# Operations on spectral partition Lie algebras and TAQ cohomology

Adela YiYu Zhang  Department of Mathematics, Massachusetts Institute of
Technology, Cambridge MA 02139<EMAIL_ADDRESS>
###### Abstract.

We determine all natural operations and their relations on the homotopy groups
of spectral partition Lie algebras, as well as mod $p$ Topological André-
Quillen cohomology operations at any prime. We construct unary operations and
a shifted restricted Lie algebra structure on the homotopy groups of spectral
partition Lie algebras. Then we prove a composition law for the unary
operations, as well as a compatibility condition between unary operations and
the shifted Lie bracket with restriction. Comparing with Brantner-Mathew’s
result on the ranks of the homotopy groups of free spectral partition Lie
algebras, we deduce that these generate all natural operations and obtain the
target category. Since $\mathbb{F}_{p}$-linear TAQ cohomology operations
coincide with operations on the homotopy groups of spectral partition Lie
algebras, we deduce the structure of $\mathbb{F}_{p}$-linear TAQ cohomology
operations, thereby clarifying unpublished results by Kriz and Basterra-
Mandell. As a corollary, we determine all natural operations on mod $p$
$\mathbb{S}$-linear TAQ cohomology, along with relations among the operations.

##  1\. Introduction

In this paper, we give explicit constructions of all natural operations on the
homotopy groups of spectral partition Lie algebras over $H\mathbb{F}_{p}$, as
well as mod $p$ Topological André-Quillen cohomology operations. We then
determine all relations among the operations.

The project started as an attempt to refine a result of Brantner and Mathew on
the ranks of the homotopy groups of the free spectral partition Lie algebras
in [BM19]. Roughly speaking, spectral partition Lie algebras over
$H\mathbb{F}_{p}$ are divided power algebras Koszul dual to non-unital
$\mathbb{E}_{\infty}$-$H\mathbb{F}_{p}$-algebras, implementing the Koszul
duality between the non-unital $\mathbb{E}_{\infty}$-operad and the spectral
Lie operad. Since spectral partition Lie algebras are algebras over a certain
monad $\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}$, the homotopy
groups of free spectral partition Lie algebras
$\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(\Sigma^{i_{1}}H\mathbb{F}_{p}\oplus\cdots\oplus\Sigma^{i_{k}}H\mathbb{F}_{p})$
parametrize all natural $k$-ary operations on the homotopy groups of spectral
partition Lie algebras as $(i_{1},\ldots,i_{k})$ varies. In [BM19, Theorem
1.20], Brantner and Mathew obtained bases for homotopy groups of free spectral
partition Lie algebras on single generators via an isotropy spectral sequence
studied by Arone, Dwyer, and Lesh ([ADL13]). Then they propagated the result
to multiple generators by means of an EHP sequence and a decomposition of the
partition complex developed by Arone and Brantner in [AB21]. Nonetheless, this
method did not provide explicit descriptions of the nature of the operations,
nor were the relations among the operations clarified.

On the other hand, $\mathbb{F}_{p}$-linear TAQ cohomology
$\mathrm{TAQ}^{*}(R,H\mathbb{F}_{p};H\mathbb{F}_{p})$ of
$\mathbb{E}_{\infty}$-$H\mathbb{F}_{p}$-algebras $R$ has representing objects
trivial square-zero extensions. Hence reduced $\mathbb{F}_{p}$-linear TAQ
cohomology groups of trivial algebras
$H\mathbb{F}_{p}\oplus\Sigma^{i_{1}}H\mathbb{F}_{p}\oplus\cdots\oplus\Sigma^{i_{k}}H\mathbb{F}_{p}$
parametrize all natural $k$-ary operations. By [BM19, Proposition 5.35], there
is an isomorphism

$\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(\Sigma^{i_{1}}H\mathbb{F}_{p}\oplus\cdots\oplus\Sigma^{i_{k}}H\mathbb{F}_{p}))\oplus\mathbb{F}_{p}\cong\mathrm{TAQ}^{-*}(H\mathbb{F}_{p}\oplus\Sigma^{-i_{1}}H\mathbb{F}_{p}\oplus\cdots\oplus\Sigma^{-i_{k}}H\mathbb{F}_{p},H\mathbb{F}_{p};H\mathbb{F}_{p}).$

Hence natural operations on the homotopy groups of spectral partition Lie
algebras agree with cohomology operations on the (reduced)
$\mathbb{F}_{p}$-linear TAQ cohomology of
$\mathbb{E}_{\infty}$-$H\mathbb{F}_{p}$-algebras. In unpublished work, Kriz
computed the $\mathbb{F}_{2}$-linear TAQ cohomology on a connective generator
in [Kri93, Theorem 1.10]. Arone and Mahowald computed the dimensions of unary
operations on $\mathbb{F}_{p}$-linear TAQ cohomology of connective objects in
terms of free modules over subalgebras of the mod $p$ Steenrod algebra in
[AM99, Theorem 3.16 and 3.17]. Around the same time, Basterra and Mandell
announced a computation of unary operations and their relations as the Koszul
dual to Dyer-Lashof operations on $\mathbb{F}_{p}$-linear TAQ cohomology of
connective objects for $p>2$ and observed a shifted restricted Lie algebra
structure, but a proof never appeared, cf. [Law20, Example 1.8.8].

In this paper, we use a dual bar spectral sequence and the machinery of
classical Koszul duality developed by Priddy ([Pri70]) and Fresse ([Fre04]) to
identify the power ring of additive operations on the homotopy groups of
spectral partition Lie algebras and $\mathbb{F}_{p}$-linear TAQ cohomology.
Roughly speaking, this power ring is given by a collection of unstable Ext
groups over the Dyer-Lashof algebra, with composition product given by a
sheared Yoneda product. The verification of the law of composition relies
crucially on a result of Brantner in [Br17], which demonstrates the
compatibility of the algebraic Koszul duality on the $E^{2}$-page of the
(dual) bar spectral sequence with the monadic Koszul duality that the
$E^{\infty}$-page assembles to. As the degree of a homotopy class gets
arbitrarily large, the algebra of additive unary operations on that class
approaches the Koszul dual algebra of the Dyer-Lashof algebra.

Then we construct a shifted Lie bracket on the homotopy groups of spectral
partition Lie algebras, and use a homotopy fixed points spectral sequence to
detect a restriction map on the shifted Lie bracket. The restriction on an odd
degree homotopy class when $p>2$ and any homotopy class when $p=2$ coincides
with the bottom unary operation up to a unit. Furthermore, brackets of unary
operations that are not iterations of the restriction always vanish. Comparing
with Brantner and Mathew’s computation ([BM19, Theorem 1.20]), we deduce that
the dual bar spectral sequence collapses in the universal cases. Hence we
conclude that these are all the natural operations and obtain the optimal
target category.

As an immediate corollary, we determine the structure of operations on the mod
$p$ TAQ cohomology $\mathrm{TAQ}^{*}(-,\mathbb{S};H\mathbb{F}_{p})$ of
$\mathbb{E}_{\infty}$-$\mathbb{S}$-algebras. The $\mathbb{S}$-linear TAQ
cohomology operations consist of $\mathbb{F}_{p}$-linear TAQ cohomology
operations, as well as mod $p$ Steenrod operations. The relations among the
unary operations are given by the Adem relations and the Nishida relations on
mod $p$ cohomology ([Nis68]).

###  1.1. Statement of results

In section 2, we summarize recent results in [BM19] on spectral partition Lie
algebras over $H\mathbb{F}_{p}$. These are algebras over a certain monad
$\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}$ on the category of
$H\mathbb{F}_{p}$-modules. Then we describe the relation between spectral
partition Lie algebras and the mod $p$ TAQ spectrum
$\mathrm{TAQ}(R,H\mathbb{F}_{p};H\mathbb{F}_{p})$ of
$\mathbb{E}_{\infty}$-$H\mathbb{F}_{p}$-algebras $R$ constructed by Basterra
([Bas99]). The $n$th mod $p$ TAQ cohomology group of $R$ is given by

$\mathrm{TAQ}^{n}(R,H\mathbb{F}_{p};H\mathbb{F}_{p})=[\Sigma^{-n}\mathrm{TAQ}(R,H\mathbb{F}_{p};H\mathbb{F}_{p}),H\mathbb{F}_{p}]_{\mathrm{Mod}_{H\mathbb{F}_{p}}},$

with $\overline{\mathrm{TAQ}}^{*}(R,H\mathbb{F}_{p};H\mathbb{F}_{p})$ the
reduced mod $p$ TAQ cohomology. For any $m$ and tuple $(i_{1},\ldots,i_{k})$
of integers, there is an isomorphism

$\pi_{m}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(\Sigma^{i_{1}}H\mathbb{F}_{p}\oplus\cdots\oplus\Sigma^{i_{k}}H\mathbb{F}_{p}))\cong\overline{\mathrm{TAQ}}^{-m}(H\mathbb{F}_{p}\oplus\Sigma^{-i_{1}}H\mathbb{F}_{p}\oplus\cdots\oplus\Sigma^{-i_{k}}H\mathbb{F}_{p},H\mathbb{F}_{p};H\mathbb{F}_{p})$

by [BM19, Proposition 5.35]. The left hand side is the group of natural
transformations

$\prod^{k}_{l=1}\pi_{i_{l}}(-)\rightarrow\pi_{m}(-)$

of functors from the category of spectral partition Lie algebras to Sets,
whereas the right hand side is the group of cohomology operations

$\prod^{k}_{l=1}\overline{\mathrm{TAQ}}^{-i_{l}}(-,H\mathbb{F}_{p};H\mathbb{F}_{p})\rightarrow\overline{\mathrm{TAQ}}^{-m}(-,H\mathbb{F}_{p};H\mathbb{F}_{p}).$

Hence the structure of natural operations on the homotopy groups of spectral
partition Lie algebras over $H\mathbb{F}_{p}$ coincide with that on the
reduced version of TAQ cohomology $\overline{\mathrm{TAQ}}^{*}(A)$ for $A$ a
non-unital $\mathbb{E}_{\infty}$-$H\mathbb{F}_{p}$-algebra.

Denote by $\mathbb{E}^{nu}_{\infty}$ the non-unital
$\mathbb{E}_{\infty}$-operad. In section 3, we examine the bar spectral
sequence

$\widetilde{E}_{s,t}^{2}=\pi_{s}(\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Poly}_{\pazocal{R}},\pi_{*}(A))_{t}\Rightarrow\pi_{s+t}(|\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbb{E}^{nu}_{\infty}\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p},A)|)=\overline{\mathrm{TAQ}}_{s+t}(A)$

and the dual bar spectral sequence

$E_{s,t}^{2}=\pi_{s}(\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Poly}_{\pazocal{R}},\pi_{*}(A))^{\vee})_{t}\Rightarrow\pi_{s+t}(\mathbb{D}|\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbb{E}^{nu}_{\infty}\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p},A)|)=\overline{\mathrm{TAQ}}^{-s-t}(A)$

for bounded above
$\mathbb{E}^{\mathrm{nu}}_{\infty}$-$H\mathbb{F}_{p}$-algebras $A$ of finite
type. Here $\mathrm{Poly}_{\pazocal{R}}$ is the monad associated to the free
functor that first takes the graded polynomial algebra on the free unstable
module over the Dyer-Lashof algebra, then identifies
$x^{\mathbin{\mathop{\otimes}\displaylimits}2}$ with the bottom operation
$Q^{|x|}$ if $p=2$ and $x^{\mathbin{\mathop{\otimes}\displaylimits}p}$ with
the bottom operation $Q^{|x|/2}(x)$ if $p>2$ and $|x|$ is even, and finally
imposes the Cartan formula.

When $A$ is a direct sum of shifts of $H\mathbb{F}_{p}$ considered as a
trivial $\mathbb{E}^{nu}_{\infty}$-$H\mathbb{F}_{p}$-algebra, the $E^{2}$-page
of the dual bar spectral sequence parametrizes natural operations and their
relations on the André-Quillen cohomology

$\mathrm{AQ}^{*}_{\mathrm{Poly}_{\pazocal{R}}}(M):=\pi_{s}(\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Poly}_{\pazocal{R}},M)^{\vee})$

of algebras $M$ over the monad $\mathrm{Poly}_{\pazocal{R}}$, whereas the
$E^{\infty}$-page is isomorphic to the homotopy groups of the free spectral
partition Lie algebra on $A^{\vee}$.

In section 4, we compute the (dual) bar spectral sequence when $A$ is a direct
sum of shifts of $H\mathbb{F}_{p}$. The main idea is the following: first we
find a suitable factorization of the indecomposables functor
$Q^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}$, which
allowas us to replace the $E^{1}$-page with a much smaller bi-simplicial
object; then we compute the $E^{2}$-page as the homology of the total complex
using the machinery of classical Koszul duality developed by Priddy ([Pri70])
and Fresse ([Fre04]). This strategy is a modification of the method in [BHK19,
section 4.4] and was employed in [Zh21, section 3.3] to compute the mod 2
André-Quillen homology of spectral Lie algebras. The additive unary operations
on $\mathrm{AQ}^{*}_{\mathrm{Poly}_{R}}(M)$ are parametrized by a ringoid
$(\pazocal{R}^{\prime})^{!}$ that encodes certain unstable Ext groups over the
Dyer-Lashof algebra $\pazocal{R}$. The composition product is given by
juxtaposition, which corresponds to the Yoneda product on (unstable) Ext
groups.

######  Definition 1.1.

(Definition 4.1 and 4.10) The ringoid $(\pazocal{R}^{\prime})^{!}$ is defined
as follows:

(1) For $p=2$, let $\pazocal{F}$ be the ringoid with objects $\mathbb{Z}_{\leq
0}\times\mathbb{Z}$ whose morphisms are freely generated over $\mathbb{F}_{2}$
under juxtaposition by the following elements: for all $i,j$ satisfying $i>-j$
and $s\leq 0$, there is an element
$(Q^{i})^{*}\in\pazocal{F}\big{(}(s,j),(s-1,j-i)\big{)}$ of weight 2.

The ringoid $(\pazocal{R}^{\prime})^{!}$ is the quotient of $\pazocal{F}$ by
the ideal generated by the Adem relations

$(Q^{a})^{*}(Q^{b})^{*}=\sum_{a+b-c>2c,c>-j}\binom{b-c-1}{a-2c-1}(Q^{a+b-c})^{*}(Q^{c})^{*}$

for all $a,b\in\mathbb{Z}$ satisfying $a\leq 2b$, $b>-j$ and $a>b-j$ in
$\pazocal{F}\big{(}(s,j),(s-2,j-a-b)\big{)}$.

(2) For $p>2$, let $\pazocal{F}$ be the ringoid with objects $\mathbb{Z}_{\leq
0}\times\mathbb{Z}$ and morphisms freely generated over $\mathbb{F}_{p}$ under
juxtaposition on the following elements: for $2i>-j$ and any $s\leq 0$ there
are elements $(Q^{i})^{*}\in\pazocal{F}\big{(}(s,j),(s-1,j-2(p-1)i)\big{)}$
and $(\beta Q^{i})^{*}\in\pazocal{F}\big{(}(s,j),(s-1,j-2(p-1)i+1)\big{)}$.

The ringoid $(\pazocal{R}^{\prime})^{!}$ is the quotient of $\pazocal{F}$ by
the ideal generated by the Adem relations

$(Q^{a})^{*}(Q^{b})^{*}=-\sum_{a+b-c>pc,\,\,2c>-k}(-1)^{a-c}\binom{(p-1)(b-c)-1}{a-pc-1}(Q^{a+b-c})^{*}(Q^{c})^{*},$

in $\pazocal{F}\big{(}(s,j),(s-2,j-2(p-1)a-2(p-1)b)\big{)}$ for all
$a,b\in\mathbb{Z}$ satisfying $a\leq pb$, $2b>-j$, $2a>2(p-1)b-j$,

$\displaystyle(\beta Q^{a})^{*}(Q^{b})^{*}=\sum_{a+b-c\geq
pc,2c>-k}(-1)^{a-c}\binom{(p-1)(b-c)}{a-pc}(Q^{a+b-c})^{*}(\beta Q^{c})^{*}$
$\displaystyle-\sum_{a+b-c>pc,2c>-k}(-1)^{a-c}\binom{(p-1)(b-c)-1}{a-pc-1}(\beta
Q^{a+b-c})^{*}(Q^{c})^{*}=0$

in $\pazocal{F}\big{(}(s,j),(s-2,j-2(p-1)a-2(p-1)b+1)\big{)}$ for all
$a,b\in\mathbb{Z}$ satisfying $a\leq pb$, $2b>-j$, $2a>2(p-1)b-j$, and

$(\beta^{\epsilon}Q^{a})^{*}(\beta Q^{b})^{*}=\sum_{a+b-c\geq
pc,2c>-k}(-1)^{a-c}\binom{(p-1)(b-c)-1}{a-pc}(\beta^{\epsilon}Q^{a+b-c})^{*}(\beta
Q^{c})^{*}$

in $\pazocal{F}\big{(}(s,j),(s-2,j-2(p-1)a-2(p-1)b+\epsilon+1\big{)}$ for
$\epsilon\in\\{0,1\\}$ and $a,b\in\mathbb{Z}$ satisfying $a<pb$, $2b>-j$,
$2a>2(p-1)b-j$.

######  Proposition 1.2.

(Proposition 4.9 and 4.15) The André-Quillen cohomology
$\mathrm{AQ}^{*}_{\mathrm{Poly}_{R}}(M)$ of algebras $M$ over the monad
$\mathrm{Poly}_{R}$ admits a shifted Lie bracket that vanishes on additive
unary operations. When $p=2$, additive unary operations on
$\mathrm{AQ}^{*}_{\mathrm{Poly}_{R}}(M)$ are parametrized by the ringoid
$(\pazocal{R}^{\prime})^{!}$. There is also a nonadditive unary operation
acting as the restriction on the shifted Lie bracket. When $p>2$, (additive)
unary operations on $\mathrm{AQ}^{*}_{\mathrm{Poly}_{R}}(M)$ of algebras $M$
are parametrized by the ringoid $(\pazocal{R}^{\prime})^{!}$ together with
self-brackets on even degree classes.

We deduce that the spectral sequences collapse on the $E^{2}$-page with no
extension problems by comparing with the bases in [BM19, Theorem 1.20].

Section 5 is devoted to determining the composition law and all relations
among the operations on the homotopy groups of free spectral partition Lie
algebras and reduced mod $p$ TAQ cohomology. While the dimensions of
operations on the homotopy groups of free spectral partition Lie algebras
agree with the dimensions of operations on the André-Quillen cohomology
$\mathrm{AQ}^{*}_{\mathrm{Poly}_{R}}(-)$, the composition product of the
former differs a priori from the composition product of the later. Inspired by
the work of Brantner regarding the structure of operations on the Lubin-Tate
theory of spectral Lie algebras in [Br17, section 4], we encode additive unary
operations on the homotopy groups of free spectral partition Lie algebras with
a power ring. Roughly speaking, this power ring arises from equipping the
ringoid $(\pazocal{R}^{\prime})^{!}$ in Definition 1.1 with a twisted
composition product.

######  Definition 1.3.

(Definition 5.4) We define the power ring $\pazocal{P}$ to be the collection
of $\mathbb{F}_{p}$-modules
$\\{\pazocal{P}^{j}_{k}[w]\\}_{(j,k,w)\in\mathbb{Z}^{2}\times\mathbb{Z}_{\geq
0}},$ where
$\pazocal{P}^{j}_{k}[w]:=(\pazocal{R}^{\prime})^{!}((0,k),(-w,j+w))$ for
$w>0$, $\pazocal{P}^{i}_{i}[0]:=\mathbb{F}_{p}\\{\iota_{i}\\}$ for all $i$,
and $\pazocal{P}^{j}_{i}[0]=\emptyset$ for all $i\neq j$. The associative and
unital composition product
$\pazocal{P}^{i}_{j}[v]\times\pazocal{P}^{j}_{k}[w]\rightarrow\pazocal{P}^{i}_{k}[v+w]$
is given by the sheared Yoneda product

${\pazocal{P}^{i}_{j}[v]\times\pazocal{P}^{j}_{k}[w]}$${(\pazocal{R}^{\prime})^{!}((0,j),(-v,i+v))\times(\pazocal{R}^{\prime})^{!}((0,k),(-w,j+w))}$${(\pazocal{R}^{\prime})^{!}((0,j+w),(-v,i+v+w))\times(\pazocal{R}^{\prime})^{!}((0,k),(-w,j+w))}$${(\pazocal{R}^{\prime})^{!}((-w,j+w),(-v-w,i+v+w))\times(\pazocal{R}^{\prime})^{!}((0,k),(-w,j+w))}$${\pazocal{P}^{i}_{k}[v+w]}$${(\pazocal{R}^{\prime})^{!}((0,k),(-v-w,i+v+w))}$$\scriptstyle{\cong}$$\scriptstyle{\mathrm{susp}^{w}\times\mathrm{id}}$$\scriptstyle{\cong}$$\scriptstyle{\mathrm{juxtaposition}}$$\scriptstyle{\cong}$

for $v,w>0$, as well as isomorphisms
$\pazocal{P}^{j}_{i}[w]\times\pazocal{P}^{i}_{i}[0]\xrightarrow{\cong}\pazocal{P}^{j}_{i}[w]$
and
$\pazocal{P}^{j}_{j}[0]\times\pazocal{P}^{j}_{i}[w]\xrightarrow{\cong}\pazocal{P}^{j}_{i}[w]$
for all $w\geq 0$.

