\jyear

2021

\equalcont

These authors contributed equally to this work. [2]\fnmRuJia \surYan \equalcontThese authors contributed equally to this work.

1]\orgdivHUA Loo-Keng Key Laboratory of Mathematics, Academy of Mathematics and Systems Science, \orgnameChinese Academy of Sciences, \orgaddress\streetNo.55, Zhongguancun East Road, \cityBeijing, \postcode100190, \countryChina [2]\orgdivAcademy of Mathematics and Systems Science, \orgnameChinese Academy of Sciences, \orgaddress\streetNo.55, Zhongguancun East Road, \cityBeijing, \postcode100190, \countryChina

The Stable Picard Group of $A(n)$

\fnmJianZhong \surPan [email protected] [email protected] [ *

Abstract

In this paper, we showed that the Stable Picard group of $A(n)$ for $n\geq 2$ is $\mathbb{Z}\oplus\mathbb{Z}$ by considering the endotrivial modules over $A(n)$ . The proof relies on reductions from a Hopf algebra to its proper Hopf subalgebras.

keywords:

Picard group, Hopf algebra, Steenrod algebra, Endotrivial module

1 Introduction

Let $H$ be a cocommutative connected finite graded Hopf algebra over a base field $k$ with characteristic $p$ . The Picard group of $Stab(H)$ , denoted by $Pic(H)$ is the group of graded stably $\otimes$ -invertible $H$ -modules,

Pic(A):=\{M\in Stab(H):\exists N\text{ such that }M\otimes N=\underline{k}\},

where $\underline{k}$ is the unit of the symmetric monoidal category $(Stab(A),\otimes,\underline{k})$ . In this paper, we are interested in the determination of the Picard group of $A(n)$ , the Hopf subalgebra of $\mathcal{A}_{2}$ generated by $Sq^{1},Sq^{2},\cdots,Sq^{2^{n}}$ (see section 2.3). The situation when $n=1$ was calculated by Adams and Priddy in Adams and Priddy (1976) to show the uniqueness of $BSO$ in 1976. It says that

Pic(A(1))=\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}/2.

The $\mathbb{C}$ -motivic version of the above result was given by Gheorghe, Isaksen and Ricka in Gheorghe et al. (2018), which is

Pic(A_{\mathbb{C}}(1))=\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}.

In 2017, Bhattacharya and Ricka determined in Bhattacharya and Ricka (2017) that

Pic(A(2))=\mathbb{Z}\oplus\mathbb{Z}.

The authors of Bhattacharya and Ricka (2017) conjectured that

Pic(A(n))=\mathbb{Z}\oplus\mathbb{Z}

for all $n>2$ , which is exactly the main result of this paper (section 4).

The ideal is to consider $\mathbb{T}(H)$ , the group of endotrivial modules over $H$ (See section 2.2). An $H$ -module is called endotrivial if $End_{k}(M)=k\oplus(free)$ . By definition if $M$ is endotrivial, then $M$ is invertible since $End_{k}(M)=M^{*}\otimes M$ . On the other hand, since $H$ is cocommutative, the stable module category $Stab(H)$ is a stable homotopy category (section 9.6 of Hovey et al. (1997)), and hence a closed symmetric monoidal category. Therefore Theorem A.2.8 of Margolis (1983) guarantees that if $M$ is invertible, then it is endotrivial. That is, the group of endotrivial modules of $H$ is isomorphic to the stable Picard group of $H$ .

The word $endotrivial$ comes from the representation theory of finite groups, since the group algebra $kG$ of a finite $p$ -group $G$ is a well-defined Hopf algebra with diagonal coproduct. The group of endotrivial modules over $kG$ has been completely figured out by joint efforts of many mathematicians including Puig, Alperin, Dade, Carlson and Thevenaz, check Carlson and Thévenaz (2000) for references.

The main object of this paper is to give a proof of the following main result:

Theorem 1.1 (Corollary 4.8).

Given $n\geq 2$ , the morphism of groups

f:\mathbb{Z}\oplus\mathbb{Z}\to Pic(A(n))

which sends $(m,l)$ to $[\sigma(m)\Omega_{A(n)}^{l}(\mathbb{Z}/2)]$ is an isomorphism of groups.

