\oplus \Lambda^{n}\moc{F\moc{1}}^{\moc{\frac{t-s+1}{2}},\sharp}& \text{if } t-s\equiv 3(4), \\
\Lambda^{n}\moc{F\moc{1}}^{t-s,\sharp} & \text{otherwise,}
\end{cases}\\
&\cong
\begin{cases}
\fh & \text{if } \alpha\moc{t-s}=n, \\
\fh & \text{if } \alpha\moc{t-s+1}=n, \text{ and }t-s\equiv 3\moc{4}, \\
0 & \text{otherwise.}
\end{cases}
\end{align*}
and
\begin{align*}
\ho{\calU}{\tilde{H}}{\partial^{s}} &\cong \ho{\calU}{\tilde{H}}{\beta^{s-t}}\\
&\cong \begin{cases}
\Id:\fh\to \fh & \text{if } t-s\equiv 0(4),\quad\alpha\moc{t-s}=n, \\
\Id:\fh\to\fh &\text{if }t-s\equiv 2(4),\quad\alpha\moc{t-s}=n, \\
0&\text{otherwise}.
\end{cases}\\[-1em]&\qedhere
\end{align*}
\end{proof}

Similarly, we have:

\begin{coro}
For all integers $t\geq 1$ and $t-1\geq s\geq \mocv{t/2}$, we have:
\begin{align*}
\ext{\calU}{s}{\tilde{\Hrm}^{*}\moc{\Sigma\moc{\mathbb{C}P^{\infty}_{+}}}}{\Sigma^{t}\fh} &=
\begin{cases}
\fh & \text{if } t-s\equiv 1(2), \\
\mocn{0} & \text{otherwise,}
\end{cases} \\
\ext{\calU}{s}{\frac{C_{n}}{C_{n-1}}}{\Sigma^{t}\fh} & =
\begin{cases}
\fh & \text{if } t-s\equiv 1(2) \text{ and }\alpha\moc{t-s-1}=n, \\
\mocn{0} & \text{otherwise.}
\end{cases}
\end{align*}
\end{coro}

We can now make the following conclusion.

\begin{coro}\label{mainThm1}
For all integers $t\geq 1$ and $t-1\geq s\geq \mocv{t/2}$, we have:
\begin{align*}
\bigoplus_{n\geq 1}\ext{\calU}{s}{\frac{R_{n}}{R_{n-1}}}{\Sigma^{t}\fh} &\cong \ext{\calU}{s}{\tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}}{\Sigma^{t}\fh},\\
\bigoplus_{n\geq 1}\ext{\calU}{s}{\frac{C_{n}}{C_{n-1}}}{\Sigma^{t}\fh} &\cong \ext{\calU}{s}{\tilde{\Hrm}^{*}\moc{\Sigma\moc{\mathbb{C}P^{\infty}_{+}}}}{\Sigma^{t}\fh}.
\end{align*}
\end{coro}


\section{Projective resolutions and the conjecture}

In this section, we study the minimal projective resolution of the objects of interest and provide some other computations supporting Conjecture~\ref{MainConj}, namely the computations of $\Hom$ and $\Ext^{1}$ groups.

\subsection{Minimal projective resolutions}

We recall the definition of the minimal projective resolution of an unstable module and some basic properties.

\begin{defi}
Let $M$ be an unstable module. A projective resolution $\moc{P_{k},\partial_{k}}_{k\geq 0}$ of $M$ is called \emph{minimal} if $P_{0}$ is the projective cover of $M$, $P_{1}$ is the projective cover of $\ke{P_{0}\twoheadrightarrow M}$, and $P_{k}$ is that of $\ke{\partial_{k-1}}$ for all integers $k\geq 2$.
\end{defi}

\begin{rema}
The projective cover of a given module is unique up to isomorphisms so it follows from standard argument that two minimal projective resolutions of the same unstable module are isomorphic.
\end{rema}

Recall that an unstable module $M$ is of finite type if $M^{n}$ is an $\fh-$vector space of finite dimension for all $n\geq 0$.
\begin{lemm}
Let $M$ be an unstable module of finite type. Suppose that the sequence
\[
P_{1}\xrightarrow{\partial}P_{0}\to M\to 0
\]
is exact where $P_{0}$ and $P_{1}$ are projective unstable modules. Then, $P_{0}$ is the projective cover of $M$ if and only if for all integers $n\geq 0$, there are no morphisms $\phi:F(n)\to P_{1}$ and $\psi:P_{0}\to F(n)$ such that $\psi\circ\partial\circ\phi$ is an isomorphism.
\end{lemm}
\begin{proof}
Suppose that $P_{0}$ is the projective cover of $M$, then $P_{0}$ is a direct sum of free unstable modules $F(m)$. Assume that there are morphisms $\phi:F(n)\to P_{1}$ and $\psi:P_{0}\to F(n)$ such that $\psi\circ\partial\circ\phi$ is an isomorphism. Then, it follows that $\psi$ is surjective. So $\psi$ has a section, called $\bar{\psi}$. Therefore, there is an exact sequence
\[
\cke{\phi}\to \cke{\bar{\psi}}\to M\to 0.
\]
On the one hand, $\cke{\bar{\psi}}$ is a projective unstable module. On the other hand, as $M$ is of finite type, then the number of summands $F(n)$ of $P_{0}$ is finite. Therefore, the number of summands $F(n)$ of $\cke{\bar{\psi}}$ is strictly less than that of $P_{0}$. It follows that there is no surjection from $\cke{\bar{\psi}}$ to $M$, whence a contradiction.