A module over the power ring $\pazocal{P}$ is a weighted, graded
$\mathbb{F}_{p}$-module $M=\bigoplus_{(j,w)\in\mathbb{Z}\times\mathbb{Z}_{\geq
0}}M_{j}(w)$, equipped with structure maps
$\pazocal{P}^{i}_{j}[v]\mathbin{\mathop{\otimes}\displaylimits}M_{j}(w)\rightarrow
M_{i}(p^{v}w)$ that are compatible with the composition product in
$\pazocal{P}$.

In Construction 5.8, we produce a shifted (restricted) Lie bracket

$[-,-]:\pi_{m}(A)\mathbin{\mathop{\otimes}\displaylimits}\pi_{n}(A)\rightarrow\pi_{m+n-1}(A)$

for any spectral partition Lie algebra $A$ at the level of
$H\mathbb{F}_{p}$-modules. Equivalently, for any
$\mathbb{E}^{\mathrm{nu}}_{\infty}$-$H\mathbb{F}_{p}$-algebra $A$, there is a
shifted (restricted) Lie bracket

$[-,-]:\overline{\mathrm{TAQ}}^{m}(A)\mathbin{\mathop{\otimes}\displaylimits}\overline{\mathrm{TAQ}}^{n}(A)\rightarrow\overline{\mathrm{TAQ}}^{m+n+1}(A).$

Now we can state the main result of this paper, which records all natural
operations and their relations on the homotopy groups of a spectral partition
Lie algebra over $H\mathbb{F}_{p}$ and the reduced TAQ cohomology of
$\mathbb{E}_{\infty}$-$H\mathbb{F}_{p}$-algebras.

######  Theorem 1.4.

(Theorem 5.5 and 5.13)

  1. (1)

The homotopy groups of a spectral partition Lie algebra over
$H\mathbb{F}_{2}$, or the reduced TAQ cohomology of an
$\mathbb{E}_{\infty}$-$H\mathbb{F}_{2}$-algebra, form a module over the power
ring $\pazocal{P}$ of additive unary operations.

  2. (2)

The weight $2$ additive operations are given by the collection
$R^{i}\in\pazocal{P}^{j-i}_{j}[1]$ for all $i,j\in\mathbb{Z}$ satisfying
$i>-j+1$, subject to the Adem relations

$R^{a}R^{b}=\sum_{a+b-c\geq 2c,\,\,c>-j+1}\binom{b-c-1}{a-2c}R^{a+b-c}R^{c}$

in $\pazocal{P}_{j}^{j-a-b}[2]$ for all $a,b\in\mathbb{Z}$ satisfying
$b-j<a<2b$ and $b>-j+1$.

  3. (3)

There is a nonadditive unary operation $R^{-|x|+1}(x)$ for any homotopy class
$x$ that serves as the restriction $x^{[2]}$ on $x$. The restriction on a sum
of classes $x$ and $y$ in different degrees is given by

$(x+y)^{[2]}=R^{-|x|+1}(x)+R^{-|y|+1}(y)+[x,y].$

The bracket is compatible with the unary operations in the sense that
$[y,\alpha(x)]=0$ for any homotopy class $x,y$ and unary operation $\alpha$ of
positive weight that is not an iteration of the restriction.

  4. (4)

The operations $R^{i}$ and the shifted restricted Lie bracket generate all
natural operations under the above relations. A basis for unary operations on
a degree $j$ class is given by the collection of all monomials
$R^{i_{1}}R^{i_{2}}\cdots R^{i_{l}}$ such that $i_{l}>-j$ and $i_{m}\geq
2i_{m+1}$ for $1\leq m<l$.

######  Theorem 1.5.

(Theorem 5.6 and 5.13)

  1. (1)

The homotopy groups of a spectral partition Lie algebra over
$H\mathbb{F}_{p}$, or the reduced TAQ cohomology of any
$\mathbb{E}_{\infty}$-$H\mathbb{F}_{p}$-algebra, form a module over the power
ring $\pazocal{P}$.

  2. (2)

The weight $p$ unary operations are given by the collection
$\beta^{\epsilon}R^{i}\in\pazocal{P}^{j-2(p-1)i-\epsilon}_{j}[1]$ for
$\epsilon=0,1$ and any $2i>-j$, subject to the Adem relations

$\beta R^{a}\beta
R^{b}=\sum_{a+b-c>pc,2c>-j}(-1)^{a-c+1}\binom{(p-1)(b-c)-1}{a-pc-1}\beta
R^{a+b-c}\beta R^{c}$

in $\pazocal{P}^{j-2(p-1)(a+b)-2}_{j}[2]$ for all $a,b\in\mathbb{Z}$
satisfying $a\leq pb$, $2b>-j$, $2a>2(p-1)b-j$ ,

$\displaystyle R^{a}\beta R^{b}=$ $\displaystyle\sum_{a+b-c\geq
pc,2c>-j}(-1)^{a-c}\binom{(p-1)(b-c)}{a-pc}\beta P^{a+b-c}R^{c}$
$\displaystyle-\sum_{a+b-c>pc,2c>-j}(-1)^{a-c}\binom{(p-1)(b-c)-1}{a-pc-1}R^{a+b-c}\beta
R^{c}$

in $\pazocal{P}^{j-2(p-1)(a+b)-1}_{j}[2]$ for all $a,b\in\mathbb{Z}$
satisfying $a\leq pb$, $2b>-j$, $2a>2(p-1)b+1-j$ ,

$\beta^{\epsilon}R^{a}R^{b}=\sum_{a+b-c\geq
pc,2c>-j}(-1)^{a-c}\binom{(p-1)(b-c)-1}{a-pc}\beta^{\epsilon}R^{a+b-c}R^{c}$

in $\pazocal{P}^{j-2(p-1)(a+b)-\epsilon}_{j}[2]$ for all $a,b\in\mathbb{Z}$
satisfying $a<pb$, $2b>-j$, $2a>2(p-1)b-j$, and $\epsilon\in\\{0,1\\}$.

  3. (3)

For all odd $j$ and $x$ a homotopy class in degree $j$, the restriction
$x^{[p]}$ is the bottom operation $R^{(-j+1)/2}(x)$ up to a unit
$\lambda_{j}$. The restriction map on a sum of classes $x$ and $y$ in odd
degrees $j\neq k$ is given by

$(x+y)^{[p]}=\lambda_{j}R^{(-j+1)/2}(x)+\lambda_{k}R^{(-k+1)/2}(y)+\sum_{i=1}^{p-1}\frac{s_{i}}{i}(x,y),$

where $s_{i}$ is the coefficient of $t^{i-1}$ in the formal expression
$\mathrm{ad}(tx+y)^{p-1}(x)$. Furthermore, $[y,\alpha(x)]=0$ for any homotopy
class $x,y$ and $\alpha$ a unary operation of positive weight, unless $x$ is
in odd degree and $\alpha$ an iteration of the restriction.

  4. (4)

The operations $\beta^{\epsilon}R^{i}$ and the shifted restricted Lie bracket
generate all natural operations under the above relations. A basis for unary
operations on a degree $j$ class with $j$ odd is given by all monomials
$\beta^{\epsilon_{1}}R^{i_{1}}\beta^{\epsilon_{2}}R^{i_{2}}\cdots\beta^{\epsilon_{l}}R^{i_{l}}$
such that $2i_{l}>-j$ and $i_{m}\geq pi_{m+1}+\epsilon_{m+1}$ for $1\leq m<l$.
If $j$ is even, a basis is given by
$\beta^{\epsilon_{1}}R^{i_{1}}\beta^{\epsilon_{2}}R^{i_{2}}\cdots\beta^{\epsilon_{l}}R^{i_{l}}B^{\epsilon}$
such that $2i_{l}>-(1+\epsilon)j-\epsilon$ and $i_{m}\geq
pi_{m+1}+\epsilon_{m+1}$ for $1\leq m<l$.

The key step in the proof of Theorem 5.5 and 5.6 is checking that the
composition product in $\pazocal{P}$ is the law of composition for additive
operations on
$\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(A))$. The
proof relies on a general result of Brantner ([Br17, Theorem 3.5.1 and
4.3.2]), which explains the compatibility of the structure of additive
operations on the $E^{2}$-page of the (dual) bar spectral sequence with that
on the homotopy groups of the monadic bar construction the $E^{\infty}$-page
assembles to.

When the degree of a class gets arbitrarily large, we observe in Corollary 5.7
that the additive operations on that class form an algebra Koszul dual to the
Dyer-Lashof algebra, which is isomorphic to the extended Steenrod algebra but
with $Sq^{0}=0$. Indeed, the Dyer-Lashof algebra is isomorphic to the extended
Steenrod algebra for restricted Lie algebras, cf. [KL83]. For filtration
reasons, the restriction on the shifted Lie bracket is not visible on the
$E^{2}$-page of the dual bar spectral sequence, i.e. the André-Quillen
cohomology associated to the monad $\mathrm{Poly}_{\pazocal{R}}$, when $p>2$.
Instead, we use a homotopy fixed points spectral sequence to detect the
restriction map in Lemma 5.12. Thus we recover and clarify the unpublished
computations of Kriz and Basterra-Mandell on connective objects via a
different method.

######  Remark 1.6.

The shifted Lie brackets on the mod $p$ homology and Lubin-Tate theory of
spectral Lie algebras always vanish on unary operations of positive weights,
as was shown by Antolín-Camerena ([AC20]), Brantner ([Br17]), and Kjaer
([Kja18]). The slight difference from our situation comes from the fact that
that all unary operations on spectral Lie algebras are additive, whereas the
restriction map for the shifted Lie bracket on the homotopy groups of spectral
partition Lie algebras is a nonadditive unary operation. See Remark 5.14.

######  Remark 1.7.

To identify the unit $\lambda_{j}$ by which the bottom operation and the
restriction on an odd class $x$ in degree $j$ differ when $p>2$, we expect
that a chain-level understanding of the operations involved is necessary. We
hope to construct these operations on the explicit chain model of spectral
partition Lie algebra obtained by Brantner, Campos, and Nuiten in [BCN21,
Definition 4.43].

######  Remark 1.8.

In [Kja18], Kjaer computed the dimensions of the odd primary homology groups
of free spectral Lie algebras and conjectured that the relations among the
unary operations are given by the mixed Adem relations in [CLM76, II.3], cf.
[Kja18, Conjecture 5.3]. One approach to proving these relations is to imitate
Behrens’ computation of the quadratic relations in [Beh12, Section 1.4-1.5]
when $p=2$. The difficulty lies in working out a formula for the induced map
of the transfer $\Sigma_{p^{2}}\rightarrow\Sigma_{p}\wr\Sigma_{p}$ on mod $p$
homology groups of the extended power constructions. This formula was obtained
by Kuhn in [Kuh85] when $p=2$, but remains elusive when $p>2$. The method in
this paper suggests an alternative approach to resolving the question. Namely,
one could construct a sifted-colimit-preserving monad $\pazocal{C}$ that
parametrizes divided power
$\mathbb{E}^{\mathrm{nu}}_{\infty}$-$H\mathbb{F}_{p}$-algebras following the
axiomatic argument in [BM19, Section 4], determine the structure of natural
operations on the homotopy groups of algebras over $\pazocal{C}$, and feed it
into the dual bar spectral sequence converging to
$\pi_{*}(\mathbb{D}|\mathrm{Bar}_{\bullet}(\mathrm{id},\pazocal{C},A)|)$ with
$A$ any trivial algebra over $\pazocal{C}$ of finite type.

Therefore the target category for the homotopy groups of a spectral partition
Lie algebra or the reduce mod $p$ TAQ cohomology is the category of
$\pazocal{P}$-$\mathrm{sLie}^{\rho}$-algebras (Definition 5.15), i.e., modules
$L$ over the power ring $\pazocal{P}$ together with a shifted restricted Lie
bracket satisfying the conditions in part (3) of Theorem 1.4 and 1.5. Denote
by $\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{P}}}$ the free
$\pazocal{P}$-$\mathrm{sLie}^{\rho}$-algebra functor.

As an immediate application, we obtain a computation of natural operations and
relations on the mod $p$ TAQ cohomology
$\mathrm{TAQ}^{*}(R,\mathbb{S};H\mathbb{F}_{p})$ of
$\mathbb{E}_{\infty}$-$\mathbb{S}$-algebras $R$, which is based on
conversations with Tyler Lawson.

Since the functor $\mathrm{TAQ}^{i}(-,\mathbb{S};H\mathbb{F}_{p})$ has
representing object the trivial square-zero extension
$\mathbb{S}\oplus\Sigma^{i}H\mathbb{F}_{p}$ for all $i$, operations and
relations are again parametrized by the mod $p$ TAQ cohomology on the trivial
square-zero extensions
$\mathbb{S}\oplus\Sigma^{i_{1}}H\mathbb{F}_{p}\oplus\cdots\oplus\Sigma^{i_{k}}H\mathbb{F}_{p}$.
Using the base change formula

$\mathrm{TAQ}(-,\mathbb{S};H\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits_{\mathbb{S}}}H\mathbb{F}_{p}\simeq\mathrm{TAQ}(-\mathbin{\mathop{\otimes}\displaylimits_{\mathbb{S}}}H\mathbb{F}_{p},H\mathbb{F}_{p};H\mathbb{F}_{p}),$

we deduce immediately from Theorem 1.4 and 1.5 the structure of natural
operations on the mod $p$ TAQ cohomology
$\mathbb{E}_{\infty}$-$\mathbb{S}$-algebras.

######  Theorem 1.9.

(Corollary 6.1, Proposition 6.2) For any tuple $(i_{1},\ldots i_{k})$ of
integers, the $k$-ary cohomology operations

$\prod^{k}_{i=1}\mathrm{TAQ}^{i_{l}}(-,\mathbb{S};H\mathbb{F}_{p})\rightarrow\mathrm{TAQ}^{m}(-,\mathbb{S};H\mathbb{F}_{p}).$

are parametrized by the homological degree $-m$ part of
$\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{P}}}(\Sigma^{-i_{1}}\pazocal{A}\oplus\cdots\oplus\Sigma^{-i_{k}}\pazocal{A})$,
where $\pazocal{A}$ is the Steenrod algebra graded homologically. All
operations vanish on the unit except for scalar multiplication. The Steenrod
operations commute with the bracket via the Cartan formula and the
$\mathbb{F}_{p}$-linear TAQ cohomology operations via the Nishida relations on
cohomology of the second extended power:

  1. (1)

For $p=2$ we have

$Sq^{a}[x,y]=\sum_{i}[Sq^{i}(x),Sq^{a-i}(y)],$
$Sq^{a}R^{-|x|+1}(x)=\sum\binom{|x|-c}{a-2c}R^{a+|x|+1-c}Sq^{c}(x)+\sum_{l<k,l+k=a}[Sq^{l}(x),Sq^{k}(x)],$
$Sq^{a}R^{b}(x)=\sum\binom{b-1-c}{a-2c}R^{a+b-c}Sq^{c}(x),\,\,\,\,b>-|x|+1.$

  2. (2)

For $p>2$ we have

$P^{a}[x,y]=\sum_{i}[P^{i}(x),P^{a-i}(y)],\,\,\,\,\beta
P^{a}[x,y]=\sum_{i}([\beta P^{i}(x),P^{a-i}(y)]+[P^{i}(x),\beta P^{a-i}(y)]).$

For any class $x$ and all $2j>-|x|+1$, the Nishida relations are

$\displaystyle P^{n}\beta R^{j}=$
$\displaystyle(-1)^{n-i}\sum_{i}\binom{(j-i)(p-1)}{n-pi}\beta
R^{n+j-i}P^{i}+(-1)^{n-i}\sum_{i}\binom{(j-i)(p-1)-1}{n-pi-1}R^{n+j-i}\beta
P^{i},$
$P^{n}R^{j}=(-1)^{n-i}\sum_{i}\binom{(j-i)(p-1)-1}{n-pi}R^{n+j-i}P^{i},$

as well as

$\displaystyle P^{n}R^{j}(x)=$
$\displaystyle(-1)^{n-i}\sum_{i}\binom{(j-i)(p-1)-1}{n-pi}R^{n+j-i}P^{i}(x)$
$\displaystyle+\frac{1}{\lambda_{|x|}}\sum_{I,\sigma\in\Sigma_{p},\sigma(1)=1}[[\cdots[[P^{i_{\sigma(1)}}(x),P^{i_{\sigma(2)}}(x)],P^{i_{\sigma(3)}}(x)]\cdots],P^{i_{\sigma(p)}}(x)]$

when the degree of $x$ is odd and $2j=-|x|+1$, where the bracket term sums
over all nondecreasing sequences $I=(0\leq i_{1}\leq i_{2}\leq\ldots\leq
i_{p})$ with $i_{1}+i_{2}+\cdots+i_{p}=n$, and $\lambda_{|x|}$ is a fixed unit
given in Theorem 1.5.(3).

######  Remark 1.10.

In parallel to the theory of spectral partition Lie algebras, Brantner and
Mathew developed a derived version called partition Lie algebras, which are
algebras over a certain monad $\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\Delta}$ on
the derived category of chain complexes over $\mathbb{F}_{p}$. They showed
that coconnective, finite type partition Lie algebras over $\mathbb{F}_{p}$
serve as the Koszul dual to complete local Noetherian simplicial commutative
$\mathbb{F}_{p}$-algebras ([BM19, Theorem 1.11 and 1.13]). They then computed
the ranks of homotopy groups of free partition Lie algebras in [BM19, Theorem
1.16]. While partition Lie algebras are more delicate than their spectral
counterparts studied in this paper, we are hopeful that a similar strategy can
be employed to obtain relations among the operations on the homotopy groups of
partition Lie algebras. This will require as input the structure of natural
operations on the homotopy groups of simplicial and cosimplicial commutative
$\mathbb{F}_{p}$-algebras. Note that for $p=2$, Goerss identified all
operations and their relations on the homotopy groups of coconnective
partition Lie algebras in [Goe90, Theorem H] via a Quillen spectral sequence
and a Hilton-Milnor type argument, with input the structure of unary
operations on the homotopy groups simplicial commutative
$\mathbb{F}_{2}$-algebras by the works of Cartan ([Car58]), Bousfield
([Bou68]), Dwyer ([Dwy80]). For $p>2$, a basis of unary operations was
computed by Nakaoka in the simplicial case ([Nak57, Nak58]), and the quadratic
relations among the operations were determined by Bousfield in [Bou68, Theorem
8.9]. In the cosimplicial case, Priddy determined the unary operations and
their relations in [Pri73] at all primes.

###  1.2. Conventions

The grading convention is homological unless for cohomology groups or
otherwise stated. We assume that every object is graded and weighted whenever
it makes sense. For instance, $\mathrm{Mod}_{\mathbb{F}_{p}}$ stands for the
ordinary category of weighted graded $\mathbb{F}_{p}$-modules, and
$\mathrm{Mod}_{H\mathbb{F}_{p}}$ the ($\infty$-)category of weighted graded
modules over $H\mathbb{F}_{p}$. A weighted graded $\mathbb{F}_{2}$-module
(resp $H\mathbb{F}_{p}$-module) $M$ is an $\mathbb{N}$-indexed collection of
$\mathbb{Z}$-graded $\mathbb{F}_{p}$-modules ($H\mathbb{F}_{p}$-modules)
$\\{M(w)\\}_{w\in\mathbb{N}}$. The weight grading of an element $x\in M(w)$ is
$w$. Morphisms are weight preserving morphisms of graded
$\mathbb{F}_{p}$-modules ($H\mathbb{F}_{p}$-modules). The Day convolution
$\mathbin{\mathop{\otimes}\displaylimits}$ makes
$\mathrm{Mod}_{\mathbb{F}_{p}}$ ($\mathrm{Mod}_{H\mathbb{F}_{p}}$) a symmetric
monoidal category. The Koszul sign rule
$x\mathbin{\mathop{\otimes}\displaylimits}y=(-1)^{|x||y|}y\mathbin{\mathop{\otimes}\displaylimits}x$
for the symmetric monoidal product $\mathbin{\mathop{\otimes}\displaylimits}$
depends only on the internal grading and not the weight grading.