This article is organized as follows. Section 2 introduces some preliminaries and conventions. In section 3, we showed that the problem of classifying endotrivial $H$ -modules can somehow be reduced to the one about each $E$ -modules for certain Hopf subalgebras $E$ (See Theorem 3.14). With help of this reduction theorem, the Picard group of $A(n)$ was given in section 4.

Throughout this paper, $k$ will denote a field with positive character $p$ . Every algebraic structure is implicitly over the base field $k$ , and tensor products are taken over $k$ . The Hopf algebras under consideration in this paper are cocommutative, unless explicitly specified otherwise.

2 Preliminaries

A Hopf algebra is a bialgebra over field $k$ (of characteristic $p>0$ ) with a compatible antipode map $S$ (Montgomery (1993)). We will adapt the following notation: If $H$ is a Hopf algebra, denote $\triangledown:H\otimes H\to H$ as its product; $\triangle:H\to H\otimes H$ as its coproduct; $\epsilon:H\to k$ and $\eta:k\to H$ are counit and unit respectively.

Suppose $M,N$ are $H$ -modules, then $M\otimes N$ is an $H$ -module defined by $h(m\otimes n)=\sum_{i}a_{i}m\otimes b_{i}n$ , where $h\in H$ , $m\in M$ and $n\in N$ with $\triangle(h)=\sum_{i}a_{i}\otimes b_{i}$ . For an $H$ -module $M$ , the dual $M^{*}$ is defined by $M^{*}=Hom_{k}(H,k)$ , with the natural $H$ -module structure

(h\cdot f)(v)=f(S(h)v)\quad\forall h,v\in H\mathchar 24635\relax\;\ f\in M^{*}

2.1 Finite connected graded Hopf algebra

If the underlying algebra of $H$ is decomposed into a direct sum of vector spaces:

H=\bigoplus_{n=0}^{\infty}H_{n}=H_{0}\oplus H_{1}\oplus H_{2}\oplus\cdots

such that $H_{i}H_{j}\subset H_{i+j}$ for all nonnegative $i$ and $j$ , we say that $H$ is a graded algebra.

Moreover, a graded $H$ -module $M$ is an $H$ -module which has a decomposition as direct sum of vector spaces:

M=\bigoplus_{-\infty}^{\infty}M_{n}

such that $H_{i}\cdot M_{j}\subset M_{i+j}$ . When $M$ , $N$ are graded $H$ -module, both $M\otimes N$ and $M^{*}$ defined above are still graded modules. In the rest of the paper, all algebras and modules are assumed to be graded.

An algebra (or module) is finite if it is a finite dimensional vector space over the base field.

Let $grH$ be the category with objects finitely generated graded $H$ -modules, and morphisms graded $H$ -homomorphisms. The $i$ th shift functor, $\sigma(i):grH\to grH$ , is defined by $\sigma(i)X=\sigma(i)(\oplus X_{n})=Y=\oplus Y_{n}$ , where $Y_{n}=X_{n-i}$ , and the obvious way on morphisms.

Suppose $H$ is a finite graded Hopf algebra over a field $k$ , then there is a minimal $n\in\mathbb{Z}^{+}$ such that $H_{j}=0$ if $j>n$ , and we say that the top degree of $H$ is $n$ and denote as $|H|=n$ .

The trivial $H$ -module $k$ can be viewed as a graded $H$ -module $\underline{k}$ where $\underline{k}_{0}=k,\underline{k}_{j}=0$ for $j\neq 0$ . We still use $k$ to denote $\underline{k}$ if there is no confusion.

Recall that a graded algebra $\Lambda$ is connected if $\Lambda_{0}=k$ , and we have

Definition 2.1.

A finite graded connected algebra $\Lambda$ is a Poincare algebra if there is a map of graded $k$ -modules $e:\Lambda\to\sigma(n)k$ for some $n$ such that the pairing $\Lambda_{q}\otimes\Lambda_{n-q}\xrightarrow{\triangledown}\Lambda_{n}\xrightarrow{e}k$ is non-singular.

In particular, this implies that if $\Lambda$ is a Poincare algebra, then $\Lambda_{i}=0$ for $i>n$ and $\Lambda_{n}\cong k$ .