Now, if $P_{0}$ is not the projective cover of $M$, then there is an epimorphism from $P_{0}$ to the projective cover $P$ of $M$ with nontrivial kernel. Let $F(n)$ be a direct summand of $P_{0}$ lying in the kernel of the projection $P_{0}\to P$, then the composition
\[
P_{1}\xrightarrow{\partial}P\to F(n)
\]
is surjective. It follows that $P_{1}$ contains a direct summand $F(n)$ such that the composition
\[
F(n)\hookrightarrow P_{1}\xrightarrow{\partial}P_{0}\twoheadrightarrow F(n)
\]
is an isomorphism. The conclusion follows.
\end{proof}

Therefore, for unstable modules of finite type, we get the following simple characterization of minimal projective resolutions.

\begin{prop}\label{minimalresolution}
Let $M$ be an unstable module of finite type. A projective resolution $\moc{P_{k},\partial_{k}}_{k\geq 0}$ of $M$ is minimal if and only if for all integers $n,k\geq 0$, there are no morphisms $\phi:F(n)\to P_{k+1}$ and $\psi:P_{k}\to F(n)$ such that $\psi\circ\partial_{k+1}\circ\phi$ is an isomorphism.
\end{prop}

\begin{coro}
Let $M$ be an unstable module of finite type and let $\moc{P_{k},\partial_{k}}_{k\geq 0}$ be the minimal projective resolution of $M$. Then, there are isomorphisms
\[
\ext{\calU}{s}{M}{\Sigma^{n}\fh}\cong \ho{\calU}{P_{s}}{\Sigma^{n}\fh}
\]
of $\fh-$vector spaces for all integers $s\geq 0$.
\end{coro}

\begin{proof}
Note that
\[
\ho{\calU}{F\moc{k}}{\Sigma^{n}\fh}\cong
\begin{cases}
\fh & \text{if }k=n, \\
\mocn{0} & \text{otherwise}.
\end{cases}
\]
Following Proposition~\ref{minimalresolution}, if $\moc{P^{k},\partial_{k}}_{k\geq 0}$ is the minimal projective resolution of \(M \), then
\[
\ho{\calU}{\partial^{\bullet}}{\Sigma^{n}\fh}=0.
\]
The conclusion follows.
\end{proof}

\begin{coro}\label{omegares}
Let $M$ be an unstable module of finite type such that $\Omega_{1}M=\mocn{0}$. If $\moc{P_{k},\partial_{k}}_{k\geq 0}$ is the minimal projective resolution of $M$, then $\moc{\Omega P_{k},\Omega\partial_{k}}_{k\geq 0}$ is the minimal projective resolution of $\Omega M$.
\end{coro}

\begin{proof}
From~\cite[Lemma~2.8]{NC19}, $\moc{\Omega P_{k},\Omega\partial_{k}}_{k\geq 0}$ is a projective resolution of $\Omega M$ because \(\Omega _{ 1 } M = 0 \). Due to Proposition~\ref{minimalresolution}, this resolution is minimal.
\end{proof}

\subsection{Pseudo-hyperresolutions}

In this paragraph, we use pseudo-hyperresolutions to construct projective resolutions of the objects of interest: $\tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}$, $\Sigma\Phi\chv{\fd{r}P^{\infty}}$, $\Lambda^{n}\moc{F\moc{1}}$ and $\Sigma\Phi\Lambda^{n}\moc{F\moc{1}}$ for all $n\geq 1$. Our aim is modest: we only compute the first and second terms of the minimal projective resolutions of these objects for they give access to the computations of $\Hom$ and $\Ext^{1}$ groups. Because of the following lemma, if we can verify Conjecture~\ref{MainConj} for \(\mathbb{R} P ^{ \infty } \), then the result for \(\mathbb{C} P ^{ \infty } \) also holds. For this reason, we focus only on \(\mathbb{R} P ^{ \infty } \).