Similarly, a shifted Lie algebra $L$ over $\mathbb{F}_{p}$ is a weighted
graded $\mathbb{F}_{p}$-module equipped with a shifted Lie bracket
$[-,-]:L_{m}\mathbin{\mathop{\otimes}\displaylimits}L_{n}\rightarrow
L_{m+n-1}$ that adds weights, as well as satisfying graded commutativity
$[x,y]=(-1)^{|x||y|}[y,x]$ and the graded Jacobi identity

$(-1)^{|x||z|}[x,[y,z]]+(-1)^{|y||x|}[y,[z,x]]+(-1)^{|z||y|}[z,[x,y]]=0.$

If $p=2$, then we further require that $[x,x]=0$ for all $x$. If $p=3$, then
we further require that $[x,[x,x]]=0$ for all $x$.

We use $\pi_{n}(-)$ to denote the functor taking the $n$th homotopy group of a
spectrum, $H\mathbb{F}_{p}$-module, or simplicial $\mathbb{F}_{p}$-module, as
well as the functor taking the $n$th homology group of a chain complex over
$\mathbb{F}_{p}$.

We use $\mathbb{E}_{\infty}$ and $\mathbb{E}^{\mathrm{nu}}_{\infty}$ to denote
respectively the unital and non-unital commutative operad in the category of
spectra, i.e. the category of $\mathbb{S}$-algebras, and
$\mathbb{P}=\mathbb{E}^{\mathrm{nu}}_{\infty}\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p}$
the nonunital commutative operad in $\mathrm{Mod}_{H\mathbb{F}_{p}}$. Often we
abuse notations and denote by $O$ the monad associated to the free $O$-algebra
functor when
$O=\mathbb{E}_{\infty},\mathbb{E}^{\mathrm{nu}}_{\infty},\mathbb{P}$.

Acknowledgements. The author would like to thank Lukas Brantner for suggesting
the project and sharing his insights, Jeremy Hahn, Tyler Lawson, and Haynes
Miller for many helpful conversations, corrections, and encouragements, as
well as Basterra and Mandell for their generosity. Special thanks to Lukas
Brantner and Tyler Lawson for pointing out several issues in earlier drafts.

##  2\. Preliminaries

###  2.1. Spectral Lie operad

We begin with a brief review of the spectral Lie operad
$\partial_{*}(\mathrm{Id})$. Ching ([Chi05]) and Salvatore showed that the
Goowillie derivatives $\\{\partial_{n}(\mathrm{Id})\\}$ of the identity
functor $\mathrm{Id}:\mathrm{Top}_{*}\rightarrow\mathrm{Top}_{*}$ form an
operad $\partial_{*}(\mathrm{Id})$ in Spectra. This operad is Koszul dual to
the nonunital commutative operad $\mathbb{E}^{\mathrm{nu}}_{\infty}$ via the
operadic bar construction

$\partial_{*}(\mathrm{Id})\simeq\mathbb{D}\mathrm{Bar}(1,\mathbb{E}_{\infty}^{\mathrm{nu}},1).$

For a description of the operadic bar construction, see [Chi05] for a
topological model using trees and [Br17, Appendix D] for an
$\infty$-categorical construction along with a comparison with the topological
model.

The $n$th-derivative $\partial_{n}(\mathrm{Id})$ admits an explicit
description due to Arone and Mahowald ([AM99]), following the work of Johnson
([Joh95]). Let $\pazocal{P}_{n}$ be the poset of partitions of the set
$\underline{n}=\\{1,2,\ldots,n\\}$ ordered by refinements, equipped with a
$\Sigma_{n}$-action induced from that on $\underline{n}$. Denote by $\hat{0}$
the discrete partition and $\hat{1}$ the partition $\\{\underline{n}\\}$. Set
$\Pi_{n}=\pazocal{P}_{n}-\\{\hat{0},\hat{1}\\}$. Regarding a poset
$\pazocal{P}$ as a category, we obtain via the nerve construction a simplicial
set $N_{\bullet}(\pazocal{P})$. The partition complex
$\Sigma|\Pi_{n}|^{\diamond}$, the reduced-unreduced suspension of the
realization $|\Pi_{n}|$, is modeled by the simplicial set

$N_{\bullet}(\pazocal{P}_{n})/(N_{\bullet}(\pazocal{P}_{n}-\hat{0})\cup
N_{\bullet}(\pazocal{P}_{n}-\hat{1}))$

for $n\geq 2$ and the simplicial 0-circle $S^{0}$ for $n=1$. Then there is an
equivalence

$\partial_{n}(\mathrm{Id})\simeq\mathbb{D}(\Sigma|\Pi_{n}|^{\diamond})$

of spectra with $\Sigma_{n}$-action, where $\mathbb{D}$ denotes the Spanier-
Whitehead dual of a spectrum.

###  2.2. Spectral partition Lie algebras

Motivated by the theory of classical operadic Koszul duality ([GK94]), the
natural next step is to formulate a Koszul duality theorem between suitable
categories of algebras over the Koszul dual pair
$\mathbb{E}^{\mathrm{nu}}_{\infty}$ and $\partial_{*}(\mathrm{Id})$. Partial
progress was achieved by Ching and Harper in [CH19], following a general
conjecture by Francis and Gaitsgory ([FG12]). Recent work of Brantner and
Mathew ([BM19]) on spectral partition Lie algebras completely resolved the
question over $H\mathbb{F}_{p}$, and we will give a very brief summary of
their results.

Let
$\mathrm{Mod}^{\mathrm{ft}}_{H\mathbb{F}_{p}}\subset\mathrm{Mod}_{H\mathbb{F}_{p}}$
be the subcategory spanned by $H\mathbb{F}_{p}$-modules of finite type, i.e.
$H\mathbb{F}_{p}$-modules with degree-wise finite homotopy groups. Denote by
$\mathrm{Mod}^{\mathrm{ft}}_{H\mathbb{F}_{p},\leq
0}\subset\mathrm{Mod}^{\mathrm{ft}}_{H\mathbb{F}_{p}}$ the subcategory spanned
by coconnective objects. Let $\mathbb{P}$ be the nonunital commutative operad
in $\mathrm{Mod}_{H\mathbb{F}_{p}}$. There is an adjunction

$\mathrm{Alg}_{\mathbb{P}}(\mathrm{Mod}_{H\mathbb{F}_{p}})$$\mathrm{Mod}_{H\mathbb{F}_{p}}$
,$\mathrm{cot}$$\mathrm{sqz}$

where the functor sqz sends an object $M$ to the $\mathbb{P}$-algebra $M$ as a
trivial square-zero extension. The restriction of this adjunction to the
subcategory $\mathrm{Mod}^{\mathrm{ft}}_{H\mathbb{F}_{p},\leq 0}$ defines a
sifted-colimit-preserving monad
$(M\mapsto\mathrm{cot}(\mathrm{sqz}(M)^{\vee})^{\vee})$ on
$\mathrm{Mod}^{\mathrm{ft}}_{H\mathbb{F}_{p},\leq 0}$.

######  Definition 2.1.

[BM19, Definition 5.32] The spectral partition Lie monad
$\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}$ is the unique
sifted-colimit-preserving monad

$\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}:\mathrm{Mod}_{H\mathbb{F}_{p}}\rightarrow\mathrm{Mod}_{H\mathbb{F}_{p}}$

extending the monad $(M\mapsto\mathrm{cot}(\mathrm{sqz}(M)^{\vee})^{\vee})$ on
$\mathrm{Mod}^{\mathrm{ft}}_{H\mathbb{F}_{p},\leq 0}$,

Algebras over the monad
$\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}$ are called spectral
partition Lie algebras. The free spectral partition Lie algebras on bounded
above objects admit an explicit description.

######  Proposition 2.2.

[BM19, Proposition 5.35] For $V$ a bounded above $H\mathbb{F}_{p}$-module,

$\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(V)\simeq\mathbb{D}|\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbb{P},V^{\vee})|\simeq\bigoplus_{n\geq
1}\big{(}(\partial_{n}(\mathrm{id})\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits}(V)^{\mathbin{\mathop{\otimes}\displaylimits}n}\big{)}^{h\Sigma_{n}}.$

The above formula makes it clear that spectral partition Lie algebras are not
algebras over the spectral Lie operad, as the structural map of an algebra $L$
over the spectral Lie operad in $\mathrm{Mod}_{H\mathbb{F}_{p}}$ is given by

$\mathrm{Free}^{\partial_{*}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p}}(L)\simeq\bigoplus_{n\geq
1}\big{(}(\partial_{n}(\mathrm{id})\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits}(L)^{\mathbin{\mathop{\otimes}\displaylimits}n}\big{)}_{h\Sigma_{n}}\rightarrow
L.$

Heuristically, spectral partition Lie algebras are the dual of divided power
coalgebras over the cooperad $\mathrm{Bar}(1,\mathbb{P},1)$, and hence
candidates for the Koszul dual of
$\mathbb{E}^{\mathrm{nu}}_{\infty}$-$H\mathbb{F}_{p}$-algebras. To formulate
the precise Koszul duality statement, we need to introduce one more technical
condition.

######  Definition 2.3.

An $\mathbb{E}_{\infty}$-H$\mathbb{F}_{p}$-algebra $A$ is complete local
Noetherian if

(1). $\pi_{0}(A)$ is a complete local Noetherian ring;

(2). $A$ is connective and $\pi_{n}(A)$ is a finitely-generated module over
$\pi_{0}(A)$ for all $n\geq 0$.

Now we can state a restricted version of of the main results by Brantner and
Mathew.

######  Theorem 2.4.

[BM19, Theorem 1.19] There is an equivalence of $\infty$-categories between
complete local Noetherian $\mathbb{E}_{\infty}$-$H\mathbb{F}_{p}$-algebras and
the $\infty$-category of coconnective spectral partition Lie algebras of
finite type.

###  2.3. Relation to TAQ cohomology

Spectral partition Lie algebras are closely related to the
$\mathbb{F}_{p}$-linear TAQ spectrum. Inspired by the unpublished work of Kriz
([Kri93]), Basterra constructed in [Bas99] the topological André-Quillen
homology object $\mathrm{TAQ}(R,A;B)$ for a fixed map of
$\mathbb{E}_{\infty}$-algebras $A\rightarrow B$ and any object $R$ in the
category of $\mathbb{E}_{\infty}$-algebras between $A$ and $B$. For any object
$R$ in the category of $\mathbb{E}_{\infty}$-$\mathbb{S}$-algebras with a map
to $H\mathbb{F}_{p}$, we obtain the TAQ spectrum

$\mathrm{TAQ}(R,\mathbb{S};H\mathbb{F}_{p})\simeq|\mathrm{Bar}_{\bullet}(H\mathbb{F}_{p}\mathbin{\mathop{\otimes}\displaylimits}(-),\mathbb{E}_{\infty},R)|,$

cf. [Bas99, section 5] and [Law20, Proposition 1.8.9]. There is a base change
formula to $\mathbb{F}_{p}$-linear TAQ spectrum

$\mathrm{TAQ}(R,\mathbb{S};H\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits_{\mathbb{S}}}H\mathbb{F}_{p}\simeq\mathrm{TAQ}(R\mathbin{\mathop{\otimes}\displaylimits_{\mathbb{S}}}H\mathbb{F}_{p},H\mathbb{F}_{p};H\mathbb{F}_{p})$

for $R$ any $\mathbb{E}_{\infty}$-$\mathbb{S}$-algebra. The $n$th
$\mathbb{F}_{p}$-linear TAQ cohomology is defined to be

$\mathrm{TAQ}^{n}(R,H\mathbb{F}_{p};H\mathbb{F}_{p})=[\Sigma^{-n}\mathrm{TAQ}(R,H\mathbb{F}_{p};H\mathbb{F}_{p}),H\mathbb{F}_{p}]_{\mathrm{Mod}_{H\mathbb{F}_{p}}}$

for $R$ any $\mathbb{E}_{\infty}$-$H\mathbb{F}_{p}$-algebra.

In this paper we work with the nonunital $\mathbb{F}_{p}$-linear version

$\overline{\mathrm{TAQ}}(A)\simeq|\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbb{P},A)|,$

where $\mathbb{P}$ is the nonunital $\mathbb{E}_{\infty}$-operad in
$\mathrm{Mod}_{H\mathbb{F}_{p}}$ and $A$ a $\mathbb{P}$-algebra. We call this
the reduced mod $p$ TAQ spectrum of $A$, since

$\overline{\mathrm{TAQ}}(A)\oplus
H\mathbb{F}_{p}\simeq\mathrm{TAQ}(H\mathbb{F}_{p}\oplus
A,H\mathbb{F}_{p};H\mathbb{F}_{p}).$

Thus the reduced mod $p$ TAQ cohomology group
$\overline{\mathrm{TAQ}}^{n}(A):=[\Sigma^{-n}\overline{\mathrm{TAQ}}(A),H\mathbb{F}_{p}]_{\mathrm{Mod}_{H\mathbb{F}_{p}}}$
differ from the $\mathbb{F}_{p}$-linear TAQ cohomology group
$\mathrm{TAQ}^{n}(A\oplus H\mathbb{F}_{p},H\mathbb{F}_{p};H\mathbb{F}_{p})$
only when $n=0$ by a copy of $\mathbb{F}_{p}$. By Proposition 2.2, when $A$ is
a bounded above $H\mathbb{F}_{p}$-module of finite type considered as a
trivial $\mathbb{P}$-algebra, there is an equivalence

$\overline{\mathrm{TAQ}}^{n}(A^{\vee})\cong\pi_{-n}(\mathbb{D}|\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbb{P},A^{\vee})|)\cong\pi_{-n}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(A)).$

Going forward we will often omit $\mathbb{F}_{p}$-linear and mod $p$ when
there is no ambiguity regarding which version of TAQ cohomology is concerned.

###  2.4. Operations

The goal of this paper is to understand natural operations and their relations
on the homotopy groups of spectral partition Lie algebras and mod $p$ TAQ
cohomology. First we record a few general remarks about operations on algebras
over a monad, adapted from Lawson’s excellent survey ([Law20, section 1.4]) on
the theory of operations for algebras over operads.

Given a monad $\mathbf{T}$ on $\mathrm{Mod}_{H\mathbb{F}_{p}}$, we define an
operation on $\mathbf{T}$-algebras to be a natural transformation
$\pi_{m}(-)\rightarrow\pi_{n}(-)$ of functors
$h\mathrm{Alg}_{\mathbf{T}}(\mathrm{Mod}_{H\mathbb{F}_{p}})\rightarrow\mathrm{Sets}$
for some $m,n$. Here
$h\mathrm{Alg}_{\mathbf{T}}(\mathrm{Mod}_{H\mathbb{F}_{p}})$ is the homotopy
category of $\mathbf{T}$-algebras over $\mathrm{Mod}_{H\mathbb{F}_{p}}$. Let
$\mathrm{Op}(m;n)$ be the set of operations for fixed $m,n$. It follows from
the universal property of free algebras that for any $\mathbf{T}$-algebra $A$,

$\pi_{m}(A)\cong\mathrm{Map}_{\mathrm{Alg}_{\mathbf{T}}(\mathrm{Mod}_{H\mathbb{F}_{p}})}(\mathrm{Free}^{\mathbf{T}}(\Sigma^{m}H\mathbb{F}_{p}),A).$

Hence $\mathrm{Free}^{\mathbf{T}}(\Sigma^{m}H\mathbb{F}_{p})$ is the
representing object for the functor $\pi_{m}(-)$ on
$h\mathrm{Alg}_{\mathbf{T}}(\mathrm{Mod}_{H\mathbb{F}_{p}})$.

By the Yoneda Lemma, the set of operations $\mathrm{Op}(m;n)$, or equivalently
natural transformations $\pi_{m}(-)\rightarrow\pi_{n}(-)$ in
$h\mathrm{Alg}_{\mathbf{T}}(\mathrm{Mod}_{H\mathbb{F}_{p}})$, is isomorphic to
$\pi_{n}(\mathrm{Free}^{\mathbf{T}}(\Sigma^{m}H\mathbb{F}_{p}))$. Explicitly,
given an operation
$\alpha\in\pi_{n}(\mathrm{Free}^{\mathbf{T}}(\Sigma^{m}H\mathbb{F}_{p}))$ and
a class $x\in\pi_{m}(A)$ with $A$ a $\mathbf{T}$-algebra, we obtain a class
$\alpha(x)$ in $\pi_{n}(A)$ via the pullback

$\pi_{m}(A)\cong\mathrm{Map}_{\mathrm{Alg}_{\mathbf{T}}(\mathrm{Mod}_{H\mathbb{F}_{p}})}(\mathrm{Free}^{\mathbf{T}}(\Sigma^{m}H\mathbb{F}_{p}),A)\xrightarrow{\alpha^{*}}\mathrm{Map}_{\mathrm{Alg}_{\mathbf{T}}(\mathrm{Mod}_{H\mathbb{F}_{p}})}(\mathrm{Free}^{\mathbf{T}}(\Sigma^{n}H\mathbb{F}_{p}),A)\cong\pi_{n}(A).$

Therefore, to understand the unary operations on $\mathbf{T}$-algebras and
their relations, we need to first compute
$\pi_{*}(\mathrm{Free}^{\mathbf{T}}(\Sigma^{m}H\mathbb{F}_{p}))$ as an algebra
for all $m$. Then we need to understand the composition product on unary
operations

$\pi_{n}(\mathrm{Free}^{\mathbf{T}}(\Sigma^{m}H\mathbb{F}_{p}))\times\pi_{m}(\mathrm{Free}^{\mathbf{T}}(\Sigma^{l}H\mathbb{F}_{p}))\rightarrow\pi_{n}(\mathrm{Free}^{\mathbf{T}}(\Sigma^{l}H\mathbb{F}_{p}))$

for all $l,m,n$, which corresponds to composing two natural transformations
$\pi_{l}(-)\rightarrow\pi_{m}(-)$ and $\pi_{m}(-)\rightarrow\pi_{n}(-)$ of
functors on $h\mathrm{Alg}_{\mathbf{T}}(\mathrm{Mod}_{H\mathbb{F}_{p}})$. In
general, natural $k$-ary operations
$\prod_{l=1}^{k}\pi_{i_{l}}(-)\rightarrow\pi_{n}(-)$ are parametrized by the
homotopy groups

$\pi_{n}(\mathrm{Free}^{\mathbf{T}}(\Sigma^{i_{1}}H\mathbb{F}_{p}\oplus\cdots\oplus\Sigma^{i_{k}}H\mathbb{F}_{p}))$

for all $k$-tuples $(i_{1},\ldots,i_{k})$.

Here we specialize to
$\mathbf{T}=\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}$. The
decomposition of the free algebra over
$\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}$ into homogeneous
pieces in Proposition 2.2 allows us to impose a weight grading on the
operations on the homotopy groups of
$\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}$-algebras in the
usual sense.

On the other hand, the mod $p$ TAQ cohomology functor
$\mathrm{TAQ}(-,H\mathbb{F}_{p};H\mathbb{F}_{p})$ on
$\mathrm{Mod}_{H\mathbb{F}_{p}}$ has as representing objects the trivial
square-zero extensions. ([Law20, section 1.8]) Therefore, for any $m$ and
tuple $(i_{1},\ldots,i_{k})$, the group of cohomology operations

$\prod^{k}_{i=1}\mathrm{TAQ}^{i_{l}}(-,H\mathbb{F}_{p};H\mathbb{F}_{p})\rightarrow\mathrm{TAQ}^{m}(-,H\mathbb{F}_{p};H\mathbb{F}_{p})$

is the given by
$\mathrm{TAQ}^{m}(H\mathbb{F}_{p}\oplus\Sigma^{i_{1}}H\mathbb{F}_{p}\oplus\cdots\Sigma^{i_{k}}H\mathbb{F}_{p},H\mathbb{F}_{p};H\mathbb{F}_{p}).$

Note that all operations vanish on the unit except for scalar multiplication.
Since

$\overline{\mathrm{TAQ}}^{n}(A^{\vee})\cong\pi_{-n}(\mathbb{D}|\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbb{P},A^{\vee})|)\cong\pi_{-n}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(A))$

for all $n$ when $A$ is a bounded above $H\mathbb{F}_{p}$-module of finite
type considered as a trivial $\mathbb{P}$-algebra, natural operations and
their relations on the reduced mod $p$ TAQ cohomology, or equivalently, the
mod $p$ TAQ cohomology away from the unit, agree with those on the homotopy
group of spectral partition Lie algebras up to a change of grading
conventions.