Theorem 2.2 (Theorem 12.2.5 & 12.2.9 of Margolis (1983)).

If $H$ is a finite graded connected Hopf algebra, then $H$ is a Poincare algebra.

So if $H$ is a finite graded connected Hopf algebra and $|H|=n$ , $H$ is a Poincare algebra and thus $H\cong\sigma(n)H^{*}$ . Namely, $H_{0}\cong H_{n}\cong k$ .

Theorem 2.3 (Proposition 12.2.8 of Margolis (1983)).

Let $\Lambda$ be a Poincare algebra and let $M$ be a $\Lambda$ -module. Then the followings are equivalent:

(1) $M$ is free

(2) $M$ is projective

(3) $M$ is flat

(4) $M$ is injective.

Another fact worth to be known is that a Hopf algebra of our interest is always free over its Hopf subalgebras:

Theorem 2.4 (Theorem 1.3 of Aguiar and Lauve (2013)).

Let $H$ be a finite graded connected Hopf algebra. If $Z\subset H$ is a Hopf subalgebra, then $H$ is a free left (and right) $Z$ -module. Moreover,

H\cong Z\otimes(H/Z^{+}H)

as left $Z$ -modules and as graded vector spaces.

Remark 2.5.

The proof of Aguiar and Lauve (2013) applies for arbitrary characteristic.

In the rest of this article, all Hopf algebras are assumed to be finite graded connected Hopf algebras.

2.2 Endotrivial modules

An $H$ -module $M$ is called endotrivial if there is an isomorphism of $H$ -modules $End_{k}(M)\cong\underline{k}\oplus F$ , where $F$ is a free $H$ -module. As an $H$ -module, $End_{k}(M)\cong M\otimes M^{*}$ , thus $M$ is endotrivial iff $M\otimes M^{*}\cong k\oplus F$ for some free module $F$ . Note that $M\otimes N$ and $M^{*}$ are endotrivial if $M$ and $N$ are endotrivial.

Two endotrivial $H$ -modules $M$ and $N$ are said to be equivalent if

either\quad M\cong N\oplus(free)\quad or\quad N\cong M\oplus(free).

Denote the equivalent class of an endotrivial module $M$ as $[M]$ , and define

\mathbb{T}(H)=[M]|\text{ M is an endotrivial H-module}.

$\mathbb{T}(H)$ is an abelian group with unit $k$ , and $[M]+[N]=[M\otimes N]$ .

Lemma 2.6.

If an endotrivial $H$ -module $M$ decomposes as $M=M_{1}\oplus M_{2}$ , then one of the summands is free and the other one is endotrivial.

Proof: By Theorem 2.3, $H$ is indecomposable as an $H$ -module. By definition,

k\oplus F\cong End_{k}(M_{1})\oplus End_{k}(M_{2})\oplus Hom_{k}(M_{1},M_{2})\oplus Hom_{k}(M_{2},M_{1}).

Krull-Schmidt Theorem shows that all the summands on the right hand side are free except one. Denote it as $L$ , one has $dimL=1$ mod $dimH$ . On the other hand, $dim(Hom_{k}(M_{1},M_{2}))=dim(Hom_{k}(M_{2},M_{1}))=dimM_{1}dimM_{2}$ , thus $L\neq Hom_{k}(M_{2},M_{1})$ and $L\neq Hom_{k}(M_{1},M_{2})$ . Without loss of generality, we assume $L=End_{k}(M_{1})=k\oplus(free)$ and $End_{k}(M_{2})=(free)$ . Therefore $End_{k}(M_{2})\otimes M_{2}=M_{2}\otimes M_{2}^{*}\otimes M_{2}$ , is free and has a direct summand $M_{2}$ . By Theorem 2.3, $M_{2}$ is free.

Therefore, in the equivalence class $[M]$ , there is, up to isomorphism, a unique indecomposable module $M_{0}$ and every module in the class is isomorphic to $M_{0}\oplus(free)$ .

Definition 2.7 (Definition 2.15 of Gheorghe et al. (2018)).

Consider the projective resolution of $k$ :

ker(\epsilon)\to H\to k.