\begin{lemm}
If the isomorphism
\[
\bigoplus_{n\geq 1}\ext{\calU}{s}{\frac{R_{n}}{R_{n-1}}}{\Sigma^{t}\fh} \cong \ext{\calU}{s}{\tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}}{\Sigma^{t}\fh},
\]
holds, then so does Conjecture~\ref{MainConj}.
\end{lemm}

\begin{proof}
It follows from~\cite[Lemma~3.4]{NC19} that we have:
\begin{align*}
\ext{\calU}{s}{\Sigma\Omega \tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}}{\Sigma^{t}\fh} &\cong \ext{\calU}{s}{\Omega \tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}}{\Sigma^{t-1}\fh}\oplus \ext{\calU}{s-1}{\Omega \Phi\tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}}{\Sigma^{t-1}\fh} \\
&\cong \ext{\calU}{s}{\tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}}{\Sigma^{t}\fh}\oplus \ext{\calU}{s-1}{\Phi\tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}}{\Sigma^{t}\fh},\\
\ext{\calU}{s}{\Sigma\Omega \frac{ R_{n}}{R_{n-1}}}{\Sigma^{t}\fh} &\cong \ext{\calU}{s}{\Omega \frac{ R_{n}}{R_{n-1}}}{\Sigma^{t-1}\fh}\oplus \ext{\calU}{s-1}{\Omega \Phi\frac{ R_{n}}{R_{n-1}}}{\Sigma^{t-1}\fh} \\
&\cong \ext{\calU}{s}{\frac{ R_{n}}{R_{n-1}}}{\Sigma^{t}\fh}\oplus \ext{\calU}{s-1}{\Phi\frac{ R_{n}}{R_{n-1}}}{\Sigma^{t}\fh}.
\end{align*}
Note that by Corollary~\ref{hq:27} and Proposition~\ref{md:28} together with the right exactness of \(\Omega \) we have
\begin{align*}
\Omega \tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}&\cong \Phi\chv{\fd{r}P^{\infty}} \\
\Omega \frac{ R_{n}}{R_{n-1}} & \cong \frac{\Phi R_{n}}{\Phi R_{n-1}}.
\end{align*}
Now, it follows from Lemma~\ref{omegaCn} that we have
\[
\frac{ \Phi R _{ n } }{ \Phi R _{ n - 1 } } \cong \frac{ C _{ n } }{ C _{ n - 1 } }
\]
Then, thanks to Corollary~\ref{omegares}, we have an isomorphism
\[
\bigoplus_{n\geq 1}\ext{\calU}{s}{\frac{C_{n}}{C_{n-1}}}{\Sigma^{t}\fh} \cong \ext{\calU}{s}{\Phi\chv{\fd{r}P^{\infty}}}{\Sigma^{t}\fh}.
\]
The conclusion follows.
\end{proof}

Thanks to this lemma, it suffices to study the minimal projective resolution of $\tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}$ and $\Lambda^{n}\moc{F\moc{1}}$ for all $n\geq 1$. Note that $\Lambda^{1}\moc{F\moc{1}}\cong F\moc{1}$ is projective, so we only need to study the cases $n\geq 2$.

Observe that if \(Sq ^{ I } \) is an admissible Steenrod operation, then \(\bullet Sq ^{ I } : J \left(m \right) \to J \left(n \right) \) is nontrivial if and only if the excess of \(Sq ^{ I } \) is less than or equal to \(n \). It follows that the equality \(\bullet \alpha = \bullet \beta : J \left(m \right) \to J \left(n \right) \) implies that \(\alpha - \beta \) is a sum of admissible operations of excess greater than \(n \). That is what we use to verify the following technical computation of Steenrod operations.

\begin{lemm}\label{admis}
Let $Sq^{a}Sq^{b}$ be admissible of excess $n$. Then, every admissible term in $Sq^{a}Sq^{b}-Sq^{a+b-n}Sq^{n}$ is of excess strictly greater than $n$.
\end{lemm}

\begin{proof}
Consider the following commutative diagram:
\[
\xymatrixcolsep{5em}\xymatrix{
\Sigma J\moc{2a+2b+2n-1}\ar[r]^-{\Sigma\bullet Sq^{2a}Sq^{2b}}\ar[d]&\Sigma J\moc{2n-1}\ar[d]\\
J\moc{2a+2b+2n}\ar[r]^-{\bullet Sq^{2a}Sq^{2b}}\ar[d]_{\bullet Sq^{a+b+n}}&J\moc{2n}\ar[d]^{\bullet Sq^{n}}\\
J\moc{a+b+n}\ar[r]^-{\bullet Sq^{a}Sq^{b}}&J(n) }
\]
Note that $a-b=n$ then $a=b+n$ and $a+b+n=2b+2n$. Therefore, we have
\[
Sq^{a+b+n}Sq^{a+b-n}=Sq^{2a}Sq^{2b}.
\]
Hence, we have
\begin{align*}
\moc{\bullet Sq^{n}}\circ\moc{\bullet Sq^{a+b-n}}\moc{\bullet Sq^{a+b+n}} & =\moc{\bullet Sq^{n}}\circ\moc{\bullet Sq^{2a}Sq^{2b}}\\
&= \moc{\bullet Sq^{a}Sq^{b}}\circ\moc{\bullet Sq^{a+b+n}}.
\end{align*}
As $\bullet Sq^{a+b+n}$ is surjective, we have:
\[
\moc{\bullet Sq^{a}Sq^{b}}=\moc{\bullet Sq^{n}}\circ\moc{\bullet Sq^{a+b-n}}=\bullet \moc{Sq^{a+b-n}Sq^{n}}
\]
whence the conclusion.
\end{proof}