Brantner and Mathew obtained bases of the homotopy groups of free spectral
partition Lie algebras on $\Sigma^{j}H\mathbb{F}_{p}$ via an isotropy spectral
sequence as in [ADL13, Example 1.3]. Then they propagated the result to any
direct sum of shifts of $H\mathbb{F}_{p}$ using a Hilton-Milnor-type
decomposition of the partition complex and an EHP sequence developed in
[AB21].

######  Definition 2.5.

We say a word $w$ in letters $\\{x_{1},\ldots,x_{k}\\}$ is a Lyndon word if it
is smaller than any of its cyclic rotations in the lexicographic order with
$x_{1}<\cdots<x_{k}$. Write $B(n_{1},\ldots,n_{k})$ for the set of Lyndon
words in which the letter $x_{i}$ appears precisely $n_{i}$ times.

Note that the collection of all Lyndon words in letters $x_{1},\ldots,x_{k}$
produces a basis for the free totally-isotropic Lie algebra over
$\mathbb{F}_{p}$ on $k$ generators, where totally-isotropic means there are no
non-vanishing self-brackets.

######  Theorem 2.6.

[BM19, Theorem 1.20] The $\mathbb{F}_{p}$-vector space
$\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(\Sigma^{l_{1}}\mathbb{F}_{p}\oplus\cdots\oplus\Sigma^{l_{k}}\mathbb{F}_{p}))$
has a basis indexed by sequences $(i_{1},\ldots,i_{k},e,w)$. Here $w\in
B(n_{1},\ldots,n_{m})$ is a Lyndon word. We have $e\in\\{0,\iota\\}$, where
$\iota=1$ if $p$ is odd and $\mathrm{deg}(w):=\sum_{i}(l_{i}-1)n_{i}+1$ is
even. Otherwise, $\iota=0$. The integers $i_{1},\ldots,i_{k}$ satisfy:

(1). Each $i_{j}$ is congruent to 0 or 1 modulo $2(p-1)$.

(2). For all $1\leq j<k$, we have $i_{j}<pi_{j+1}$.

(3). We have $i_{k}\leq(p-1)(1+e)\mathrm{deg}(w)-\iota$. The homological
degree of $(i_{1},\ldots,i_{k},e,w)$ is
$((1+e)\mathrm{deg}(w)-e)+i_{1}+\ldots+i_{k}-k$.

Nonetheless, their method does not yield explicit descriptions of the nature
of the operations, nor are the composition products or the relations among the
operations clarified. In the rest of the paper, we use a dual bar spectral
sequence to compute the relations among the unary operations. By comparing
with the bases in the above theorem, we show that the spectral sequences of
interest degenerate. Then we define a composition product among unary
operations and construct a shifted Lie bracket with restrictions on the
homotopy group of spectral partition Lie algebras. Finally we deduce that
these are all the natural operations and construct the target category.

##  3\. A bar spectral sequence

This section serves as a preliminary examination of the bar spectral sequence
for a $\mathbb{P}$-algebra $A$, where $\mathbb{P}$ is the
$\mathbb{E}^{\mathrm{nu}}_{\infty}$-operad in
$\mathrm{Mod}_{H\mathbb{F}_{p}}$, obtained by the skeletal filtration of the
geometric realization of the bar construction

$\widetilde{E}^{2}_{s,t}=\pi_{s}(\pi_{t}(\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbb{P},A))\Rightarrow\pi_{s+t}(|\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbb{P},A)|)\cong\overline{\mathrm{TAQ}}_{s+t}(A).$

Similarly, there is a dual bar spectral sequence

$E_{s,t}^{2}=\pi_{s}(\pi_{t}(\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbb{P},A)^{\vee}))\Rightarrow\pi_{s+t}(\mathbb{D}|\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbb{P},A)|)$

When $A=\Sigma^{-j}H\mathbb{F}_{p}$ is a trivial $\mathbb{P}$-algebra, the
$E^{\infty}$-page records unary operations on a degree $j$ class in the
homotopy group of any spectral partition Lie algebra and those on a degree
$-j$ cohomology class in the reduced mod p TAQ cohomology.

###  3.1. The Dyer-Lashof algebra

Dyer-Lashof operations are natural unary operations on the mod $p$ homology of
infinite loop spaces and $\mathbb{E}_{\infty}$-algebras in Spectra. These
operations and their relations were computed by Araki-Kudo ([KA56]), Dyer-
Lashof ([DL62]), Cohen-Lada-May ([CLM76]), and Bruner-May-McCLure-Steinberger
([BMMS86]). Denote by $\pazocal{R}$ the non-unital mod $p$ Dyer-Lashof
algebra.

######  Proposition 3.1.

[CLM76, I.1], [BMMS86, III.1] At $p=2$, the Dyer-Lashof algebra $\pazocal{R}$
is generated by operations $Q^{i}$ in degree $i$ and weight $2$ subject to the
Adem relations

$Q^{r}Q^{s}=\sum_{r+s-i\leq 2i}\binom{i-s-1}{2i-r}Q^{r+s-i}Q^{i}$

for $r>2s$.

For $p$ an odd prime, the mod $p$ Dyer-Lashof algebra is generated by
operations $\beta^{\epsilon}Q^{i}$ in degree $2(p-1)i-\epsilon$ and weight $p$
for $\epsilon\in\\{0,1\\}$ and all $i$, subject to the Adem relations

$Q^{r}Q^{s}=\sum_{r+s-i\leq
pi}(-1)^{r+i}\binom{(p-1)(i-s)-1}{pi-r}Q^{r+s-i}Q^{i},$ $\beta
Q^{r}Q^{s}=\sum_{r+s-i\leq pi}(-1)^{r+i}\binom{(p-1)(i-s)-1}{pi-r}\beta
Q^{r+s-i}Q^{i}$

for $r>ps$ and

$\displaystyle Q^{r}\beta Q^{s}=$
$\displaystyle\sum_{r+s-i<pi}(-1)^{r+i}\binom{(p-1)(i-s)}{pi-r}\beta
Q^{r+s-i}Q^{i}$
$\displaystyle-\sum_{r+s-i<pi}(-1)^{r+i}\binom{(p-1)(i-s)-1}{pi-r-1}Q^{r+s-i}\beta
Q^{i},$

$\beta Q^{r}\beta
Q^{s}=-\sum_{r+s-i<pi}(-1)^{r+i}\binom{(p-1)(i-s)-1}{pi-r-1}\beta
Q^{r+s-i}\beta Q^{i}$

for $r\geq ps$.

We say that a modules $M$ over the Dyer-Lashof algebra $\pazocal{R}$ is
unstable (or allowable) if the following conditions holds:

1\. When $p=2$, for any nonempty sequence of operation $Q^{I}=Q^{i_{1}}\cdots
Q^{i_{k}}$ and $x\in M$ of degree $j$, if $i_{l}-i_{l+1}-\ldots-i_{k}<j$ for
some $1\leq l\leq k$ then $Q^{I}(x)=0$.

2\. When $p>2$, for any $x\in M$ of degree $j$ and any nonempty sequence of
operation
$\alpha=\beta^{\epsilon_{1}}Q^{i_{1}}\cdots\beta^{\epsilon_{k}}Q^{i_{k}}$, if
$2i_{m}-\epsilon_{m}<j+2(p-1)i_{m+1}+\ldots
2(p-1)i_{k}-\epsilon_{1}-\cdots\epsilon_{k}+|x|$ for some $1\leq m\leq k$ then
$\alpha(x)=0$.

Denote by $\mathrm{Mod}_{\pazocal{R}}$ the category of unstable
$\pazocal{R}$-modules.

######  Definition 3.2.

A $\mathrm{Poly}_{\pazocal{R}}$-algebra $M$ is a (graded weighted) polynomial
algebra over $\mathbb{F}_{p}$ with an unstable $\pazocal{R}$-module structure
that is compatible with the commutative product
$\mathbin{\mathop{\otimes}\displaylimits}$ in the sense that

(1) The Cartan formula is satisfied:
$Q^{i}(x\mathbin{\mathop{\otimes}\displaylimits}y)=\sum_{j}Q^{j}(x)\mathbin{\mathop{\otimes}\displaylimits}Q^{i-j}(y)$;

(2) We further require that
$Q^{|x|/2}(x)=x^{\mathbin{\mathop{\otimes}\displaylimits}p}$ for all even
degree $x\in M$ when $p>2$, and
$Q^{|x|}(x)=x^{\mathbin{\mathop{\otimes}\displaylimits}2}$ when $p=2$.

Let $\mathrm{Poly}_{\pazocal{R}}$ be the category of
$\mathrm{Poly}_{\pazocal{R}}$-algebras. A classical result by May and McClure
tells us that this is the target category for the mod $p$ homology of non-
unital $\mathbb{E}_{\infty}$-$H\mathbb{F}_{p}$-algebras.

######  Theorem 3.3.

[May72, BMMS86] For any $\mathbb{P}$-algebra $X$, there is an isomorphism

$\pi_{*}(\mathrm{Free}^{\mathbb{P}}(X))\cong\mathrm{Free}^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}(\pi_{*}(X))$

of free $\mathrm{Poly}_{\pazocal{R}}$-algebras.

The free $\mathrm{Poly}_{\pazocal{R}}$-algebra on an $\mathbb{F}_{p}$-module
$M$ can be obtained by taking the polynomial algebra on
$\mathrm{Free}^{\mathrm{Mod}_{\pazocal{R}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}(M)$,
identifying $Q^{|x|/2}(x)=x^{\mathbin{\mathop{\otimes}\displaylimits}p}$ for
all even $x\in M$ when $p>2$ and
$Q^{|x|}(x)=x^{\mathbin{\mathop{\otimes}\displaylimits}2}$ when $p=2$, and
finally imposing the Cartan formula. Denote again by
$\mathrm{Poly}_{\pazocal{R}}$ the monad coming from the free-forgetful
adjunction

$\mathrm{Mod}_{\mathbb{F}_{p}}$$\mathrm{Poly}_{\pazocal{R}}$
.$\mathrm{Free}^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}$$U^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}$

###  3.2. A bar spectral sequence

By repeatedly applying the equivalence in Theorem 3.3, the bar spectral
sequence for a $\mathbb{P}$-algebra $A$ with $M=\pi_{*}(A)$ can be rewritten
as

$\widetilde{E}^{2}_{s,t}=\pi_{s}(\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Poly}_{\pazocal{R}},M))_{t}\Rightarrow\pi_{s+t}(|\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbb{P},A)|)=\overline{\mathrm{TAQ}}_{s+t}(A).$

Similarly, the dual bar spectral sequence takes the form

$E_{s,t}^{2}=\pi_{s}(\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Poly}_{\pazocal{R}},M)^{\vee})_{t}\Rightarrow\pi_{s+t}(\mathbb{D}|\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbb{P},A)|).$

The $E^{\infty}$-page is the reduced mod $p$ $\mathrm{TAQ}$ cohomology
$\overline{\mathrm{TAQ}}^{-*}(A)$, or the homotopy group of the spectral
partition Lie algebra
$\mathbb{D}|\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbb{P},A)|$ when $A$ is a
trivial bounded below $\mathbb{P}$-algebra of finite type. Since we will only
be concerned with bounded above objects of finite type over $H\mathbb{F}_{p}$,
we can switch freely between the two version by taking linear dual when
computing the second page. It is easier to work with the bar construction, so
we will focus on the bar spectral sequence.

Observe that the $E^{2}$-page of the bar spectral sequence

$\widetilde{E}^{2}_{s,t}=\pi_{s}(\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Poly}_{\pazocal{R}},M)_{t}=\pi_{s}(\mathbb{L}Q^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}(M))_{t}$

is the André-Quillen homology of $M$ with respect to the monad
$\mathrm{Poly}_{\pazocal{R}}$. Our plan is to find a suitable factorization of
the indecomposable functor
$Q^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}$ to separate
the unary and binary structures of the monad $\mathrm{Poly}_{\pazocal{R}}$.
This will allow us to replace the bar construction computing its total left
derived functor by a smaller double complex that is amenable to Koszul-
duality-type computations. We will follow the same strategy in [Zh21, section
3.3], which is inspired by [BHK19, section 4.4]. The subtlety lies in the
bottom non-vanishing Dyer-Lashof operations, which appear in both the unary
and binary structures of $\mathrm{Poly}_{\pazocal{R}}$.

###  3.3. The derived indecomposable functor

Before diving into the computation, we briefly recall without proof the
homotopy theory of monads on the category of weighted graded
$\mathbb{F}_{p}$-modules and especially the two-sided bar construction for
simplicial objects, following closely Sections 3.1, 4.2 and 4.3 in [BHK19].
For the general theory, see for instance Sections 3.1 and 3.2 of [JN14].

Let $\mathbf{T}$ be an augmented monad on the category
$\mathrm{Mod}_{\mathbb{F}_{p}}$ of weighted graded $\mathbb{F}_{p}$-modules.
Denote by $\mathrm{Alg}_{\mathbf{T}}(\mathrm{Mod}_{\mathbb{F}_{p}})$ the
category of $\mathbf{T}$-algebras. The forgetful functor
$U:\mathrm{Alg}_{\mathbf{T}}(\mathrm{Mod}_{\mathbb{F}_{p}})\rightarrow\mathrm{Mod}_{\mathbb{F}_{p}}$
admits a left adjoint, the free functor
$\mathrm{Free}^{\mathbf{T}}:\mathrm{Mod}_{\mathbb{F}_{p}}\rightarrow\mathrm{Alg}_{\mathbf{T}}(\mathrm{Mod}_{\mathbb{F}_{p}})$.

Levelwise application of this pair of adjunction gives rise to a adjunction
between the corresponding categories of simplicial objects, denoted again by
$\mathrm{Free}^{\mathbf{T}}\dashv U$, as well as a monad $\mathbf{T}$ on the
category of simplicial weighted graded $\mathbb{F}_{p}$-modules with the
standard cofibrantly generated model structure. Then this adjunction induces a
right transferred model structure on the category of simplicial
$\mathbf{T}$-algebras, with weak equivalences and fibrations defined on the
underlying simplicial weighted graded $\mathbb{F}_{p}$-modules.

Denote by
$\mathrm{triv}:\mathrm{Mod}_{\mathbb{F}_{p}}=\mathrm{Alg}_{\mathbf{\mathrm{Id}}}(\mathrm{Mod}_{\mathbb{F}_{p}})\rightarrow\mathrm{Alg}_{\mathbf{T}}(\mathrm{Mod}_{\mathbb{F}_{p}})$
the inclusion of trivial $\mathbf{T}$-algebras, which is induced by the
augmentation. It has a left adjoint
$Q^{\mathbf{T}}:\mathrm{Alg}_{\mathbf{T}}(\mathrm{Mod}_{\mathbb{F}_{p}})\rightarrow\mathrm{Mod}_{\mathbb{F}_{p}}$,
the indecomposable functor with respect to the $\mathbf{T}$-algebra structure.
Applying this pair of adjunction levelwise to the corresponding categories of
simplicial objects, we obtain a Quillen adjunction denoted again as
$Q^{\mathbf{T}}\dashv\mathrm{triv}$. The total left derived functor
$\mathbb{L}Q^{\mathbf{T}}$ of $Q^{\mathbf{T}}$ can be computed by the
following standard recipe.

######  Construction 3.4.

Given a right functor $R:\mathrm{Mod}_{\mathbb{F}_{p}}\rightarrow\mathscr{D}$
over $\mathbf{T}$, and a simplicial object $A_{\bullet}$ in
$\mathrm{Alg}_{\mathbf{T}}(\mathrm{Mod}_{\mathbb{F}_{p}})$, one can apply the
two-sided bar construction $\mathrm{Bar}_{\bullet}(R,\mathbf{T},-)$ levelwise
to $A_{\bullet}$. The diagonal of the resulting bisimplicial complex is a
simplicial object in $\mathscr{D}$, denoted by
$\mathrm{Bar}_{\bullet}(R,\mathbf{T},A_{\bullet})$.

In particular, if we regard a $\mathbf{T}$-algebra $A$ as a simplicial
$\mathbf{T}$-algebra concentrated in simplicial degree 0, denoted also as $A$
by abuse of notation, then $\mathrm{Bar}_{\bullet}(R,\mathbf{T},A)$ agrees
with the usual two-sided bar construction.

The free resolution
$\mathrm{Bar}_{\bullet}(\mathrm{Free}^{\mathbf{T}},\mathbf{T},A)$ is a
cofibrant replacement of $A$ in the category of simplicial
$\mathbf{T}$-algebras. The left derived functor of a functor $F$ can be
computed by applying $F$ levelwise to a cofibrant replacement. Hence

$\mathbb{L}Q^{\mathbf{T}}(A)\simeq
Q^{\mathbf{T}}\mathrm{Bar}_{\bullet}(\mathrm{Free}^{\mathbf{T}},\mathbf{T},A)=\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbf{T},A).$

###  3.4. A smaller complex for the $E^{1}$-page

We disentangle the unary and binary structure by defining a structure that
heuristically discard the bottom non-vanishing Dyer-Lashof operations on any
element in a module over the Dyer-Lashof algebra when $p=2$ and on any even
class when $p>2$.

Let $\mathrm{Mod}_{\pazocal{R}^{\prime}}$ be category of modules over the
Dyer-Lashof algebra $\pazocal{R}$ with unstability conditions $Q^{i}(x)=0$ for
$i\leq|x|$ when $p=2$ and $2i\leq|x|$ when $p>2$. Then the indecomposables
functor $Q^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}$
factors as
$Q^{\mathrm{Mod}_{\pazocal{R}^{\prime}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}\circ
Q^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\pazocal{R}^{\prime}}}$ sitting
in the composite adjunction with the inclusion/trivial functor.

$\mathrm{Mod}_{\mathbb{F}_{p}}$$\mathrm{Mod}_{\pazocal{R}^{\prime}}$$\mathrm{Poly}_{\pazocal{R}}$.$T^{\mathrm{Mod}_{\pazocal{R}^{\prime}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}$$Q^{\mathrm{Mod}_{\pazocal{R}^{\prime}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}$$T^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\pazocal{R}^{\prime}}}$$Q^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\pazocal{R}^{\prime}}}$

In particular, there is an isomorphism

$U^{\mathrm{Mod}_{\pazocal{R}^{\prime}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}\circ
Q^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\pazocal{R}^{\prime}}}\cong
Q^{\mathrm{Poly}_{\mathbb{F}_{p}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}\circ
U^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Poly}_{\mathbb{F}_{p}}}.$

Denote by $\mathrm{Poly}_{\mathbb{F}_{p}}$ the monad corresponding to the free
graded polynomial algebra functor on $\mathrm{Mod}_{\mathbb{F}_{p}}$. We want
to use the factorization
$Q^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}\cong
Q^{\mathrm{Mod}_{\pazocal{R}^{\prime}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}\circ
Q^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\pazocal{R}^{\prime}}}$ to
obtain a double complex that is more computable than the bar complex
$\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Poly}_{\pazocal{R}},M)$.

######  Lemma 3.5.

There is a weak equivalence of simplicial $\pazocal{R}^{\prime}$-modules

$\mathbb{L}Q^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\pazocal{R}^{\prime}}}(M)\simeq\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Poly}_{\mathbb{F}_{p}},M)$

for any $\mathbb{F}_{p}$-module $M$ considered as a trivial
$\mathrm{Poly}_{\pazocal{R}}$-algebra.

###### Proof.

There is a map of augmented monads
$\mathrm{Poly}_{\pazocal{R}}\rightarrow\mathrm{Poly}_{\mathbb{F}_{p}}\rightarrow\mathrm{id}$,
the first of which kills all Dyer-Lashof operations that are not the bottom
operations $Q^{|x|}(x)=x^{\mathbin{\mathop{\otimes}\displaylimits}2}$ when
$p=2$ or the bottom operations
$Q^{/2}(x)=x^{\mathbin{\mathop{\otimes}\displaylimits}p}$ on even classes when
$p>2$. When $M$ is an $\mathbb{F}_{p}$-module considered as a trivial
$\mathrm{Poly}_{\pazocal{R}}$-algebra, the map
$\mathrm{Poly}_{\pazocal{R}}(M)\rightarrow\mathrm{Poly}_{\mathbb{F}_{p}}(M)\rightarrow
M$ is a map of $\mathrm{Poly}_{\pazocal{R}}$-algebras if we regard
$\mathrm{Poly}_{\mathbb{F}_{p}}(M)$ as a $\mathrm{Poly}_{\pazocal{R}}$-algebra
where all Dyer-Lashof operations vanish except for the bottom operations when
$p=2$ and the bottom operations on even classes when $p>2$. Therefore we
obtain a map of free bar resolutions

$\Psi:\mathrm{Bar}_{\bullet}(\mathrm{Free}^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}},\mathrm{Poly}_{\pazocal{R}},M)\rightarrow\mathrm{Bar}_{\bullet}(\mathrm{Free}^{\mathrm{Poly}_{\mathbb{F}_{p}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}},\mathrm{Poly}_{\mathbb{F}_{p}},M),$

which is a weak equivalence of simplicial
$\mathrm{Poly}_{\pazocal{R}}$-algebras.