The operator $\Omega_{H}$ is given by

\Omega_{H}M=ker(\epsilon)\otimes M,

where $\epsilon:H\to k$ is the augmentation of $H$ . For $m\geq 0$ , define $\Omega^{m}_{H}M$ inductively to be $\Omega_{H}(\Omega_{H}^{m-1}M)$ . For $m<0$ , define $\Omega_{H}^{m}M$ to be $(\Omega_{H}^{-m}(M^{*}))^{*}$ .

As showed in section 2.3 of Gheorghe et al. (2018), $\Omega_{H}^{m}(k)$ is endotrivial and different choices of resolution will give the same equivalence class in $\mathbb{T}(H)$ , so the choice we made is harmless.

When $H=kG$ , the group algebra of a finite $p$ -group $G$ , we know well about endotrivial modules on $kG$ (Carlson and Thévenaz (2000)). With help of this example, we get:

Proposition 2.8 (Prop3.2 of Bhattacharya and Ricka (2017)).

Given an elementray Hopf algebra $B$ over $k$ generated by $\{x_{1},x_{2},\cdots,x_{t}\}$ where $t\geq 2$ . Then $f:\mathbb{Z}\oplus\mathbb{Z}\to\mathbb{T}(B)$ with $f(m,n)=[\sigma(m)\Omega_{B}^{n}(k)]$ is an isomorphism of groups.

Remark 2.9.

A Hopf algebra $H$ over $k$ is called elementary if it is bicommunicative and has $x^{p}=0$ for all $x\in ker\epsilon$ .

Remark 2.10.

The proof of Bhattacharya and Ricka (2017) still works for $p>2$ .

2.3 Mod 2 Steenrod algebra

We now turn to the algebra of central interest to us, the mod $2$ Steenrod algebra $\mathcal{A}_{2}$ and its subalgebra $A(n)$ . The background knowledge about $\mathcal{A}_{2}$ in this section mainly comes from 15.1 of Margolis (1983).

The mod $2$ Steenrod algebra $\mathcal{A}_{2}$ is a graded vector space over $\mathbb{Z}/2$ with basis all formal symbols $Sq(r_{1},r_{2},\cdots)$ , called Milnor basis, where $r_{i}\geq 0$ and $r_{i}>0$ only finitely often. If $r_{j}=0$ for $j>i$ we will also write $Sq(r_{1},\cdots,r_{i})$ and $|Sq(r_{1},r_{2},\cdots)|$ = $\sum(2^{i}-1)r_{i}$ . The product of $\mathcal{A}_{2}$ are defined as follows:

Sq(r_{1},r_{2},\cdots)\cdot Sq(s_{1},s_{2},\cdots)=\sum_{X}\beta(X)Sq(t_{1},t_{2},\cdots)

the summation being over all matrices

X=\left[\begin{matrix}*&x_{01}&x_{02}&\cdots\\ x_{10}&x_{11}&\cdots&\\ x_{20}&\vdots&&\\ \vdots&&&\end{matrix}\right]

with nonnegative $x_{ij}$ satisfying $\sum_{i}x_{ij}=s_{j}$ and $\sum_{j}2^{j}x_{ij}=r_{i}$ , and we will call such matrix allowable.

Then take $t_{k}=\sum_{i+j=k}x_{ij}$ and $\beta(X)=\prod_{k}(x_{k0},x_{(k-1)1},\cdots x_{0k})\in\mathbb{Z}/2$ , where $(n_{1},\cdots,n_{r})=(n_{1}+\cdots n_{r})!/n_{1}!\cdots n_{r}!$ reduced mod $2$ . In particular, there is always an allowable matrix with $x_{i0}=r_{i}$ and $x_{0j}=s_{j}$ and all the other entries are zero, we call such allowable matrix trivial.

The coproduct is defined as follow:

\triangle Sq(r_{1},r_{2},\cdots)=\sum_{r_{i}=s_{i}+t_{i}}Sq(s_{1},s_{2},\cdots)\otimes Sq(t_{1},t_{2},\cdots).

Here is a useful Lemma for the calculation of $(n_{1},\cdots,n_{r})$ .

Lemma 2.11.