\begin{lemm}\label{odd}
If $a,b$ are two odd numbers such that $a<2b$, then
\[
Sq^{a}Sq^{b}=\sum_{i=0}^{\moc{\mocv{a/2}-1}/2}\binom{b-2i-2}{a-4i-2}Sq^{a+b-2i-1}Sq^{2i+1}.
\]
\end{lemm}
\begin{proof}
Let $m>a+b$ be an integer and let $n=2m+b$ and $2k=n+a$. Consider the following commutative diagram:
\[
\xymatrixcolsep{5em}\xymatrix{ J\moc{2k}\ar[r]^-{\bullet Sq^{a}}\ar[d]_{\bullet Sq^{k}}&J(n)\ar[r]^{\bullet Sq^{b}}&J\moc{2m}\ar[d]^{\bullet Sq^{m}}\\
J\moc{k}\ar[r]_-{\tilde{\Phi}\moc{\bullet Sq^{a}}}&0\ar[r]_-{\tilde{\Phi}\moc{\bullet Sq^{b}}}&J\moc{m} }
\]
Note that, we have:
\begin{enumerate}\romanenumi
\item $\tilde{\Phi}\moc{\bullet Sq^{a}}=\tilde{\Phi}\moc{\bullet Sq^{b}}=0$;
\item $\tilde{\Phi}\moc{\bullet Sq^{b}}\circ \tilde{\Phi}\moc{\bullet Sq^{a}}=\tilde{\Phi}\moc{\bullet Sq^{a}Sq^{b}}$.
\end{enumerate}
The conclusion follows.
\end{proof}

\begin{lemm}\label{connected}
Let $M$ be a connected unstable module, that is $M^{0}=\mocn{0}$. If $\lambda_{M}:\Phi M\to M,x\mapsto Sq_{0}x$ is surjective then $M=\mocn{0}$.
\end{lemm}
\begin{proof}
As $\Phi M$ is concentrated in even degree and \(\lambda _{ M } : \Phi M \to M \) is surjective, then $M$ is also concentrated in even degree. A simple induction on \(k \geq 1 \) shows that \(M \) is concentrated in degree divisible by \(2 ^{ k } \). Hence, \(M ^{ n } = \left\{ 0 \right\} \) for all \(n \geq 1 \). Moreover, as \(M \) is connected, we have \(M ^{ 0 } = \left\{ 0 \right\} \). Then, we have \(M = \left\{ 0 \right\} \).
\end{proof}

The following result is a mean to construct a projective resolution for $M$ from that of $\Omega M$.

\begin{coro}\label{augmentation}
Let $M$ be a connected unstable module such that $\Omega_{1}M=\mocn{0}$. Let $\calP$ be a complex
\begin{equation}\label{omegatoID}
P_{n}\xrightarrow{\partial_{n}}P_{n-1}\xrightarrow{\partial_{n}}\cdots \xrightarrow{\partial_{n}}P_{0}\to M
\end{equation}
in which each $P_{i}$, $0\leq i\leq n$, is projective. If\/ $\Omega\calP$ is exact, then so is $\calP$.
\end{coro}

\begin{proof}
Note that the functors $\Phi$ and $\Sigma$ are exact. Then, the long exact sequence associated with the following short exact sequence of complexes
\[
0\to \Phi \calP\to \calP \to\Sigma\Omega \calP\to 0
\]
yields the triviality of the homology groups $\ch{s}{\calP}$ for all $n-1\geq s\geq 1$. Moreover, we get the following commutative diagram
\[
\xymatrixrowsep{1.3pc}
\xymatrix{
\Phi \ch{0}{\calP}\ar[r]^{\Phi f}\ar[d] &\Phi M\ar[d]\\
\ch{0}{\calP}\ar[r]^{f}\ar[d] &M \ar[d]\\
\Sigma\ch{0}{\Omega \calP}\ar[r]^{\Sigma\Omega f}& \Sigma\Omega M }
\]
where both columns are short exact sequences. Now, it follows from the Snake lemma that both $\ke{f}$ and $\cke{f}$ are connected. Moreover, by assumption \(\Omega \mathcal{P} \) is exact so $\Omega\moc{f}$ is an isomorphism. Therefore, we get the surjectivity of $\Phi \ke{f}\to\ke{f}$ and $\Phi\cke{f}\to\cke{f}$. It follows from Lemma~\ref{connected} that $f$ is an isomorphism whence the conclusion.
\end{proof}