Next we want to show that applying
$Q^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\pazocal{R}^{\prime}}}$
preserves this weak equivalence, i.e.

$\mathbb{L}Q^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\pazocal{R}^{\prime}}}(M)\simeq
Q^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\pazocal{R}^{\prime}}}\mathrm{Bar}_{\bullet}(\mathrm{Free}^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}},\mathrm{Poly}_{\pazocal{R}},M)\rightarrow
Q^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\pazocal{R}^{\prime}}}\mathrm{Bar}_{\bullet}(\mathrm{Free}^{\mathrm{Poly}_{\mathbb{F}_{p}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}},\mathrm{Poly}_{\mathbb{F}_{p}},M)$

is a weak equivalence of simplicial $\pazocal{R}^{\prime}$-modules. This is
equivalent to showing that the underlying map of simplicial
$\mathbb{F}_{p}$-modules
$U^{\mathrm{Mod}_{\pazocal{R}^{\prime}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}(\Psi)$
is a weak equivalence.

Using the isomorphism

$U^{\mathrm{Mod}_{\pazocal{R}^{\prime}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}\circ
Q^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\pazocal{R}^{\prime}}}\cong
Q^{\mathrm{Poly}_{\mathbb{F}_{p}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}\circ
U^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Poly}_{\mathbb{F}_{p}}},$

we can rewrite
$U^{\mathrm{Mod}_{\pazocal{R}^{\prime}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}(\Psi)$
as

$\displaystyle
U^{\mathrm{Mod}_{\pazocal{R}^{\prime}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}\circ
Q^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\pazocal{R}^{\prime}}}\mathrm{Bar}_{\bullet}(\mathrm{Free}^{\mathrm{Mod}_{\pazocal{R}^{\prime}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}},\mathrm{Poly}_{\pazocal{R}},M)\simeq
Q^{\mathrm{Poly}_{\mathbb{F}_{p}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}\circ
U^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Poly}_{\mathbb{F}_{p}}}\mathrm{Bar}_{\bullet}(\mathrm{Free}^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}},\mathrm{Poly}_{\pazocal{R}},M)$
$\displaystyle\rightarrow$ $\displaystyle
U^{\mathrm{Mod}_{\pazocal{R}^{\prime}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}\circ
Q^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\pazocal{R}^{\prime}}}\mathrm{Bar}_{\bullet}(\mathrm{Free}^{\mathrm{Poly}_{\mathbb{F}_{p}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}},\mathrm{Poly}_{\mathbb{F}_{p}},M)$
$\displaystyle\simeq$ $\displaystyle
Q^{\mathrm{Poly}_{\mathbb{F}_{p}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}\circ
U^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Poly}_{\mathbb{F}_{p}}}\mathrm{Bar}_{\bullet}(\mathrm{Free}^{\mathrm{Poly}_{\mathbb{F}_{p}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}},\mathrm{Poly}_{\mathbb{F}_{p}},M)\simeq\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Poly}_{\mathbb{F}_{p}},M).$

This is indeed a weak equivalence since we are applying
$Q^{\mathrm{Poly}_{\mathbb{F}_{p}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}$ to a free
simplicial $\mathrm{Poly}_{\mathbb{F}_{p}}$-algebra on both sides. Therefore
we obtain a weak equivalence of simplicial $\pazocal{R}^{\prime}$-modules

$\mathbb{L}Q^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}(M))\cong\mathrm{Bar}_{\bullet}(\mathrm{Free}^{\mathrm{Mod}_{\pazocal{R}^{\prime}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}},\mathrm{Poly}_{\pazocal{R}},M)\simeq\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Poly}_{\mathbb{F}_{p}},M)$

as desired. ∎

Let $\pazocal{A}_{\pazocal{R}^{\prime}}$ be the additive monad associated to
the free $\pazocal{R}^{\prime}$-module functor. Therefore the André-Quillen
homology of an algebra $M$ over the monad $\mathrm{Poly}_{\pazocal{R}}$, i.e.
the $E^{2}$-page of the bar spectral sequence

$\displaystyle\widetilde{E}^{2}_{*,*}\cong\pi_{*}(\mathbb{L}Q^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}(M))$
$\displaystyle\cong\pi_{*}\big{(}Q^{\mathrm{Mod}_{\pazocal{R}^{\prime}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}\circ
Q^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\pazocal{R}^{\prime}}}\mathrm{Bar}_{\bullet}(\mathrm{Free}^{\mathrm{Poly}_{\pazocal{R}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}},\mathrm{Poly}_{\pazocal{R}},M)\big{)}$
$\displaystyle\cong\pi_{*}\big{(}Q^{\mathrm{Mod}_{\pazocal{R}^{\prime}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}\mathrm{Bar}_{\bullet}(\mathrm{Free}^{\mathrm{Mod}_{\pazocal{R}^{\prime}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}},\mathrm{Poly}_{\pazocal{R}},M)\big{)}$
$\displaystyle\cong\pi_{*}\big{(}\mathbb{L}Q^{\mathrm{Mod}_{\pazocal{R}^{\prime}}}_{\mathrm{Mod}_{\mathbb{F}_{p}}}(\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Poly}_{\mathbb{F}_{p}},M))\big{)},$

can be computed as the homotopy group of the double complex
$\mathrm{Bar}_{\bullet}(\mathrm{id},\pazocal{A}_{\pazocal{R}^{\prime}},\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Poly}_{\mathbb{F}_{p}},M)).$

##  4\. Computing the dual bar spectral sequence

In this section, we compute the $E^{2}$-page of the (dual) bar spectral
sequences in the universal case, i.e., when
$A=\Sigma^{i_{1}}H\mathbb{F}_{p}\oplus\cdots\oplus\Sigma^{i_{l}}H\mathbb{F}_{p}$.
Then a comparison with Theorem 2.6 allows us to deduce the degeneration of the
spectral sequence in these cases. There is a non trivial distinction between
the cases $p=2$ and $p>2$ regarding the restriction on shifted Lie algebra
structures and the notations are rather different, so we record them
separately.

###  4.1. The $E^{2}$-page at $p=2$

We will utilize Priddy’s machinery on algebraic Koszul duality in [Pri70,
Theorem 2.5]. Since the Dyer-Lashof algebra $\pazocal{R}$ is a Koszul algebra,
the Ext group

$\mathrm{Ext}^{*,*}_{\pazocal{R}}(\mathbb{F}_{2},\mathbb{F}_{2})=\pi_{*}((\mathrm{Bar}_{\bullet}(\mathbb{F}_{2},\pazocal{R},\mathbb{F}_{2}))^{\vee})$

is the Koszul dual algebra of $\pazocal{R}$. The Koszul generators are given
by the collection

$(Q^{i})^{*}:=[(Q^{i})^{\vee}]1\in\mathrm{Ext}^{-1,*}_{\pazocal{R}}(\mathbb{F}_{2},\mathbb{F}_{2})$

with homological bidegree $(-1,-i)$ and weight 2, with composition given by
juxtaposition, which corresponds to the Yoneda product on Ext groups, cf.
[Pri70, p.42] and [McC01, Theorem 9.8]. The quadratic relations among the
generators are the Koszul dual of the Adem relations (Proposition 3.1), i.e.

(1)
$(Q^{a})^{*}(Q^{b})^{*}=\sum_{a+b-c>2c}\binom{b-c-1}{a-2c-1}(Q^{a+b-c})^{*}(Q^{c})^{*}$

for $a\leq 2b$.

We are interested in the unstable $\mathrm{Ext}$ group

$\mathrm{UnExt}^{*,*}_{\pazocal{R}^{\prime}}(\mathbb{F}_{2},\Sigma^{j}\mathbb{F}_{2})=\pi_{*}(\mathrm{Bar}_{\bullet}(\mathrm{id},\pazocal{A}_{\pazocal{R}^{\prime}},\Sigma^{-j}\mathbb{F}_{2})^{\vee}),$

which is a variant of the Ext group

$\mathrm{Ext}^{*,*}_{\pazocal{R}}(\mathbb{F}_{2},\mathbb{F}_{2})=\pi_{*}((\mathrm{Bar}_{\bullet}(\mathbb{F}_{2},\pazocal{R},\mathbb{F}_{2}))^{\vee})$

obtained by regardding $\Sigma^{-j}\mathbb{F}_{2}$ as an unstable trivial
module over $\pazocal{R}$ and imposing the unstablitly conditions
$[Q^{j}|\alpha]=0$ for $j\leq|\alpha|$ in the bar complex, cf. [BC70, §3]. To
incorporate the Koszul dual of the unstability conditions as well as the
simplicial grading, we introduce the following ringoid.

######  Definition 4.1.

Let $\pazocal{F}$ be the ringoid with objects the $\mathbb{Z}_{\leq
0}\times\mathbb{Z}$ and morphisms freely generated over $\mathbb{F}_{2}$ under
juxtaposition by the following elements: for any $s\leq 0$ and all $i,j$
satisfying $i>-j$, there is an element
$(Q^{i})^{*}\in\pazocal{F}\big{(}(s,j),(s-1,j-i)\big{)}$ of weight 2. Let
$(\pazocal{R}^{\prime})^{!}$ be the quotient of $\pazocal{F}$ by the ideal
generated by the relations

(2)
$(Q^{a})^{*}(Q^{b})^{*}=\sum_{a+b-c>2c,c>-j}\binom{b-c-1}{a-2c-1}(Q^{a+b-c})^{*}(Q^{c})^{*}$

for all $a,b$ satisfying $a\leq 2b$, $b>-j$ and $a>b-j$ in
$\pazocal{F}\big{(}(s,j),(s-2,j-a-b)\big{)}$.

The first grading corresponds to the homological degree in Ext, or
equivalently the filtration degree in the dual bar spectral sequence. The
second grading is the topological degree.

######  Remark 4.2.

There is an evident isomorphism
$(\pazocal{R}^{\prime})^{!}((s,i),(s^{\prime},j))\cong(\pazocal{R}^{\prime})^{!}((s-r,i),(s^{\prime}-r,j))$
for any $i,j,s,s^{\prime}$ and $r$ such that $s-r<0$. For any $t>0,$ there is
an injection

$\mathrm{susp^{t}}:(\pazocal{R}^{\prime})^{!}((s,i),(s^{\prime},j))\hookrightarrow(\pazocal{R}^{\prime})^{!}((s,i+t),(s^{\prime},j+t)),$

since more operations are defined on classes with higher homological degree.

######  Remark 4.3.

Note the relations are always well defined on both sides: if $b\geq-j+1,$ then
$a\geq b-j+1\geq-2j+2$, so $\lfloor\frac{a+b}{3}\rfloor\geq-j+1$ and the right
hand side is never empty.

The unstable $\mathrm{Ext}$ group
$\mathrm{UnExt}^{*,*}_{\pazocal{R}^{\prime}}(\mathbb{F}_{2},\Sigma^{j}\mathbb{F}_{2})$
is thus the underlying bigraded $\mathbb{F}_{2}$-module of the free
$(\pazocal{R}^{\prime})^{!}$-module $(\pazocal{R}^{\prime})^{!}((0,j),-)$. We
grade the Ext groups homologicially.

On the other hand, the Tor group
$\mathrm{Tor}^{*}_{\pazocal{R}^{\prime}}(\mathbb{F}_{2},\mathbb{F}_{2})$ is a
coalgebra ${(\pazocal{R})^{\vee}}$ generated by classes in

$\mathrm{Tor}^{1,*}_{\pazocal{R}^{\prime}}(\mathbb{F}_{2},\mathbb{F}_{2})=\mathbb{F}_{2}\\{[Q^{i}]1,i\in\mathbb{N}\\}.$

The unstable Tor group

$\mathrm{UnTor}^{*,*}_{\pazocal{R}^{\prime}}(\mathbb{F}_{2},\Sigma^{j}\mathbb{F}_{2})=\pi_{*}(\mathrm{Bar}_{\bullet}(\mathrm{id},\pazocal{A}_{\pazocal{R}^{\prime}},\Sigma^{j}\mathbb{F}_{2}))$

is thus a coalgebra generated under juxtaposition by elements in

$\mathrm{UnTor}_{\pazocal{R}^{\prime}}^{1,*}(\mathbb{F}_{2},\Sigma^{j}\mathbb{F}_{2})=\\{[Q^{i}]1,i>j\\},$

which is a comodule over the co-ringoid ${(\pazocal{R}^{\prime})^{\vee}}$.

Back to computing the $E^{2}$-page. Denote by
$co\mathrm{Free}^{(\pazocal{R}^{\prime})^{\vee}}$ the functor that takes the
cofree ${(\pazocal{R}^{\prime})^{\vee}}$-comodule. Similarly,
$\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}$ is the functor that takes the
free $(\pazocal{R}^{\prime})^{!}$-module with a simplicial grading, i.e. the
class
$(Q^{i_{1}})^{*}(Q^{i_{2}})^{*}\cdots(Q^{i_{k}})^{*}(x)\in\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}(M)$
has simplicial degree $-k$ for $x\in M$.

Since $\pazocal{A}_{\pazocal{R}^{\prime}}$ is an additive monad, there is no
nontrivial simplicial operations on the total left derived functor.
Furthermore, the homological (vertical) and simplicial (horizontal)
differentials do not mix, i.e. the targets of the vertical differential never
involve elements from the inner bar complex and vice versa. Therefore we can
deduce the following:

######  Lemma 4.4.

Suppose that $V_{\bullet}$ is a trivial simplicial
$\pazocal{R}^{\prime}$-module. Then

$\pi_{*}(\mathrm{Bar}_{\bullet}(\mathrm{id},\pazocal{A}_{\pazocal{R}^{\prime}},V_{\bullet}))=co\mathrm{Free}^{(\pazocal{R}^{\prime})^{\vee}}(\pi_{*}(V_{\bullet})).$

In our case, we are interested in the trivial simplicial
$\pazocal{R}^{\prime}$-module
$V_{\bullet}=\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Poly}_{\mathbb{F}_{2}},M)$
where $M$ is a direct sum of shifts of $\mathbb{F}_{2}$ as trivial
$\mathrm{Poly}_{\mathbb{F}_{2}}$-modules.

######  Definition 4.5.

(cf. [Jac41], [Fre00] for the unshifted version.) A shifted restricted Lie
algebra over $\mathbb{F}_{2}$, denoted as a
$\mathrm{sLie}^{\rho}_{\mathbb{F}_{2}}$-algebra, is a graded
$\mathbb{F}_{2}$-module $L=L_{\bullet}$ with a shifted Lie bracket
$L_{m}\mathbin{\mathop{\otimes}\displaylimits}L_{n}\rightarrow L_{m+n-1}$ and
a restriction map $x\mapsto x^{[2]}$ with $x^{[2]}\in L_{2|x|-1}$, satisfying
the following identities:

  1. (1)

$\mathrm{ad}(x^{[2]})=\mathrm{ad}(x)^{2}$ for all $x\in L$;

  2. (2)

For all $x,y\in L$, $(x+y)^{[p]}=x^{[p]}+y^{[p]}+[x,y]$.

Here $\mathrm{ad}(x)$ stands for the adjoint representation, i.e., the self-
map $y\mapsto[y,x]$ on $L$.

Let $\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\mathbb{F}_{2}}}$ be the associated
free functor. Given an $\mathbb{F}_{p}$-module $M$ with basis
$\\{x_{1},\ldots,x_{k}\\}$, a basis for
$\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\mathbb{F}_{2}}}(M)$ is given by

$\\{u,u^{[2]},(u^{[2]})^{[2]},\ldots\\},$

where $u$ ranges over Lyndon words in letters $x_{1},\ldots,x_{k}$. (See, for
instance, [BKS05, section 2].)

Note that the monad $\mathrm{Poly}_{\mathbb{F}_{2}}$ is the monad associated
with the free functor of the commutative operad $\mathrm{Comm}$ in
$\mathrm{Mod}_{\mathbb{F}_{2}}$. Denote by $\mathrm{sLie}$ the shifted Lie
operad in the category chain complexes over $\mathbb{F}_{2}$, i.e.
$\mathrm{sLie}(n)$ sits in homological dimension $1-n$ with trivial
differentials. The comparison morphism
$\bigoplus_{n}(\mathrm{sLie}(n)^{\vee}\mathbin{\mathop{\otimes}\displaylimits}M)^{\mathbin{\mathop{\otimes}\displaylimits}n})_{\Sigma_{n}}\rightarrow\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Comm},M)$
of the shifted Harrison complex and the bar construction (cf. [Fre04, 5.2.3,
6.6]) is an isomorphism on homotopy when $M$ is an $\mathbb{F}_{2}$-module
considered as a trivial algebra over Comm: any cycle on the left-hand side is
a sum of all way to put $n-1$ nested parenthesis on a fixed sequence
$x_{1},\ldots x_{n}$ of $n$ classes in $M$, such that each nesting represents
taking the polynomial product with one more class on a different simplicial
level and no nesting is trivial, and this cycle has preimage the bracket
$[\cdots[[x_{1},x_{2}],x_{3}]\ldots],x_{n}]$. Taking linear dual, self-
brackets become restrictions by [Fre00, Theorem 0.1], so we have

$\pi_{*}(\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Comm},M))=\pi_{*}(\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Poly}_{\mathbb{F}_{2}},M))\cong
co\mathrm{Free}^{\mathrm{co-sLie_{\mathbb{F}_{2}}}}(M),$
$\pi_{*}(\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Comm},M)^{\vee})=\bigoplus_{n}(\mathrm{sLie}(n)\mathbin{\mathop{\otimes}\displaylimits}(M^{\vee})^{\mathbin{\mathop{\otimes}\displaylimits}n})^{\Sigma_{n}}\cong\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\mathbb{F}_{2}}}(M^{\vee})$

for any trivial algebra $M$ over $\mathrm{Comm}$. On the other hand, a shifted
Lie coalgebra over $\mathbb{F}_{2}$ has a shifted Lie cobracket
$L_{m+n+1}\rightarrow L_{m}\mathbin{\mathop{\otimes}\displaylimits}L_{n}$
satisfying the co-Jacobi identity. The shift reflects the simplicial degree in
the (co)bar resolution. Similarly a
co-$\mathrm{sLie}_{\mathbb{F}_{2}}$-algebra stands for shifted coLie-algebra
over $\mathbb{F}_{2}$.

###  4.2. The $E^{\infty}$-page for $p=2$

In the case where $A=\Sigma^{j}H\mathbb{F}_{p}$, the $E^{\infty}$-page records
all unary operations on a degree $-j$ class in the homotopy groups of spectral
partition Lie algebras. Note that

$\pi_{*}(\mathbb{D}\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Poly}_{\mathbb{F}_{2}},\pi_{*}(\Sigma^{j}H\mathbb{F}_{p})))=\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\mathbb{F}_{2}}}(\Sigma^{-j}\mathbb{F}_{2})$

has exactly one class $x^{[2]^{s}}\in\pi_{s}$ of weight $2^{s}$ for all $s\geq
0$. Therefore the dual bar spectral sequence simplifies to

$E^{2}_{s,t}=\mathrm{Free}^{({\pazocal{R}^{\prime})^{!}}}(\mathbb{F}_{2}\\{x^{[2]^{s}},s\geq
0\\})\Rightarrow\pi_{s+t}(\mathbb{D}|\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbb{P},\Sigma^{j}H\mathbb{F}_{p})|)$

Note that the $E^{2}$-page is concentrated in weight $2^{k}$ for
$k\in\mathbb{N}$, and the weight $2^{k}$ part is concentrated on a single line
$s=-k$. Hence the spectral sequences collapses on the second page and there
are no extension problems. Therefore we have found all the unary operations on
a degree $j$ class for any $j$.

In particular, we see that the restriction $x^{[2]}$ is represented by the
cycle $(Q^{|x|})^{*}|x$ on the $E^{1}$-page
$\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Poly}_{\pazocal{R}},\Sigma^{j}\mathbb{F}_{2})^{\vee}$.
On the other hand, the relations in (1) never involve the bottom operation: on
a class $x$ of degree $j$, if $c=-j$ then the coefficient
$\binom{b+j-1}{a+2j-1}$ of $(Q^{a+b-c})^{*}(Q^{c})^{*}$ is nonzero only if
$a+2j-1\leq b+j-1$, which is impossible since $a>b-j$.

Hence the ringoid $(\pazocal{R}^{\prime})^{!}$ is the ringoid of additive
operations. Now we define a new ringoid that takes into account the
restriction being an unary operation.