Suppose the unique dyadic expansion of the natural number $n$ is $n=\sum_{j}a_{j}2^{j}$ , then we say $2^{i}\in n$ if $a_{i}=1$ ; and $2^{i}\notin n$ if $a_{i}=0$ .
Then $(n_{1},\cdots,n_{r})=1$ if and only if $2^{i}\in n_{j}$ implies $2^{i}\notin n_{k}$ for $k\neq j$ .

We denote the Milnor basis element $Sq(0,\cdots,0,r)$ with $r$ at the $t$ -th position as $P_{t}(r)$ , and denote $P^{s}_{t}=P_{t}(2^{s})$ . The homogeneous primitives of $\mathcal{A}_{2}$ are precisely the $P^{0}_{t}$ for all $t\geq 1$ .

Lemma 2.12.

Here are some direct consequences about the product in $\mathcal{A}_{2}$ :

(1) If $r<2^{v}$ and $u<2^{t}$ then $[P_{t}(r),P_{v}(u)]=0$ .

(2) If $r<2^{t}$ then $P_{t}(r)P_{t}(s)=(r,s)P_{t}(r+s)$ , in particular $P_{t}(r)^{2}=0$ .

(3) $P^{t}_{t}\cdot P^{t}_{t}=P_{t}(2^{t}-1)P^{0}_{2t}$ .

(4) If $1\leq r<2^{t}$ then $[P^{t}_{t},P_{t}(r)]=P_{t}(r-1)P^{0}_{2t}$

(5) If $1\leq r\leq 2^{i}\leq 2^{t}$ and $r<2^{t}$ , then $[P_{i}(r),P^{i}_{t}]=P_{i}(r-1)P_{t+i}^{0}$

(6) If $\leq 2^{i}\leq r\leq 2^{t}$ , then $[P_{i}^{0},P_{t}(r)]=P_{t}(r-2^{i})P_{t+i}^{0}$

Proof: (1)-(4) comes from Lemma 1.3 of Adams and Margolis (1992).

By definition, if $X=(x_{ij})$ is an allowable matrix for the product $Sq(r_{1},r_{2}\cdots)\cdot Sq(s_{1},s_{2}\cdots)$ , then the entries satisfies

	$\displaystyle\sum_{i}x_{ij}=s_{j}$
	$\displaystyle\sum_{j}2^{j}x_{ij}=r_{i}.$

As for (5) $P_{i}(r)P^{i}_{t}$ only has trivial allowable matrix, while beside the trivial matrix, $P^{i}_{t}P_{i}(r)$ has another allowable matrix $X=(x_{uv})$ where $x_{0i}=r-1$ , $x_{ti}=1$ and all the other entries are zero. Note that $\beta(X)=1$ , therefore $[P^{i}_{t},P_{i}(r)]=Sq(r_{1},r_{2}\cdots)$ with $r_{i}=r-1$ , $r_{t+i}=1$ and all the other coordinates are zero. On the other hand $P_{i}(r-1)P_{t+i}^{0}$ only has trivial allowable matrix, thus $P_{i}(r-1)P_{t+i}^{0}=Sq(r_{1},r_{2}\cdots)=[P^{i}_{t},P_{i}(r)]$ .

As for (6), $P^{0}_{i}P_{t}(r)$ only has trivial allowable matrix, while beside the trivial matrix, $P_{t}(r)P^{0}_{i}$ has another allowable matrix $X=(x_{uv})$ where $x_{t0}=r-2^{i}$ , $x_{ti}=1$ and all the other entries are zero. Note that $\beta(X)=1$ , therefore $[P^{0}_{i},P_{t}(r)]=Sq(r_{1},r_{2}\cdots)$ with $r_{t}=r-2^{i}$ , $r_{t+i}=1$ and all the other components are zero. Meanwhile, $P_{t}(r-2^{i})P_{t+i}^{0}=Sq(r_{1},r_{2}\cdots)$ with $r_{t}=r-2^{i}$ , $r_{t+i}=1$ and all the other components are zero, since the product only has trivial allowable matrix, we are done.

Lemma 2.13.

Suppose $r=\sum_{i=0}^{t}\alpha_{i}2^{i}$ is the dyadic decomposition of $r$ and $r<2^{t+1}$ , then $P_{t}(r)=\prod_{\alpha_{i}=1}P^{i}_{t}$ .