\begin{rema}\label{rmk}
Note that we have \(\Omega \Phi F \left(n \right) \cong \Sigma \Phi F \left(n - 1 \right) \). Moreover, \(\Omega _{ 1 } \Phi F \left(n \right) = \left\{ 0 \right\} \). Therefore, we can use Lemma~\ref{augmentation} to construct projective resolutions of \(\Phi F \left(n \right) \) by induction on \(n \). We start with the case \(n = 1 \).
\end{rema}

\begin{lemm}
The following sequence is exact
\[
\cdots\to F(n)\xrightarrow{Sq^{1}}F\moc{n-1}\xrightarrow{Sq^{1}}\cdots\xrightarrow{Sq^{1}}F\moc{3}\xrightarrow{Sq^{1}}F\moc{2}\to \Phi F\moc{1}\to 0
\]
\end{lemm}

\begin{proof}
Recall that the module \(F \left(n \right) \) has an additive basis
\[
\left\{ Sq ^{ I }\imath _{ n } \,\middle|\, I \text{ is admissible and }ex \left(I \right) \leq n \right\}
\]
and the morphism \(Sq ^{ 1 } : F \left(n \right) \to F \left(n - 1 \right) \) sends \(Sq ^{ I } \imath _{ n } \) to \(Sq ^{ I } Sq ^{ 1 } \imath _{ n - 1 } \). If \(I = \left(I _{ 1 }, 1 \right) \) then \(Sq ^{ I } Sq ^{ 1 } = 0 \). Otherwise, if \(I = \left(i _{ 1 }, i _{ 2 }, \ldots, i _{ k } \right) \) is admissible of excess less than or equal to \(n \) and \(i _{ k } \geq 2 \) then \(Sq ^{ I } Sq ^{ 1 } \) is admissible of excess less than or equal to \(n - 1 \). Therefore, the kernel of \(Sq ^{ 1 } : F \left(n \right) \to F \left(n - 1 \right) \) has an additive basis
\[
\left\{ Sq ^{ I }\imath _{ n } \,\middle|\, I = \left(i _{ 1 }, i _{ 2 }, \ldots, i _{ k }, 1 \right) \text{ is admissible and }ex \left(I \right) \leq n \right\}
\]
This is also an additive basis of the image of the morphism \(Sq ^{ 1 } : F \left(n + 1 \right) \to F \left(n \right) \). Therefore, the following complex is acyclic
\[
\cdots\to F(n)\xrightarrow{Sq^{1}}F\moc{n-1}\xrightarrow{Sq^{1}}\cdots\xrightarrow{Sq^{1}}F\moc{2}\xrightarrow{Sq^{1}}F\moc{1}
\]
Because the image of \(Sq ^{ 1 } : F \left(2 \right) \to F \left(1 \right) \) is \(\Phi F \left(1 \right) \), then we get the desired conclusion.
\end{proof}

The sequence
\[
\cdots\to F(n)\xrightarrow{Sq^{1}}F\moc{n-1}\xrightarrow{Sq^{1}}\cdots\xrightarrow{Sq^{1}}F\moc{3}\xrightarrow{Sq^{1}}F\moc{2}\to \Phi F\moc{1}\to 0
\]
is the minimal projective resolution of $\Phi F\moc{1}$. We now use pseudo-hyperresolutions to give access to projective resolutions of \(\Phi F \left(n \right) \) for \(n \geq 2 \). For this, we introduce the following notations.

\begin{defi}
Let $P\moc{1}=F\moc{4}$, $Q\moc{1}=F\moc{3}$. For $n\geq 2$, define:
\[
\begin{aligned}
A_{n} &= \mocn{4n+1,4n+5,\ldots,8n-7}, \\
B_{n} &= \mocn{4n,4n+2,\ldots,8n-4}, &D_{n} &= \mocn{4n-1},\\
C_{n} &= \mocn{k+2\,|\,k \in X_{n-1}}, &E_{n} &= \mocn{k+2\,|\,k \in Q_{n-1}},\\
P_{n} &= A_{n}\sqcup B_{n} \sqcup C_{n}, &Q_{n} &= D_{n}\sqcup E_{n},
\end{aligned}
\]
and for $T\in\mocn{A,B,C,D,E,P,Q}$, let $T(n)=F\moc{T_{n}}$. We also write:
\[
\begin{array}{lccl}
a_{n}:=\moc{Sq^{2},Sq^{6},\ldots,Sq^{4n-6}}&:&A(n)&\to D(n), \\
d_{n}:=\diag\moc{Sq^{2n},Sq^{2n+2},\ldots,Sq^{4n-4}}&:&A(n)&\to E(n),\\
\tau_{n}:=\moc{Sq^{1},Sq^{3},\ldots,Sq^{2n-1}}&:&Q(n)&\to F\moc{2n}.
\end{array}
\]
\end{defi}