######  Definition 4.6.

Let $\pazocal{F}$ be the ringoid with objects $\mathbb{Z}_{\leq
0}\times\mathbb{Z}$ whose morphisms are freely generated over $\mathbb{F}_{2}$
under juxtaposition by the following elements: for all $i,j$ satisfying
$i\geq-j$ and $s\leq 0$, there is an element
$(Q^{i})^{*}\in\pazocal{F}\big{(}(s,j),(s-1,j-i)\big{)}$ of weight 2.

Let $\pazocal{R}^{!}$ be the quotient of $\pazocal{F}$ by the ideal generated
by the relations $(Q^{a-j})^{*}(Q^{a})^{*}=0$ for all $j,s$ and
$(Q^{a})^{*}\in\pazocal{F}\big{(}(s,j),(s-1,j-a)\big{)}$ with $a>-j$, and the
Adem relations

(3)
$(Q^{a})^{*}(Q^{b})^{*}=\sum_{a+b-c>2c,c>-j}\binom{b-c-1}{a-2c-1}(Q^{a+b-c})^{*}(Q^{c})^{*}$

for all $a,b$ satisfying $a\leq 2b$, $b>-j$ and $a>b-j$ in
$\pazocal{F}\big{(}(s,j),(s-2,j-a-b)\big{)}$.

The $E^{2}$-page of the dual bar spectral sequence on one generator has the
following structure.

######  Definition 4.7.

An $\mathrm{sLie}^{\rho}_{\pazocal{R}^{!}}$-algebra is an
$\mathbb{F}_{2}$-module $M$ with an $\pazocal{R}^{!}$-module structure and a
shifted Lie bracket

$[\,\,,\,\,]:M_{s,t}\mathbin{\mathop{\otimes}\displaylimits}M_{s^{\prime},t^{\prime}}\rightarrow
M_{s+s^{\prime}-1,t+t^{\prime}}$

with restriction $(-)^{[2]}$ satisfying the following conditions:

(1) The bottom operation $(Q^{-|x|})^{*}(x)=x^{[2]}$ is the restriction for
any $x$;

(2) $[x,\alpha(y)]=0$ for any $x,y$ and non-empty sequence $\alpha$ of
$(Q^{i})^{*}$’s unless $\alpha$ is an iteration of the restriction map.

Let $\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{R}^{!}}}$ denote the
functor that first takes the free
$\mathrm{sLie}^{\rho}_{\mathbb{F}_{2}}$-algebra on a bigraded
$\mathbb{F}_{2}$-module $M$, then takes the free $\pazocal{R}^{!}$-module on
the underlying graded $\mathbb{F}_{2}$-module of
$\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\mathbb{F}_{2}}}(M)$, and finally
identifies the restriction with the bottom operation
$x^{[2]}=(Q^{-|x|})^{*}(x)$ for all
$x\in\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\mathbb{F}_{2}}}(M)$. Therefore we
can deduce that:

######  Proposition 4.8.

For $A=\Sigma^{j}H\mathbb{F}_{2}$, the bar spectral sequence has
$E^{\infty}$-page

$\widetilde{E}^{\infty}_{*,*}\cong\widetilde{E}^{2}_{*,*}\cong
co\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}co\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\mathbb{F}_{2}}}(\Sigma^{j}\mathbb{F}_{2}).$

The dual bar spectral sequence for $A$ has $E^{\infty}$-page

$E^{\infty}_{*,*}\cong
E^{2}_{*,*}\cong\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{R}^{!}}}(\Sigma^{-j}\mathbb{F}_{2}).$

A basis is given by the monomials
$(Q^{i_{1}})^{*}(Q^{i_{2}})^{*}\cdots(Q^{i_{s}})^{*}(x)$, where
$i_{s}\geq|x|=-j$ and $i_{l}>2i_{l+1}$ for all $1\leq l<s$.

Now we can compute the $E^{\infty}$-page of the (dual) bar spectral sequences
in the universal case, and deduce the set of $k$-ary natural operations of all
$k$. A priori, knowing the composition product and relations among operations
on the André-Quillen cohomology
$\mathrm{AQ}^{*}_{\mathrm{Poly}_{\pazocal{R}}}(-)$ does not imply knowledge of
the relations on the homotopy groups of spectral partition Lie algebra and mod
$2$ TAQ cohomology. The composition product on the later differs from that on
the former, as we will see in Theorem 5.5.

######  Proposition 4.9.

Let
$A=\Sigma^{j_{1}}H\mathbb{F}_{2}\oplus\Sigma^{j_{2}}H\mathbb{F}_{2}\oplus\cdots\oplus\Sigma^{j_{k}}H\mathbb{F}_{2}$
be a trivial $\mathbb{P}$-algebra. Then the dual bar spectral sequence for
$\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{2},\mathbb{E}_{\infty}}(A^{\vee}))\cong\overline{\mathrm{TAQ}}^{*}(A)$
has $E^{\infty}$-page

$\displaystyle E^{\infty}_{*,*}\cong
E^{2}_{*,*}\cong\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{R}^{!}}}(\Sigma^{-j_{1}}\mathbb{F}_{2}\oplus\cdots\oplus\Sigma^{-j_{k}}\mathbb{F}_{2}).$

###### Proof.

The dual bar spectral sequence simplifies to

$E^{2}_{*,*}=\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{R}^{!}}}(\Sigma^{-j_{1}}\mathbb{F}_{2}\oplus\cdots\oplus\Sigma^{-j_{k}}\mathbb{F}_{2})\Rightarrow\pi_{s+t}(\mathbb{D}|\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbb{P},A)|).$

A priori we can’t deduce that the spectral sequence collapses using a sparsity
argument when $k>1$, since the $E^{2}$-page is concentrated on multiple lines
at $p^{m}$ when $m>1$. However, the $E^{\infty}$-page of the dual bar spectral
sequence is the homotopy group of

$\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(A^{\vee})\simeq\mathbb{D}|\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbb{P},A)|,$

the free spectral partition Lie algebra on $A^{\vee}$. Comparing the basis of
the $E^{2}$-page with Theorem 2.6, which is a basis of
$\mathrm{Lie}^{\pi}_{\mathbb{F}_{2},\mathbb{E}_{\infty}}(A^{\vee})$, we deduce
that $E^{2}\cong E^{\infty}$, so the spectral sequence collapses on the second
page and there are no extension problems. ∎

###  4.3. The dual bar spectral sequence for odd primes

In the section, we apply the same analysis to the odd primary case.

Again utilizing Priddy’s machinary on algebraic Koszul duality in [Pri70,
Theorem 2.5], we deduce that
$\mathrm{Ext}_{\pazocal{R}}(\mathbb{F}_{p},\mathbb{F}_{p})=\pi_{*}((\mathrm{Bar}_{\bullet}(\mathbb{F}_{p},\pazocal{R}^{\prime},\mathbb{F}_{p}))^{\vee})$
is isomorphic as an algebra to the Koszul dual $(\pazocal{R}^{\prime})^{!}$ of
$\pazocal{R}^{\prime}$. The Koszul generators are given by the collection

$(\beta^{\epsilon}Q^{i})^{*}:=[(\beta^{\epsilon}Q^{i})^{\vee}]1\in\mathrm{Ext}^{1}_{\pazocal{R}^{\prime}}(\mathbb{F}_{p},\mathbb{F}_{p}),\epsilon\in\\{0,1\\},i\in\mathbb{N}$

where $(\beta^{\epsilon}Q^{i})^{*}$ has homological bidegree
$(-1,-2(p-1)i+\epsilon)$ and weight $p$. Composition is given by juxtaposition
corresponding to the Yoneda product. The quadratic relations are the Koszul
dual of the Adem relations (Proposition 3.1), i.e.

$(Q^{a})^{*}(Q^{b})^{*}+\sum_{a+b-c>pc}(-1)^{a-c}\binom{(p-1)(b-c)-1}{a-pc-1}(Q^{a+b-c})^{*}(Q^{c})^{*}=0$

for $a\leq pb$ ,

$\displaystyle(\beta Q^{a})^{*}(Q^{b})^{*}-\sum_{a+b-c\geq
pc}(-1)^{a-c}\binom{(p-1)(b-c)}{a-pc}(Q^{a+b-c})^{*}(\beta Q^{c})^{*}$
$\displaystyle+\sum_{a+b-c>pc}(-1)^{a-c}\binom{(p-1)(b-c)-1}{a-pc-1}(\beta
Q^{a+b-c})^{*}(Q^{c})^{*}=0$

for $a\leq pb,$

$(\beta^{\epsilon}Q^{a})^{*}(\beta Q^{b})^{*}-\sum_{a+b-c\geq
pc}(-1)^{a-c}\binom{(p-1)(b-c)-1}{a-pc}(\beta^{\epsilon}Q^{a+b-c})^{*}(\beta
Q^{c})^{*}=0$

for $\epsilon\in\\{0,1\\}$ and $a<pb$, cf. [KL83].

Analogous to the case $p=2$, we dualize the unstability condition and record
the simplicial grading using a ringoid.

######  Definition 4.10.

Let $\pazocal{F}$ be the ringoid with objects $\mathbb{Z}_{\leq
0}\times\mathbb{Z}$ and morphisms freely generated over $\mathbb{F}_{p}$ under
juxtaposition by the following elements: for $2i>-j$ and any $s\leq 0$ there
are elements $(Q^{i})^{*}\in\pazocal{F}\big{(}(s,j),(s-1,j-2(p-1)i)\big{)}$
and $(\beta Q^{i})^{*}\in\pazocal{F}\big{(}(s,j),(s-1,j-2(p-1)i+1)\big{)}$. We
suppress the first grading for ease of notation when there is no ambiguity.

Let $(\pazocal{R}^{\prime})^{!}$ be the quotient of $\pazocal{F}$ by the ideal
generated by the following quadratic relations for all $s\leq 0$:

$(Q^{a})^{*}(Q^{b})^{*}=-\sum_{a+b-c>pc,\,\,2c>-k}(-1)^{a-c}\binom{(p-1)(b-c)-1}{a-pc-1}(Q^{a+b-c})^{*}(Q^{c})^{*},$

in $\pazocal{F}\big{(}j,j-2(p-1)a-2(p-1)b\big{)}$ for all $a,b\in\mathbb{Z}$
and satisfying $a\leq pb$, $2b>-j$, $2a>2(p-1)b-j$,

$\displaystyle(\beta Q^{a})^{*}(Q^{b})^{*}=\sum_{a+b-c\geq
pc,2c>-k}(-1)^{a-c}\binom{(p-1)(b-c)}{a-pc}(Q^{a+b-c})^{*}(\beta Q^{c})^{*}$
$\displaystyle-\sum_{a+b-c>pc,2c>-k}(-1)^{a-c}\binom{(p-1)(b-c)-1}{a-pc-1}(\beta
Q^{a+b-c})^{*}(Q^{c})^{*}=0$

in $\pazocal{F}\big{(}j,j-2(p-1)a-2(p-1)b+1\big{)}$ for all $a,b\in\mathbb{Z}$
satisfying $a\leq pb$, $2b>-j$, $2a>2(p-1)b-j$, and

$(\beta^{\epsilon}Q^{a})^{*}(\beta Q^{b})^{*}=\sum_{a+b-c\geq
pc,2c>-k}(-1)^{a-c}\binom{(p-1)(b-c)-1}{a-pc}(\beta^{\epsilon}Q^{a+b-c})^{*}(\beta
Q^{c})^{*}$

in $\pazocal{F}\big{(}j,j-2(p-1)a-2(p-1)b+\epsilon+1\big{)}$ for
$\epsilon\in\\{0,1\\}$ and $a,b\in\mathbb{Z}$ satisfying $a<pb$, $2b>-j$,
$2a>2(p-1)b-j$.

A basis for $(\pazocal{R}^{\prime})^{!}\big{(}(s,j),(s-k,-)\big{)}$ is given
by sequences
$(\beta^{\epsilon_{1}}Q^{i_{1}})^{*}(\beta^{\epsilon_{2}}Q^{i_{2}})^{*}\cdots(\beta^{\epsilon_{k}}Q^{i_{k}})^{*}$
where $2i_{k}>-j$ and $i_{l}>pi_{l+1}-\epsilon$ for $1\leq l<k$.

######  Remark 4.11.

Similar to the case $p=2$, there is an isomorphism

$(\pazocal{R}^{\prime})^{!}((s,i),(s^{\prime},j))\cong(\pazocal{R}^{\prime})^{!}((s-r,i),(s^{\prime}-r,j))$

for any $i,j,s,s^{\prime}$ and $r$ such that $s-r<0$. For any $t>0,$ there is
an injection

$\mathrm{susp}^{t}:(\pazocal{R}^{\prime})^{!}((s,i),(s^{\prime},j))\hookrightarrow(\pazocal{R}^{\prime})^{!}((s,i+t),(s^{\prime},j+t)),$

since more operations are defined on classes with higher homological degree.

For $x$ in degree $2d+1$, the relations never involve the bottom operations on
$x$, i.e. the terms $(\beta^{\epsilon}Q^{a+b+d})^{*}(\beta Q^{-d})^{*}$: the
coefficient is nonzero only if $(p-1)(b+d)-1\geq a+pd$, i.e., $(p-1)b-d-1\geq
a$. But $2a\geq 2(p-1)b-2d$ by assumption.

Hence the unstable $\mathrm{Ext}$ group

$\mathrm{UnExt}^{*,*}_{\pazocal{R}}(\mathbb{F}_{p},\Sigma^{-j}\mathbb{F}_{p})=\pi_{*}(\mathrm{Bar}_{\bullet}(\mathrm{id},\pazocal{A}_{\pazocal{R}^{\prime}},\Sigma^{-j}\mathbb{F}_{p})^{\vee})$

is the free $(\pazocal{R}^{\prime})^{!}$-module
$(\pazocal{R}^{\prime})^{!}((0,j),-)$. Whereas the Tor group
$\mathrm{Tor}^{*,*}_{\pazocal{R}^{\prime}}(\mathbb{F}_{p},\mathbb{F}_{p})\cong{({\pazocal{R}}^{!})}^{\vee}$
is a coalgebra generated by classes in

$\mathrm{Tor}^{1,*}_{\pazocal{R}^{\prime}}(\mathbb{F}_{p},\mathbb{F}_{p})=\mathbb{F}_{p}\\{[\beta^{\epsilon}Q^{i}]1,i\in\mathbb{N},\epsilon=0,1\\}.$

The unstable Tor groups are cofree comodules over the co-ringoid
${((\pazocal{R}^{\prime})^{!})^{\vee}}$.

The functor $\mathrm{Free}^{({\pazocal{R}}^{\prime})^{!}}$ takes the free
$(\pazocal{R}^{\prime})^{!}$-module with a dual simplicial grading that counts
the number of generators, i.e., the element
$(\beta^{\epsilon_{1}}Q^{i_{1}})^{*}(\beta^{\epsilon_{2}}Q^{i_{2}})^{*}\cdots(\beta^{\epsilon_{k}}Q^{i_{k}})^{*}(x)\in\mathrm{Free}^{{\pazocal{R}}^{!}}(M)$
with $x\in M$ has simplicial degree $-k$. Denote by
$co\mathrm{Free}^{{(\pazocal{R}^{\prime})^{\vee}}}$ the functor that takes the
cofree ${(\pazocal{R}^{\prime})^{\vee}}$-comodule. Hence
$\pi_{*}(\mathrm{Bar}_{\bullet}(\mathrm{id},\pazocal{A}_{\pazocal{R}^{\prime}},\Sigma^{-j}\mathbb{F}_{p})^{\vee})$
is the free $\pazocal{R}^{!}$-module on a single generator $x$ in degree $j$,
and
$\pi_{*}(\mathrm{Bar}_{\bullet}(\mathrm{id},\pazocal{A}_{\pazocal{R}^{\prime}},\Sigma^{j}\mathbb{F}_{p}))$
is the free ${(\pazocal{R}^{\prime})^{\vee}}$-comodule on $x$.

Since $\pazocal{A}_{\pazocal{R}^{\prime}}$ is an additive monad, there is no
nontrivial simplicial operations on the total left derived functor.
Furthermore the vertical and horizontal differentials act strictly within
their complexes, i.e., the targets of the vertical differential never involve
elements from the inner bar complex and vice versa. Therefore we can deduce
the following:

######  Lemma 4.12.

Suppose that $V_{\bullet}$ is a trivial simplicial
$\pazocal{R}^{\prime}$-module. Then

$\pi_{*}(\mathrm{Bar}_{\bullet}(\mathrm{id},\pazocal{A}_{\pazocal{R}^{\prime}},V_{\bullet}))=co\mathrm{Free}^{{((\pazocal{R}^{\prime})^{!})^{\vee}}}(\pi_{*}(V_{\bullet})).$

Now we run the spectral sequence for a trivial $\mathbb{P}$-algebra $A$. First
we look at the dual bar spectral sequences for $A=\Sigma^{j}H\mathbb{F}_{p}$,
which parametrizes unary operations on a degree $-j$ class the homotopy groups
of spectral partition Lie algebra.

######  Proposition 4.13.

Let $A=\Sigma^{j}H\mathbb{F}_{p}$. If $j$ is odd, then
$\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(A^{\vee}))\cong\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}(\Sigma^{-j}\mathbb{F}_{p})$.
If $j$ is even, then
$\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(A^{\vee}))\cong\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}(\Sigma^{-j}\mathbb{F}_{p}\oplus\Sigma^{-2j-1}\mathbb{F}_{p})$.

###### Proof.

If $A=\Sigma^{j}H\mathbb{F}_{p}$ with $j$ odd, then
$\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Poly}_{\mathbb{F}_{p}},\Sigma^{j}\mathbb{F}_{p})$
is the constant simplicial object on $\Sigma^{j}\mathbb{F}_{p}$ due to graded
commutativity. The dual bar spectral sequence simplifies to

$E^{2}_{*,*}=\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}(\Sigma^{-j}\mathbb{F}_{p})\Rightarrow\pi_{s+t}(\mathbb{D}|\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbb{P},A)|)\cong\pi_{s+t}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(A^{\vee})).$

Note that the $E^{2}$-page is concentrated in weight $p^{k}$ for
$k\in\mathbb{N}$, and the weight $p^{k}$ part is concentrated on a single line
$s=-k$. Hence the spectral sequence collapses on the second page and there are
no extension problems.

If $A=\Sigma^{j}H\mathbb{F}_{p}$ with $j$ even, then

$\pi_{*}(\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Poly}_{\mathbb{F}_{p}},\Sigma^{j}\mathbb{F}_{p})^{\vee})\cong\mathbb{F}_{p}\\{x,[x,x]\\}$

with a weight 1 class $x\in\pi_{0}$ and a weight 2 class $[x,x]\in\pi_{-1}$.
On the $E^{2}$-page of the dual bar spectral sequence

$E^{2}_{*,*}=\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}(\mathbb{F}_{p}\\{x,[x,x]\\})\Rightarrow\pi_{s+t}(\mathbb{D}|\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbb{P},A)|)$

is concentrated in weights $p^{k}$ and $2p^{k}$, and at each $k$ concentrated
on the line $s=-k$ at weight $p^{k}$ and the line $s=-k-1$ at weight $2p^{k}$.
Again there are no further differentials or extension problems. ∎

######  Remark 4.14.

(1) As in the case $p=2$, knowing the relations among unary operations on the
André-Quillen cohomology $\mathrm{AQ}^{*}_{\mathrm{Poly}_{\pazocal{R}}}(-)$
does not immediately yield knowledge of the relations on the homotopy groups
of spectral partition Lie algebra and mod $p$ TAQ cohomology. The composition
product on the later differs from that on the former, as we will see in
Theorem 5.6.

(2) While we expect to see a shifted restricted Lie algebra, the restriction
on an odd class is not detected on the $E^{2}$-page for filtration reasons, as
we shall see in Lemma 5.12 in the next section.

In general, we have the odd primary counterpart of Proposition 4.9.