\begin{lemm}\label{bigRes}
For all $n\geq 1$, there exists an exact sequence
\begin{equation}\label{PhiRes}
P(n)\xrightarrow{\sigma_{n}}Q(n)\xrightarrow{\tau_{n}}F\moc{2n}\to\Phi F(n)\to 0
\end{equation}
such that \(\sigma _{ 1 } = Sq ^{ 1 } \) and for \(n \geq 2 \), we have
\[
\sigma_{n}=
\begin{pmatrix}
\tau_{2n-1}& *_{1}\\ a_{n} &d_{n}\\ *_{2} &\sigma_{n-1}
\end{pmatrix}:B(n)\oplus A(n)\oplus C(n)\to D(n)\oplus E(n).
\]
\end{lemm}
\begin{proof}
We proceed by induction on $n$. The case \(n = 1 \) holds by Remark~\ref{rmk}. The induction argument goes as follows. Assume that the result holds for $n\leq m$. We now consider the case $n=m+1$. The sequence
\[
0\to \Phi F\moc{q}\xrightarrow{\Phi x\mapsto Sq_{0}x}F\moc{q}\xrightarrow{\imath_{q}\mapsto \Sigma\imath_{q-1}}\Sigma F\moc{q-1}\to 0
\]
is exact by Proposition~\ref{exactseq}. Therefore, the following sequence
\[
F\moc{2q}\xrightarrow{\imath_{2q}\mapsto Sq^{q}\imath_{q}}F\moc{q}\to\Sigma F\moc{q-1}\to 0
\]
provides the first and the second term in the minimal projective resolution of $\Sigma F\moc{q-1}$ for all integers $q\geq 1$. If $X$ is a set of integers, and $s\in \fd{z}$, then we write $s+X=\mocn{s+k\nmoct{k\in X}}$. Consider the following diagram:
\[
\xymatrix{ &&F\moc{1+B_{m+1}}\ar[d]^{\tau_{2m+1}}\\
&F\moc{-1+A_{m+1}}\ar[r]^{a_{m+1}}\ar[d]_{d_{m+1}}&F\moc{4m+2}\ar[d]^{Sq^{2m+1}}\\
F\moc{1+P_{m}}\ar[r]_{\sigma_{m}}\ar[d]&F\moc{1+Q_{m}}\ar[d]\ar[r]_{\tau_{m}}&F\moc{2m+1}\ar[d]\\
\Sigma P\moc{m}\ar[r]_{\Sigma \sigma_{m}}& \Sigma Q\moc{m} \ar[r]_{\Sigma\tau_{m}}&\Sigma F\moc{2m} }
\]
Denote
\[
P:=F\moc{1+B_{m+1}}\oplus F\moc{-1+A_{m+1}}\oplus F\moc{1+P_{m}}.
\]
It follows from~\cite[Corollary~4.8]{NC16} that there exists a morphism
\[
\partial: F\moc{1+P_{m}}\to F\moc{4n+2}
\]
such that the sequence
\begin{equation}\label{aug}
P\xrightarrow{
\left(\begin{smallmatrix}
\tau_{2m+1}& 0\\a_{m+1} &d_{m+1}\\\partial &\sigma_{m}
\end{smallmatrix}\right) }F\moc{4n+2}\oplus F\moc{1+Q_{m}}\xrightarrow{
\begin{pmatrix}
Sq^{2m+1}&\tau_{m}
\end{pmatrix} }F\moc{2m+1}\to 0
\end{equation}
is exact. Remark that $\Omega P\moc{m+1}\cong P$ and $\Omega Q_{m+1}\cong F\moc{4n+2}\oplus F\moc{1+Q_{m}}$. Now, note that
\[
\tau_{2m+1}=\moc{ Sq^{1},Sq^{3},\ldots,Sq^{4m+1} }.
\]
It follows from Lemma~\ref{odd} that there exists $\delta:F\moc{1+B_{m+1}}\to F\moc{1+Q_{m}}$ such that
\[
Sq^{2m+1}\circ \tau_{2m+1}=\tau_{m}\circ \delta.
\]
Therefore, the sequence
\[
P\moc{m+1}\xrightarrow{
\left(\begin{smallmatrix}
\tau_{2m+1}& \delta\\a_{m+1} &d_{m+1}\\\partial &\sigma_{m}
\end{smallmatrix}\right) }Q\moc{m+1}\xrightarrow{\tau_{m+1}}F\moc{2m+2}\to 0
\]
is an exact sequence after Corollary~\ref{aug}.
\end{proof}
\begin{rema}
A simple induction on $n$ shows that all coefficients in the matrix form of $\sigma_{n}$ are Steenrod squares.
\end{rema}

\begin{lemm}\label{generators}
Let $n\geq 1$ be an odd integer, then we can write $Sq^{n}$ as a sum of products $Sq^{a}Sq^{2b+1}$, where $a>0$, if and only if $n+1$ is not a power of $2$.
\end{lemm}
\begin{proof}
Remark that if $n+1$ is not a power of $2$ then $n$ is of the form $2^{\ell}k+2^{\ell-1}-1$ for some integers $\ell\geq 2$ and $k\geq 1$. We will now show that:
\begin{equation}\label{equation415}
Sq^{2^{\ell}k+2^{\ell-1}-1}=Sq^{2^{\ell}k}Sq^{2^{\ell-1}-1}+\sum_{s=1}^{\ell-1}Sq^{2^{\ell}-2^{s}}Sq^{2^{\ell}(k-1)+{2^{\ell-1}+2^{s}-1}}
\end{equation}
In fact, for $1\leq s\leq \ell-1$, we have
\[
Sq^{2^{\ell}-2^{s}}Sq^{2^{\ell}(k-1)+2^{\ell-1}+2^{s}-1}=\sum_{t=\max\moc{2^{\ell-1}-2^{s+1}+2,0}}^{2^{\ell-1}-2^{s-1}}\binom{2^{\ell}(k-1)+2^{\ell-1}+2^{s}-2-t}{2^{\ell}-2^{s}-2t}Sq^{2^{\ell}+2^{\ell-1}-1-t}Sq^{t}.
\]
On the other hand, we have
\[
\binom{{2^{\ell-1}+2^{s}-2-t}}{2^{\ell}-2^{s}-2t}=\binom{{2^{\ell-1}+2^{s-1}-2-t}}{2^{\ell}-2^{s-1}-2t}
\]
for all $2^{\ell-1}-2^{s}+2\leq t\leq 2^{\ell-1}-2^{s-1}$, and
\[
\binom{{2^{\ell}k-2}}{2^{\ell-1}}=1.
\]
Then, Equation~\ref{equation415} follows. The remaining conclusion follows from the fact that
\[
\binom{2b}{1+2^{k}-2b}=0
\]
for all $1\leq b$.
\end{proof}

\begin{prop}[Compare with {\cite[Proposition~6.2.1]{LZ86}}]
The following sequence provides the first two terms of the minimal projective resolution of $\Phi F(n)$ for $n\geq 1$:
\[
\bigoplus_{1\leq d< \log_{2}(n)}F\moc{2n+2^{d}-1}\xrightarrow{\omega}F\moc{2n}\xrightarrow{p} \Phi F(n)\to 0.
\]
where $\omega\moc{\imath_{2n+2^{d}-1}}=Sq^{2^{d}-1}\imath_{2n}$, and $p\moc{\imath_{2n}}=\Phi \imath_{n}$.
\end{prop}
\begin{proof}
It follows from Lemma~\ref{bigRes} that $\bigl\{Sq^{2k-1}\imath_{n}\bigm|1\leq k\leq n\bigr\}$ is a set of generators of $\ke{p}$. By Lemma~\ref{generators}, $\bigl\{Sq^{2^{d}-1}\imath_{n}\bigm| 1\leq d\leq \log_{2}(n)\bigr\}$ form a minimal set of generators of $\ke{p}$, whence the conclusion.
\end{proof}

The first two terms of the minimal projective resolution of \(\Phi F \left(n \right) \) can also be found in~\cite[Proposition~6.2.1]{LZ86}. However, in~\cite{LZ86}, instead of \(\omega \), the morphism from the second to the first term of the resolution sends \(\imath _{ 2n + 2 ^{ d } - 1 } \) to \(Q _{ d - 1 } \imath _{ 2n } \). In this paper, we choose \(Sq ^{ 2 ^{ d } - 1 } \) instead of the Milnor operation \(Q _{ d - 1 } \) because it gives us the following result, which can also be found in~\cite[Theorem~5]{Woo95}.
\begin{coro}
For every odd number $n$, the operation $Sq^{n}$ is equal to a sum of products of the form $Sq^{n-2^{d}+1}Sq^{2^{d}-1}$.
\end{coro}

Similarly, we get the following result.

\begin{prop}\label{md:420}
The following sequence provides the first two terms of the minimal projective resolution of $\Lambda^{n}\moc{F\moc{1}}$ for $n\geq 1$:
\[
\bigoplus_{0\leq d\leq n-2}F\moc{2^{n}+2^{d}-1}\xrightarrow{\partial}F\moc{2^{n}-1}\to \Lambda^{n}\moc{F\moc{1}}\to 0.
\]
where $\partial\moc{\imath_{2^{n}+2^{d}-1}}=Sq^{2^{d}}\imath_{2^{n}-1}$.
\end{prop}

In what follows, denote by $\moc{\mathfrak{P}_{k},\mathfrak{d}_{k}}_{k\geq 0}$ the minimal projective resolution of $\tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}$.