######  Proposition 4.15.

Suppose that
$A=\Sigma^{j_{1}}H\mathbb{F}_{p}\oplus\Sigma^{j_{2}}H\mathbb{F}_{p}\oplus\cdots\oplus\Sigma^{j_{l}}H\mathbb{F}_{p}$
is a trivial $\mathbb{P}$-algebra. Then the $E^{\infty}$-page of the dual bar
spectral sequence for
$\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(A))\cong\overline{\mathrm{TAQ}}^{-*}(A)$
has $E^{\infty}$-page

$E^{\infty}_{*,*}\cong
E^{2}_{*,*}\cong\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}\mathrm{Free}^{\mathrm{sLie}^{\mathbb{F}_{p}}}(\Sigma^{j_{1}}\mathbb{F}_{p}\oplus\cdots\oplus\Sigma^{j_{l}}\mathbb{F}_{p}).$

###### Proof.

The comparison morphism
$\bigoplus_{n}(\mathrm{sLie}(n)^{\vee}\mathbin{\mathop{\otimes}\displaylimits}M)^{\mathbin{\mathop{\otimes}\displaylimits}n})_{\Sigma_{n}}\rightarrow\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Comm},M)$
of the shifted Harrison complex and the bar construction (cf. [Fre04, 5.2.3,
6.2]) is surjective on homotopy when $M$ is an $\mathbb{F}_{p}$-module
considered as a trivial algebra: any cycle on the left-hand side is an
alternating sum of all way to put $n-1$ nested parenthesis on a fixed sequence
$x_{1},\ldots x_{n}$ of $n$ classes in $M$, such that each nesting represents
taking the polynomial product with one more class on a different simplicial
level and no nesting is trivial, and this cycle has preimage the bracket
$[\cdots[[x_{1},x_{2}],x_{3}]\ldots],x_{n}]$. The graded commutativity and the
vanishing of self-brackets on odd degree classes correspond to the graded
commutativity of the polynomial product. Since the restriction on odd classes
are zero by Proposition 4.13, taking linear dual yields

$\pi_{*}(\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Comm},M)^{\vee})\cong\mathrm{Free}^{\mathrm{sLie}_{\mathbb{F}_{p}}}(M^{\vee})$

for any trivial algebra $M$ over $\mathrm{Comm}$ and the dual bar spectral
sequence simplifies to

$E^{2}_{*,*}=\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}\mathrm{Free}^{\mathrm{sLie}_{\mathbb{F}_{p}}}(\Sigma^{j_{1}}\mathbb{F}_{p}\oplus\cdots\oplus\Sigma^{j_{l}}\mathbb{F}_{p})\Rightarrow\pi_{s+t}(\mathbb{D}|\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbb{P},A^{\vee})|).$

As in the case $p=2$, a priori we can’t deduce that the spectral sequence
collapses using a sparsity argument when $k>1$, since the $E^{2}$-page is
concentrated on multiple lines at $p^{m}$ when $m>1$. Nonetheless, comparing
the basis of the $E^{2}$-page with Theorem 2.6, we deduce that $E^{2}\cong
E^{\infty}$, so the spectral sequence collapses on the second pages and there
are no extension problems. ∎

##  5\. Operations and their relations

In this section, we construct all natural operations on the homotopy groups of
spectral partition Lie algebras and mod $p$ TAQ (co)homology. It follows from
a general result of Brantner ([Br17, Theorem 3.5.1 and 4.3.2]) that
composition product of additive operations on the homotopy groups of spectral
partition Lie algebras agrees, up to a shearing, with the Yoneda product on
the $E^{2}$-page of the dual bar spectral sequence. This allows us to deduce
all relations among the unary operations in Theorem 5.5 and 5.6. Then we
construct a shifted Lie algebra structure and prove the existence of a
restriction map in Lemma 5.12 when $p>2$. Finally we deduce all relations
among unary operations and the bracket in Theorem 5.13.

###  5.1. Unary operations

In the dual bar spectral sequence

$E^{2}_{s,t}\cong\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{R}^{!}}}(\Sigma^{j}\mathbb{F}_{2})_{s,t}\Rightarrow\pi_{s+t}(\mathbb{D}|\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbb{P},\Sigma^{-j}H\mathbb{F}_{2})|)\simeq\pi_{s+t}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{2},\mathbb{E}_{\infty}}(\Sigma^{j}H\mathbb{F}_{2})),\,\,\,\,p=2$
$E^{2}_{s,t}\cong\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}\mathrm{Free}^{\mathrm{sLie}_{\mathbb{F}_{p}}}(\Sigma^{j}\mathbb{F}_{p})_{s,t}\Rightarrow\pi_{s+t}(\mathbb{D}|\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbb{P},\Sigma^{-j}H\mathbb{F}_{p})|)\simeq\pi_{s+t}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(\Sigma^{j}H\mathbb{F}_{p})),\,\,\,\,p>2,$

the $E^{2}$-page is generated by a single class $x\in E_{2}^{0,j}$ under unary
operations

$(Q^{i})^{*}:E^{2}_{s,t}\rightarrow E^{2}_{s-1,t-i},\,\,\,\,i>-t$

for $p=2$, and

$(\beta^{\epsilon}Q^{i})^{*}:E^{2}_{s,t}\rightarrow
E^{2}_{s-1,t-2(p-1)i+\epsilon},\,\,\,\,2i>-t$

where $\epsilon\in\\{0,1\\}$, as well as a shifted self-bracket if $j$ is even
for $p>2$. The additive unary operations, excluding the self-bracket, and
their relations are encoded by the ringoid $(\pazocal{R}^{\prime})^{!}$ in
Definition 4.1 and 4.10.

The $E^{\infty}$-page is the homotopy groups of the free spectral partition
Lie algebra on $\Sigma^{j}H\mathbb{F}_{p}$. Hence it parametrizes unary
operations on a degree $j$ homotopy class of a spectral partition Lie algebra
and a degree $-j$ cohomology class in mod $p$ TAQ cohomology. Now we give a
concrete description of unary operations on the the homotopy groups of any
spectral partition Lie algebra $A$.

######  Construction 5.1.

Suppose that $\xi:\Sigma^{j}H\mathbb{F}_{p}\rightarrow A$ represents a
homotopy class $x\in\pi_{j}(A)$.

(1) Suppose that $p=2$. For any sequence

$\alpha=(Q^{i_{1}})^{*}(Q^{i_{2}})^{*}\cdots(Q^{i_{k}})^{*}\in\pazocal{R}^{!}((0,j),(-k,j-m))$

with $i_{1}+\cdots+i_{k}=m,$ there is a unique class
$R^{(i_{1}+1,i_{2}+1,\ldots,i_{k}+1)}(x)\in\pi_{j-m-k}(A)$ given by

$\displaystyle\Sigma^{j-m-k}\mathbb{F}_{2}\xrightarrow{\alpha}\mathrm{Free}^{{\pazocal{R}}^{!}}(\Sigma^{j}\mathbb{F}_{2})$
$\displaystyle\hookrightarrow\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{R}^{!}}}(\Sigma^{j}\mathbb{F}_{2})$
$\displaystyle\cong\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{2},\mathbb{E}_{\infty}}(\Sigma^{j}H\mathbb{F}_{2}))\xrightarrow{\xi_{*}}\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{2},\mathbb{E}_{\infty}}(A))\rightarrow\pi_{*}(A).$

(2) Suppose that $p>2$. For any sequence

$\alpha=(\beta^{\epsilon_{1}}Q^{i_{1}})^{*}(\beta^{\epsilon_{2}}Q^{i_{2}})^{*}\cdots(\beta^{\epsilon_{k}}Q^{i_{k}})^{*}\in(\pazocal{R}^{\prime})^{!}((0,j),(-k,j-m))$

with $m=2(p-1)i_{1}+\cdots+2(p-1)i_{k}-\epsilon_{1}-\cdots-\epsilon_{k},$
there is a unique class
$R^{(i_{1},\ldots,i_{k},1-\epsilon_{1},\ldots,1-\epsilon_{k})}(x)\in\pi_{j-m-k}(A)$
given by

$\displaystyle\Sigma^{j-m-k}\mathbb{F}_{p}\xrightarrow{\alpha}\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}(\Sigma^{j}\mathbb{F}_{p})$
$\displaystyle\hookrightarrow\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}\mathrm{Free}^{\mathrm{sLie}_{\mathbb{F}_{p}}}(\Sigma^{j}\mathbb{F}_{p})$
$\displaystyle\cong\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(\Sigma^{j}H\mathbb{F}_{p}))\xrightarrow{\xi_{*}}\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(A))\rightarrow\pi_{*}(A).$

If $j$ is even, then there is a unique class $B(x)\in\pi_{2j-1}(A)$ given by

$\Sigma^{2j-1}\xrightarrow{[x,x]}\mathrm{Free}^{\mathrm{sLie}_{\mathbb{F}_{p}}}(\Sigma^{j}\mathbb{F}_{p})\hookrightarrow\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}\mathrm{Free}^{\mathrm{sLie}_{\mathbb{F}_{p}}}(\Sigma^{j}\mathbb{F}_{p})\cong\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(\Sigma^{j}H\mathbb{F}_{p}))\rightarrow\pi_{*}(A)$

Translating to cohomological grading, for any
$\mathbb{E}_{\infty}$-$H\mathbb{F}_{p}$-algebra $A$, there are unary
operations

$R^{(i_{1},i_{2},\ldots,i_{k})}:\overline{\mathrm{TAQ}}^{j}(A)\rightarrow\overline{\mathrm{TAQ}}^{j+i_{1}+\cdots+i_{k}}(A),\,\,\,\,i>j$

for $p=2$ and

$R^{(i_{1},\ldots,i_{k},\epsilon_{1},\ldots,\epsilon_{k})}:\overline{\mathrm{TAQ}}^{j}(A)\rightarrow\overline{\mathrm{TAQ}}^{j+2(p-1)(i_{1}+\cdots+i_{k})+\epsilon_{1}+\cdots+\epsilon_{k}}(A),\,\,\,\,2i>j$

where $\epsilon\in\\{0,1\\}$ as well as a self-bracket

$B:\overline{\mathrm{TAQ}}^{j}(A)\rightarrow\overline{\mathrm{TAQ}}^{2j+1}(A)$

when $j$ is even for $p>2$.

######  Remark 5.2.

These operations are stable in the sense that any cohomology operation
$\alpha:\overline{\mathrm{TAQ}}^{m}(A)\rightarrow\overline{\mathrm{TAQ}}^{m+|\alpha|}(A)$
agrees with
$\alpha:\overline{\mathrm{TAQ}}^{m-1}(A)\rightarrow\overline{\mathrm{TAQ}}^{m-1+|\alpha|}(A)$
under cohomological desuspension, or equivalently
$\alpha:\pi_{m}(L)\rightarrow\pi_{m-|\alpha|}(L)$ agrees with
$\alpha:\pi_{m+1}(L)\rightarrow\pi_{m+1-|\alpha|}(L)$ for any spectral
partition Lie algebra $L$. This is straightforward to check on the
$E^{2}$-page of the dual bar spectral sequence in the universal cases using
the fact that Dyer-Lashof operations satisfy this notion of stability.

A convenient way to encode the structure of additive operations is via a power
ring, as was introduced in [Br17] to encode additive unary operations on the
Lubin-Tate theory of spectral Lie algebras. Note that our convention differs
in that we switch to a logarithmic grading convention for the weight grading.

######  Definition 5.3.

[Br17, Definition 3.17] A power ring is a collection
$P=\\{P^{k}_{j}(w)\\}_{(j,k,w)\in\mathbb{Z}^{2}\times\mathbb{Z}_{\geq 0})}$ of
abelian groups with elements $\iota_{i}\in P^{i}_{i}[0]$ for all $i$, equipped
with associative and unital composition maps $P^{i}_{j}[v]\times
P^{j}_{k}[w]\rightarrow P^{i}_{k}[v+w]$.

A module over the power ring $P$ is a (weighted) $\mathbb{F}_{p}$-module
$M=\bigoplus_{(j,w)\in\mathbb{Z}\times\mathbb{Z}_{\geq 0}}M_{j}(w)$ with
structure maps
$P^{i}_{j}[v]\mathbin{\mathop{\otimes}\displaylimits}M_{j}(w)\rightarrow
M_{i}(p^{v}w)$ that are compatible with the composition maps in $P$.

######  Definition 5.4.

The collection
$\\{\pazocal{P}^{j}_{k}[w]:=(\pazocal{R}^{\prime})^{!}((0,k),(-w,j+w)),w>0\\}$,
along with $\pazocal{P}^{i}_{i}[0]:=\mathbb{F}_{p}\\{\iota_{i}\\}$ for all $i$
and $\pazocal{P}^{j}_{i}[0]=\emptyset$ for $i\neq j$, defines a power ring
$\pazocal{P}$, with composition product given by the sheared Yoneda product

${\pazocal{P}^{i}_{j}[v]\times\pazocal{P}^{j}_{k}[w]}$${(\pazocal{R}^{\prime})^{!}((0,j),(-v,i+v))\times(\pazocal{R}^{\prime})^{!}((0,k),(-w,j+w))}$${(\pazocal{R}^{\prime})^{!}((0,j+w),(-v,i+v+w))\times(\pazocal{R}^{\prime})^{!}((0,k),(-w,j+w))}$${(\pazocal{R}^{\prime})^{!}((-w,j+w),(-v-w,i+v+w))\times(\pazocal{R}^{\prime})^{!}((0,k),(-w,j+w))}$${\pazocal{P}^{i}_{k}[v+w]}$${(\pazocal{R}^{\prime})^{!}((0,k),(-v-w,i+v+w)),}$$\scriptstyle{\cong}$$\scriptstyle{\mathrm{susp}^{w}\times\mathrm{id}}$$\scriptstyle{\cong}$$\scriptstyle{\mathrm{juxtaposition}}$$\scriptstyle{\cong}$

for $v,w>0$, as well as isomorphisms
$\pazocal{P}^{j}_{i}[w]\times\pazocal{P}^{i}_{i}[0]\xrightarrow{\cong}\pazocal{P}^{j}_{i}[w]$
and
$\pazocal{P}^{j}_{j}[0]\times\pazocal{P}^{j}_{i}[w]\xrightarrow{\cong}\pazocal{P}^{j}_{i}[w]$
exhibiting $\iota_{i}$ as a two-sided unit.

The first map is an injection on the left factor because operations are stable
under suspension and here $w\geq 0$, cf. Remark 4.2 and 4.11. The last map is
the composition in the ringoid $(\pazocal{R}^{\prime})^{!}$, i.e.,
juxtaposition corresponding to the Yoneda product on Ext groups.

Explicitly, when $p=2$, the $\mathbb{F}_{2}$-module $\pazocal{P}^{k}_{j}[w]$
consists of operations $R^{i_{1},\ldots,i_{w}}$ such that $j-i_{1}-\ldots
i_{w}=k$ and $i_{l}-1>i_{l+1}+\cdots+i_{w}-j-(w-l)$ for all $1\leq l\leq w$,
subject to the relations in $(\pazocal{R}^{\prime})^{!}((0,j),(-w,k+w))$. The
composition product sends $R^{(i_{1},\ldots,i_{v})}\circ
R^{(j_{1},\ldots,j_{w}})$ to the juxtaposition
$R^{(i_{1},\ldots,i_{v},j_{1},\ldots,j_{w})}.$ The weight $2$ additive
operations are given by the collection $R^{i}\in\pazocal{P}^{j-i}_{i}[1]$ for
any $i>-j+1$.

######  Theorem 5.5.

The homotopy groups of a spectral partition Lie algebra over
$H\mathbb{F}_{2}$, or the reduced TAQ cohomology of an
$\mathbb{E}_{\infty}$-$H\mathbb{F}_{2}$-algebra form a left module over the
power ring $\pazocal{P}$ of additive unary operations. The relations among the
weight $2$ additive operations are given by the Adem relations

$R^{a}R^{b}=\sum_{a+b-c\geq 2c,\,\,c>-j+1}\binom{b-c-1}{a-2c}R^{a+b-c}R^{c}$

in $\pazocal{P}^{j-a-b}_{j}[2]$ for all $a,b\in\mathbb{Z}$ satisfying
$b-j<a<2b$ and $b>-j+1$.

A basis for unary operations on a degree $j$ class is given by all monomials
$R^{i_{1}}R^{i_{2}}\cdots R^{i_{l}}$ such that $i_{l}>-j$ and $i_{m}\geq
2i_{m+1}$ for $1\leq m<l$.

When $p>2$, the weight $p$ unary operations are given by the collection of
elements
$\beta^{\epsilon}R^{i}:=R^{(i,\epsilon)}\in\pazocal{P}^{j-2(p-1)i-\epsilon}_{j}$
for $\epsilon=0,1$ and any $2i>-j$.

######  Theorem 5.6.

The homotopy groups of a spectral partition Lie algebra $A$ over
$H\mathbb{F}_{p}$, or the reduced TAQ cohomology of any
$\mathbb{E}_{\infty}$-$H\mathbb{F}_{p}$-algebra, form a module over the power
ring $\pazocal{P}$ of unary operations. The relations among the weight $p$
operations are given by the Adem relations

$\beta R^{a}\beta
R^{b}=\sum_{a+b-c>pc,2c>-j}(-1)^{a-c+1}\binom{(p-1)(b-c)-1}{a-pc-1}\beta
R^{a+b-c}\beta R^{c}$

in $\pazocal{F}_{p}(j,j-2(p-1)a-2(p-1)b-2)$ for all $a,b\in\mathbb{Z}$
satisfying $a\leq pb$, $2b>-j$, and $2a>2(p-1)b-j,$

$\displaystyle R^{a}\beta R^{b}=$ $\displaystyle\sum_{a+b-c\geq
pc,2c>-j}(-1)^{a-c}\binom{(p-1)(b-c)}{a-pc}\beta P^{a+b-c}R^{c}$
$\displaystyle-\sum_{a+b-c>pc,2c>-j}(-1)^{a-c}\binom{(p-1)(b-c)-1}{a-pc-1}R^{a+b-c}\beta
R^{c}$

in $\pazocal{F}_{p}(j,j-2(p-1)a-2(p-1)b-1)$ for all $a,b\in\mathbb{Z}$
satisfying $a\leq pb$, $2b>-j$, and $2a>2(p-1)b+1-j$,

$\beta^{\epsilon}R^{a}R^{b}=\sum_{a+b-c\geq
pc,2c>-j}(-1)^{a-c}\binom{(p-1)(b-c)-1}{a-pc}\beta^{\epsilon}R^{a+b-c}R^{c}$

in $\pazocal{P}^{j-2(p-1)a-2(p-1)b-\epsilon}_{j}[2]$ for all
$a,b\in\mathbb{Z}$ satisfying $a<pb$, $2b>-j$, $2a>2(p-1)b-j$, and
$\epsilon\in\\{0,1\\}$.

A basis for unary operations on a degree $j$ class with $j$ odd is given by
the collection of all monomials
$\beta^{\epsilon_{1}}R^{i_{1}}\beta^{\epsilon_{2}}R^{i_{2}}\cdots\beta^{\epsilon_{l}}R^{i_{l}}$
such that $2i_{l}>-j$ and $i_{m}\geq pi_{m+1}+\epsilon_{m+1}$ for $1\leq m<l$.
If $j$ is even, a basis is given by the collection
$\beta^{\epsilon_{1}}R^{i_{1}}\beta^{\epsilon_{2}}R^{i_{2}}\cdots\beta^{\epsilon_{l}}R^{i_{l}}B^{\epsilon}$
such that $2i_{l}>-(1+\epsilon)j-\epsilon$ and $i_{m}\geq
pi_{m+1}+\epsilon_{m+1}$ for $1\leq m<l$.

###### Proof of Theorem 5.5 and 5.6.

By construction, the set $\pazocal{P}^{i}_{j}[w]$ is isomorphic to the image
of

$(\pazocal{R}^{\prime})^{!}((0,j),(-w,w+i))\subset E^{2}_{*,*}$

in $\pi_{i}(\pazocal{L}(\Sigma^{j}H\mathbb{F}_{p}))$ via the collapse of the
dual bar spectral sequence

$E^{2}_{*,*}\cong\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{R}^{!}}}(\Sigma^{j}\mathbb{F}_{p})\Rightarrow\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{2},\mathbb{E}_{\infty}}(\Sigma^{j}H\mathbb{F}_{p}))$

and for $p>2$

$E^{2}_{*,*}\cong\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}\mathrm{Free}^{\mathrm{sLie}_{\mathbb{F}_{p}}}(\Sigma^{j}\mathbb{F}_{p})\Rightarrow\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{2},\mathbb{E}_{\infty}}(\Sigma^{j}H\mathbb{F}_{p})).$

We need to verify that compositions of unary operations on the homotopy groups
of spectral partition Lie algebras is reflected by the composition product of
the power ring $\pazocal{P}$.