\begin{lemm}
The first term $\mathfrak{P}_{0}$ of the minimal projective resolution of $\tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}$ is isomorphic to the direct sum of $F\moc{2^{n}-1}$ with $n\geq 1$.
\end{lemm}

\begin{proof}
It is clear that the minimal set of $\calA-$generators of $\tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}$ is $\bigl\{u^{2^{n}-1}\bigm|n\geq 1\bigr\}$. The conclusion follows.
\end{proof}

\begin{prop}
There exists a projective unstable module $Q$ such that
\[
\mathfrak{P}_{1}\cong Q\oplus\moc{\bigoplus_{2\leq n}\moc{\bigoplus_{0\leq d\leq n-2}F\moc{2^{n}+2^{d}-1}}}.
\]
\end{prop}

\begin{proof}
Denote by $\varphi$ the projection
\[
\begin{array}{ccc}
\mathfrak{P}_{0}\cong\displaystyle\bigoplus_{1\leq n} F\moc{2^{n}-1}&\longrightarrow& \tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}} \\
\imath_{2^{n}-1} &\longmapsto& u^{2^{n}-1}
\end{array}
\]
and by $\omega_{n,d}$ the element:
\[
\omega_{n,d}:=Sq^{2^{d}}\imath_{2^{n}-1}+Sq^{2^{n-1}}\cdots Sq^{2^{d+1}}Sq^{2^{d}}\imath_{2^{d+1}-1}
\]
for all $2\leq n$ and $0\leq d\leq n-2$. It is clear that $\varphi\moc{\omega_{n,d}}=0$. As $Sq^{2^{d}}$ is indecomposable, $\omega_{n,d}$ is $\calA-$indecomposable. It follows that \(\omega _{ n, d } \) must be hit by the generator of a free module in a minimal resolution, whence the conclusion.
\end{proof}

\begin{coro}\label{hq:423}
The following sequence provides the first two terms of the minimal projective resolution of $\tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}$:
\[
\bigoplus_{2\leq n}\moc{\bigoplus_{0\leq d\leq n-2}F\moc{2^{n}+2^{d}-1}}\xrightarrow{\partial}\bigoplus_{1\leq n} F\moc{2^{n}-1}\to \tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}\to 0.
\]
where
\[
\partial\moc{\imath_{2^{n}+2^{d}-1}}=Sq^{2^{d}}\imath_{2^{n}-1}+Sq^{2^{n-1}}\cdots Sq^{2^{d+1}}Sq^{2^{d}}\imath_{2^{d+1}-1}.
\]
\end{coro}

\begin{proof}
We will show that $Q\cong \mocn{0}$. Let $I$ be a set of integers such that
\[
Q\cong \underset{k\in I}{\bigoplus}F\moc{k}.
\]
From the connectivity of the kernel of the map
\[
\bigoplus_{1\leq n} F\moc{2^{n}-1}\to \tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}},
\]
the set $I$ cannot contain $0,1,2,3$. By constructing the pseudo-hyperresolution for $\Phi\chv{\fd{r}P^{\infty}}$, it is easy to see that
\[
\{n-1\,|\,n\in I\}=\{2n\,|\,n\in I\}.
\]
Therefore, if $I$ is not empty, then it must contain $1$, whence a contradiction. It follows that $I=\emptyset$, whence $Q\cong\mocn{0}$.
\end{proof}

\begin{coro}\label{mainThm2}
Conjecture~\ref{MainConj} holds for $s\leq 1$.
\end{coro}

\begin{proof}
Note that we need to verify the following isomorphisms for \(s \leq 1 \):
\[
\displaystyle\bigoplus_{n\geq 1}\ext{\calU}{s}{\frac{R_{n}}{R_{n-1}}}{\Sigma^{t}\fh} \cong \ext{\calU}{s}{\tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}}}{\Sigma^{t}\fh},\\
\]
As \(\Sigma ^{ t } \mathbb{F} _{ 2 } \) is a simple object, the Ext-group \(\ext{\calU}{s}{M}{\Sigma^{t}\fh} \) is isomorphic to the Hom-group \(\Hom _{ \calU } \left(P _{ s }, \Sigma ^{ t } \fh \right) \) where \(P _{ s } \) is the \(s- \)th term of the minimal projective resolution of the unstable module \(M \). On the one hand, Proposition~\ref{md:420} provides the first two terms of the minimal projective resolution of \(\Lambda ^{ n } \left(F \left(1 \right) \right) \cong R _{ n } / R _{ n - 1 } \) for \(n \geq 1 \). On the other hand, Corollary~\ref{hq:423} provides the first two terms of the minimal projective resolution of \(\tilde{\Hrm}^{*}\moc{\fd{r}P^{\infty}} \). Then, the conclusion follows from counting the dimension of both sides of the isomorphisms.
\end{proof}

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