For ease of notations, we will use $\pazocal{L}$ to denote the monad
$\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}$ throughout this
proof. The unary operations on the homotopy groups of algebras over
$\pazocal{L}$, other than the self-brackets on even classes when $p>2$, are
concentrated in weights $p^{n}$ for $n\geq 1$ by Proposition 4.8 and 4.13.
When $A$ is bounded above, they live in the homotopy groups of the summands

$\pazocal{L}[n](A):=(\partial_{p^{n}}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits^{h\Sigma_{p^{n}}}}(A)^{\mathbin{\mathop{\otimes}\displaylimits}p^{n}}\xhookrightarrow{\iota_{p^{n}}}\pazocal{L}(A)$

by Proposition 2.2. The composition $\beta\circ\alpha$ of two unary operations

$\alpha\in\pazocal{P}^{j}_{k}[w]\subseteq\pi_{j}(\pazocal{L}[w](\Sigma^{k}H\mathbb{F}_{p})),\,\,\,\,\beta\in\pazocal{P}^{i}_{j}[v]\subseteq\pi_{i}(\pazocal{L}[v](\Sigma^{j}H\mathbb{F}_{p})),$

considered as maps
$\alpha:\Sigma^{j}H\mathbb{F}_{p}\rightarrow\pazocal{L}[w](\Sigma^{k}H\mathbb{F}_{p})$
and
$\beta:\Sigma^{i}H\mathbb{F}_{p}\rightarrow\pazocal{L}[v](\Sigma^{j}H\mathbb{F}_{p})$,
is given by

(4)
$\Sigma^{i}H\mathbb{F}_{p}\xrightarrow{\beta}\pazocal{L}[v](\Sigma^{j}H\mathbb{F}_{p})\xrightarrow{\pazocal{L}[v](\alpha)}\pazocal{L}[v]\circ\pazocal{L}[w](\Sigma^{k}H\mathbb{F}_{p})\rightarrow\pazocal{L}[v+w](\Sigma^{k}H\mathbb{F}_{p}).$

The last map is induced by the weight $p^{v+w}$ summand

$(\partial_{p^{v}}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits^{h\Sigma_{p^{v}}}}\big{(}(\partial_{p^{w}}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits^{h\Sigma_{p^{w}}}}A^{\mathbin{\mathop{\otimes}\displaylimits}p^{w}})\big{)}^{\mathbin{\mathop{\otimes}\displaylimits}p^{v}}\rightarrow(\partial_{p^{(v+w)}}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits^{h\Sigma_{p^{v+w}}}}A^{\mathbin{\mathop{\otimes}\displaylimits}p^{v+w}}$

of the structure map of the monad
$\pazocal{L}\circ\pazocal{L}\rightarrow\pazocal{L}$ on any bounded above
object $A$.

Let $a\in\pazocal{R}^{!}((0,k),(-w,j+w)),b\in\pazocal{R}^{!}((0,j),(-v,i+v))$
be the unique preimages under the isomorphisms in Proposition 4.8 and 4.13 of
$\alpha$ and $\beta$ on the $E^{2}$-pages of the dual bar spectral sequences
converging respectively to $\pi_{*}(\pazocal{L}(\Sigma^{k}H\mathbb{F}_{p}))$
and $\pi_{*}(\pazocal{L}(\Sigma^{j}H\mathbb{F}_{p}))$.

Since $\pi_{*}(\pazocal{L}(\Sigma^{k}H\mathbb{F}_{p}))$ is bounded above and
of finite type, we can run the dual bar spectral sequence for the
$H\mathbb{F}_{p}$-module $A=\pazocal{L}(\Sigma^{k}H\mathbb{F}_{p})$ converging
to $\pi_{*}(\pazocal{L}\circ\pazocal{L}(\Sigma^{k}H\mathbb{F}_{p}))$. The
spectral sequence collapses on the $E^{2}$-page

$E^{2}\cong\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{R}^{!}}}\big{(}\pi_{*}(\pazocal{L}(\Sigma^{k}\mathbb{F}_{p}))\big{)}\cong\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{R}^{!}}}\circ\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{R}^{!}}}(\Sigma^{k}\mathbb{F}_{p}),\,\,p=2,$
$E^{2}\cong\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}\mathrm{Free}^{\mathrm{sLie}_{\mathbb{F}_{p}}}\big{(}\pi_{*}(\pazocal{L}(\Sigma^{k}\mathbb{F}_{p}))\big{)}\cong\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}\mathrm{Free}^{\mathrm{sLie}_{\mathbb{F}_{p}}}\circ\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}\mathrm{Free}^{\mathrm{sLie}_{\mathbb{F}_{p}}}(\Sigma^{k}\mathbb{F}_{p}),\,\,p>2$

by comparing with Theorem 2.6 in the limiting case.

The map
$\pazocal{L}\circ\pazocal{L}(\Sigma^{k}H\mathbb{F}_{p})\rightarrow\pazocal{L}(\Sigma^{k}H\mathbb{F}_{p})$
coming from the monad composition induces a map of the $E^{2}$-pages of the
dual bar spectral sequences

$\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{R}^{!}}}\circ\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{R}^{!}}}(\Sigma^{k}\mathbb{F}_{p})\rightarrow\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\pazocal{R}^{!}}}(\Sigma^{k}\mathbb{F}_{p}),\,\,p=2,$
$\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}\mathrm{Free}^{\mathrm{sLie}_{\mathbb{F}_{p}}}\circ\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}\mathrm{Free}^{\mathrm{sLie}_{\mathbb{F}_{p}}}(\Sigma^{k}\mathbb{F}_{p})\rightarrow\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}\mathrm{Free}^{\mathrm{sLie}_{\mathbb{F}_{p}}}(\Sigma^{k}\mathbb{F}_{p}),\,\,p>2.$

We need to understand the restriction of the above maps to the additive part,
i.e., the horizontal maps of the diagram

${\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}\circ\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}(\Sigma^{k}\mathbb{F}_{p})}$${\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}(\Sigma^{k}\mathbb{F}_{p})}$${\pazocal\pi_{*}(\mathrm{Bar}_{\bullet}(\mathrm{id},\pazocal{A}_{\pazocal{R}^{\prime}},\pi_{*}(\mathrm{Bar}_{\bullet}(\mathrm{id},\pazocal{A}_{\pazocal{R}^{\prime}},\Sigma^{-k}\mathbb{F}_{p})^{\vee})^{\vee})}$${\pi_{*}(\mathrm{Bar}_{\bullet}(\mathrm{id},\pazocal{A}_{\pazocal{R}^{\prime}},\Sigma^{-k}\mathbb{F}_{p})^{\vee})}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

.

Recall that
$\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}(\Sigma^{k}\mathbb{F}_{p})$ is by
construction isomorphic to the unstable Ext group

$\mathrm{UnExt}^{*,*}_{\pazocal{R}^{\prime}}(\mathbb{F}_{p},\Sigma^{k}\mathbb{F}_{p})\cong\pi_{*}(\mathrm{Bar}_{\bullet}(\mathrm{id},\pazocal{A}_{\pazocal{R}^{\prime}},\Sigma^{-k}\mathbb{F}_{p})^{\vee}).$

We will make use of a general result of Brantner that follows from [Br17,
Theorem 3.5.1 and 4.3.2]: Suppose that $\mathbf{T}$ is an additive monad on
$\mathrm{Mod}_{\mathbb{F}_{p}}$ associated to the free (unstable) module
functor over an algebra $R$. Then composition map of the monad
$|\mathrm{Bar}_{\bullet}(\mathrm{id},\mathbf{T},-)|^{\vee}$ is compatible with
the Yoneda product on the (unstable) Ext groups over $R$ up to a shearing.

Here $\pazocal{A}_{\pazocal{R}^{\prime}}$ is an additive monad associated with
the free functor that takes the unstable module over the Koszul algebra
$\pazocal{R}$. It follows that the top map is a sheared Yoneda product on
(unstable) Ext groups. More precisely, given
$b\in\pazocal{R}^{\prime}((0,j),(-v,i+v))\cong\mathrm{UnExt}^{-v,i+v}_{\pazocal{R}^{\prime}}(\mathbb{F}_{p},\Sigma^{j}\mathbb{F}_{p})$
and
$a\in\pazocal{R}^{\prime}((0,k),(-w,j+w))\cong\mathrm{UnExt}^{-w,j+w}_{\pazocal{R}^{\prime}}(\mathbb{F}_{p},\Sigma^{k}\mathbb{F}_{p}),$
the top map produces an element

$b\circ a|x_{k}\mapsto
b|a|x_{k}\in(\pazocal{R}^{\prime})^{!}((0,k),(-v-w,i+v+w))\cong\mathrm{UnExt}^{-v-w,i+v+w}_{\pazocal{R}^{\prime}}(\mathbb{F}_{p},\Sigma^{k}\mathbb{F}_{p})$

via the composite

${(\pazocal{R}^{\prime})^{!}((0,j),(-v,i+v))\times(\pazocal{R}^{\prime})^{!}((0,k),(-w,j+w))}$${(\pazocal{R}^{\prime})^{!}((0,j+w),(-v,i+v+w))\times(\pazocal{R}^{\prime})^{!}((0,k),(-w,j+w))}$${(\pazocal{R}^{\prime})^{!}((-w,j+w),(-v-w,i+v+w))\times(\pazocal{R}^{\prime})^{!}((0,k),(-w,j+w))}$${(\pazocal{R}^{\prime})^{!}((0,k),(-v-w,i+v+w)).}$$\scriptstyle{\mathrm{susp}^{w}\times\mathrm{id}}$$\scriptstyle{\cong}$

The first map is an injection on the left factor because operations are stable
under suspension and here $w\geq 0$, cf. Remark 4.2 and 4.11. The last map is
the composition in the ringoid $(\pazocal{R}^{\prime})^{!}$, i.e.
juxtaposition corresponding to the Yoneda product on unstable Ext groups. This
is exactly the composition product in $\pazocal{P}$.

Therefore the map (4) lifts to a map along the $E^{2}$-pages of the respective
dual bar spectral sequences, given explicitly by $b|x_{j}\mapsto
b\circ(a|x_{k})\mapsto b|a|x_{k}.$ Here we use $|$ to denote juxtaposition in
$(\pazocal{R}^{\prime})^{!}$ and $x_{k}$ the generator for
$\Sigma^{k}\mathbb{F}_{p}$. Passing to the $E^{\infty}$-pages, we deduce that
there is a commutative diagram

${\pazocal{P}^{i}_{j}[v]\times\pazocal{P}^{j}_{k}[w]}$${\pazocal{P}^{i}_{k}[v+w]}$${(\pazocal{R}^{\prime})^{!}((0,j),(-v,i+v))\times(\pazocal{R}^{\prime})^{!}((0,k),(-w,j+w))}$${(\pazocal{R}^{\prime})^{!}((0,k),(-v-w,i+v+w))}$${(\pazocal{R}^{\prime})^{!}((-w,j+w),(-v-w,i+v+w))}$${\times(\pazocal{R}^{\prime})^{!}((0,k),(-w,j+w))}$${(\pazocal{R}^{\prime})^{!}((0,k),(-v-w,i+v+w))}$${\pi_{i}(\pazocal{L}(\Sigma^{j}H\mathbb{F}_{p}))\times\pi_{j}(\pazocal{L}(\Sigma^{k}H\mathbb{F}_{p}))}$${\pi_{t+l}(\pazocal{L}(\Sigma^{k}H\mathbb{F}_{p}))}$$\scriptstyle{*}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cup}$$\scriptstyle{\cong}$$\scriptstyle{|}$$\scriptstyle{\circ}$

as desired. The first two horizontal maps, i.e. the composition product in the
power ring $\pazocal{P}$, are given by the sheared Yoneda product, and the
bottom horizontal map is given explicitly by the map (4).

In particular, the composition product
$\pazocal{P}^{j-a-b}_{j-a}[1]\times\pazocal{P}^{j-a}_{j}[1]\rightarrow\pazocal{P}^{j-a-b}_{j}[2]$
sends $(R^{b},R^{a})$ to $R^{(b,a)}=(Q^{b-1})^{*}(Q^{a-1})^{*}$ for $p=2$, and
$(\beta^{\epsilon_{1}}R^{b^{\prime}-\epsilon_{1}},\beta^{\epsilon_{2}}R^{a^{\prime}-\epsilon_{2}})$
to

$R^{(b^{\prime}-\epsilon_{1},a^{\prime}-\epsilon_{2},\epsilon_{1},\epsilon_{2})}=(\beta^{1-\epsilon_{1}}Q^{b^{\prime}-1})^{*}(\beta^{1-\epsilon_{2}}Q^{a^{\prime}-1})^{*}$

for $p>2$ where $2(p-1)a^{\prime}=a,2(p-1)b^{\prime}=b$. The two classes in
the composition are defined whenever $a>-j+1$ and $b>a-j$ for $p=2$, and
$2a^{\prime}>-j$ and $2b^{\prime}>a+\epsilon_{2}-j$ for $p>2$. Hence all the
Adem relations hold.

When $p=2$, a basis for additive unary operations on a degree $j$ class is
given by all monomials
$(Q^{i_{1}})^{*}(Q^{i_{2}})^{*}\cdots(Q^{i_{s}})^{*}\in(\pazocal{R}^{\prime})^{!}((0,j),(s,m))$
such that $i_{s}\geq-j+1$ and $i_{l}>2i_{l+1}$ for all $1\leq l<s$, cf.
Proposition 4.8. Any such monomial is the image of the (well-defined) iterated
composition $R^{i_{1}+1}R^{i_{2}+1}\cdots R^{i_{s}+1}$ in
$\pazocal{P}^{j-m-s}_{j}[s]$. Hence every additive unary operation
$R^{(i_{1},\ldots,i_{s})}$ can be written as a linear combination of
compositions $R^{j_{1}}\ldots R^{j_{s}}$ of operations in
$\pazocal{P}^{*}_{*}[1]$. The case $p>2$ is analogous. ∎

######  Corollary 5.7.

When $j$ gets arbitrarily large, we deduce that the algebra of unary
operations on a degree $j$ homotopy class of a spectral partition Lie algebra,
or equivalently a degree $-j$ class in mod $p$ TAQ cohomology, is the Koszul
dual algebra of the mod $p$ Dyer-Lashof algebra.

This is because when the degree $j$ of a class $x$ gets arbitrarily large, all
unary operations and relations are defined on $x$, while the shifted Lie
brackets and restrictions vanish for degree reasons. For $p>2$ one needs be
careful about the precise duality. The Dyer-Lashof operation $Q^{i}$ is sent
to $\beta R^{i}=\beta P^{i}$ in cohomological degree $2(p-1)i+1$ and $\beta
Q^{i}$ to $R^{i}=P^{i}$ in cohomological degree $2(p-1)i$ since the Bockstein
homomorphism increases cohomological degree by one.

###  5.2. Shifted restricted Lie algebra structure

Next we examine the shifted restricted Lie algebra structure on the homotopy
groups of spectral partition Lie algebras and the reduced mod $p$ TAQ
cohomology. Some of the methods in this section are inspired by the thesis
works of Antolín-Camarena and Brantner regarding the shifted Lie algebra
structure on the mod 2 homology and the Lubin Tate theory of spectral Lie
algebras in [AC20, Br17].

We showed in Proposition 4.9 and 4.15 that in the case of a free
$\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}$-algebra on a bounded
$\mathbb{F}_{p}$-module $A$ as a trivial $\mathbb{P}$-algebra, there is a
shifted (restricted) Lie bracket on the André-Quillen cohomology of the
trivial $\mathrm{Poly}_{\pazocal{R}}$-algebra $\pi_{*}(A)$, which is the
$E^{2}$-page of the dual bar spectral sequence that converges to
$\overline{\mathrm{TAQ}}^{-*}(A)\cong\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(A^{\vee}))$.
Now we show that there is a shifted restricted Lie algebra structure on the
homotopy groups of any spectral partition Lie algebra, or the TAQ cohomology
of any $\mathbb{E}_{\infty}$-$H\mathbb{F}_{p}$-algebra $A$. The shifted Lie
algebra structure exists at the level of $H\mathbb{F}_{p}$-modules and agrees
with the shifted Lie algebra structure on the $E^{2}$-page.

Recall that the second Goodwillie derivative of the identity functor
$\partial_{2}(\mathrm{Id})\simeq\mathbb{S}^{-1}$ is a naïve
$\Sigma_{2}$-spectrum with trivial $\Sigma_{2}$-action. By Proposition 2.2,
the weight 2 part of the free spectral partition Lie algebra on a bounded
$H\mathbb{F}_{p}$-module $A$ is

$\big{(}(\partial_{2}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits^{h\Sigma_{2}}}A^{\mathbin{\mathop{\otimes}\displaylimits}2}\big{)}\xhookrightarrow{\iota_{2}}\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(A)\simeq\bigoplus_{n}(\partial_{n}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits^{h\Sigma_{n}}}(A)^{\mathbin{\mathop{\otimes}\displaylimits}n}.$

Here $\iota_{n}$ denotes the inclusion of the weight $n$ homogeneous piece.

The binary operation $[-,-]$ representing the shifted Lie bracket is encoded
by the weight two part of the structure map

$\xi_{2}:\mathbb{S}^{-1}\mathbin{\mathop{\otimes}\displaylimits}(A^{\mathbin{\mathop{\otimes}\displaylimits}2})^{h\Sigma_{2}}\simeq(\partial_{2}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}H\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits^{h\Sigma_{2}}}A^{\mathbin{\mathop{\otimes}\displaylimits}2}\xrightarrow{\iota_{2}}\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(A)\rightarrow
A,$

explicitly given as follows.

######  Construction 5.8.

For $x:\Sigma^{j}H\mathbb{F}_{p}\rightarrow
A,y:\Sigma^{k}H\mathbb{F}_{p}\rightarrow A$ representing two homotopy classes
of a spectral partition Lie algebra $A$, we have a map of
$\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}$-algebras

$\theta:\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(\Sigma^{j}H\mathbb{F}_{p}\oplus\Sigma^{k}H\mathbb{F}_{p})\rightarrow\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(A)\rightarrow
A,$

where the second map is the
$\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}$-algebra structural
map of $A$. Write $X=\Sigma^{j}H\mathbb{F}_{p}$ and
$Y=\Sigma^{k}H\mathbb{F}_{p}$. There is a binary operation $[-,-]$ on
$\pi_{*}(A)$ represented by the map

(5)
$\mathbb{S}^{-1}\mathbin{\mathop{\otimes}\displaylimits}X\mathbin{\mathop{\otimes}\displaylimits}Y\simeq\mathbb{S}^{-1}\mathbin{\mathop{\otimes}\displaylimits}(X\mathbin{\mathop{\otimes}\displaylimits}Y\oplus
Y\mathbin{\mathop{\otimes}\displaylimits}X)^{h\Sigma_{2}}\hookrightarrow\mathbb{S}^{-1}\mathbin{\mathop{\otimes}\displaylimits}\big{(}(X\oplus
Y)^{\mathbin{\mathop{\otimes}\displaylimits}2}\big{)}^{h\Sigma_{2}}\simeq\partial_{2}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits^{h\Sigma_{2}}}(X\oplus
Y)^{\mathbin{\mathop{\otimes}\displaylimits}2}$

followed by the composite

$\theta\circ\iota_{2}:\partial_{2}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits^{h\Sigma_{2}}}(X\oplus
Y)^{\mathbin{\mathop{\otimes}\displaylimits}2}\hookrightarrow\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(\Sigma^{j}H\mathbb{F}_{p}\oplus\Sigma^{k}H\mathbb{F}_{p})\rightarrow
A,$

which sends the pair of homotopy classes $x$ and $y$ to a class
$[x,y]\in\pi_{*}(A)$.

At the level of the dual bar spectral sequence, the binary operation
$[x,y]\in\pi_{*}(A)$ is represented uniquely up to a nonzero scalar by the
weight 2 part of the composite

$\displaystyle\Sigma^{-1}(\Sigma^{j}\mathbb{F}_{2}\oplus\Sigma^{k}\mathbb{F}_{2})\rightarrow\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\mathbb{F}_{2}}}(\Sigma^{j}\mathbb{F}_{2}\oplus\Sigma^{k}\mathbb{F}_{2})$
$\displaystyle\hookrightarrow\mathrm{Free}^{\mathrm{sLie}^{\rho}_{\mathbb{F}_{2}}}(\Sigma^{j}\mathbb{F}_{p}\oplus\Sigma^{k}\mathbb{F}_{p})$
$\displaystyle\cong\pi_{*}(\mathrm{Lie}^{\pi}_{\mathbb{F}_{p},\mathbb{E}_{\infty}}(\Sigma^{j}H\mathbb{F}_{2}\oplus\Sigma^{k}H\mathbb{F}_{2}))\xrightarrow{\theta_{*}}\pi_{*}(A),$
$\displaystyle\Sigma^{-1}(\Sigma^{j}\mathbb{F}_{p}\oplus\Sigma^{k}\mathbb{F}_{p})\rightarrow\mathrm{Free}^{\mathrm{sLie}_{\mathbb{F}_{p}}}(\Sigma^{j}\mathbb{F}_{p}\oplus\Sigma^{k}\mathbb{F}_{p})$
$\displaystyle\hookrightarrow\mathrm{Free}^{(\pazocal{R}^{\prime})^{!}}\mathrm{Free}^{\mathrm{sLie}_{\mathbb{F}_{p}}}(\Sigma^{j}\mathbb{F}_{p}\oplus\Sigma^{k}\mathbb{F}_{p})$