Proof: The lemma follows from (2) of Lemma 2.12 and Lemma 2.11.

We define the excess of a Milnor basis element by

ex(Sq(r_{1},r_{2},\cdots))=r_{1}+r_{2}+\cdots.

This relates well to the product:

Lemma 2.14.

[Lemma 15.1.2 of Margolis (1983)]

Sq(r_{1},r_{2},\cdots\cdot)Sq(s_{1},s_{2},\cdots)=bSq(r_{1}+s_{1},r_{2}+s_{2},\cdots)+\Sigma Sq(t_{1},t_{2},\cdots)

with $b=\prod(r_{i},s_{i})$ and $ex(Sq(t_{1},t_{2},\cdots))<ex(Sq(r_{1}+s_{1},r_{2}+s_{2},\cdots))$

Definition 2.15.

Let $B$ be a Hopf subalgebra of $\mathcal{A}_{2}$ and define the profile function of $B$ , $h_{B}:\{1,2,\cdots\}\to\{0,1,\cdots,\infty\}$ by

h_{B}(t)=\min\{s|r_{t}<2^{s}\ for\ all\ Sq(r_{1},r_{2},\cdots)\in B\},

and $h_{B}(t)=\infty$ if no such $s$ exists.

The importance of profile functions and the $P^{s}_{t}$ ’s can be seen in the following classification theorem:

Theorem 2.16 (Theorem 15.1.6 of Margolis (1983)).

Given a Hopf subalgebra $B$ of $\mathcal{A}_{2}$ , then

(a) $B$ is spanned by the Milnor basis elements in it; precisely, $B$ has a $\mathbb{Z}/2$ -basis $\{Sq(r_{1},r_{2},\cdots)|r_{t}<2^{h_{B}(t)}\}$ .

(b) $B$ is generated as an algebra by $\{P^{s}_{t}|s<h_{B}(t)\}$ .

(c) $h:\{1,2,\cdots\}\to\{0,1,\cdots,\infty\}$ is the profile function of a Hopf subalgebra if and only if for all $u,v\geq 1$ , $h(u)\leq v+h(u+v)$ or $h(v)\leq h(u+v)$ . Moreover, the algebra being normal if the latter condition is always satisfied.

Remark 2.17.

Suppose $h_{A}(t)$ and $h_{B}(t)$ are two profile functions. Then $A$ is a Hopf subalgebra of $B$ if and only if $h_{A}(t)\leq h_{B}(t)$ for all $t$ .

Remark 2.18.

Suppose $h_{A}(t)$ and $h_{B}(t)$ are two profile functions for graded Hopf subalgebras. Then $m(t)=min\{h_{A}(t),h_{B}(t)\}$ is the profile function of $A\cap B$ .

Definition 2.19.

Here are two examples:

(1) For each $t$ define a Hopf subalgebra $J(t)$ by the profile function $h(u)=0,u\neq t$ and $h(t)=t$ . Then $J(t)$ is an exterior algebra on generators $P^{0}_{t},\cdots,P^{t-1}_{t}$ .

(2) $A(n)$ is defined by the profile function $h(t)=max\{n+2-t,0\}$ . It has a minimal generating set $\{P^{0}_{1},P^{1}_{1},\cdots P^{n}_{1}\}.$

Remark 2.20.

$A(n)$ is the Hopf subalgebra generated by $Sq^{1},Sq^{2},\cdots,Sq^{2^{n}}$ .

Corollary 2.21.

Suppose $B$ is a finite Hopf subalgebra of $\mathcal{A}_{2}$ , then

\prod_{t=1}\left(\prod_{s=0}^{h_{B}(t)-1}P^{s}_{t}\right)\neq 0.

Proof: Iterated applications of Lemma 2.14 tell that

\prod_{t=1}\left(\prod_{s=0}^{h_{B}(t)-1}P^{s}_{t}\right)=Sq(r_{1},r_{2},\cdots)+(\text{terms with smaller excess})\neq 0

where $r_{i}=2^{h_{B}(i)}-1$ . Next, we introduce conditions for a function to be a profile function.

Lemma 2.22.

If $h_{2}$ is a profile function, $h_{1}:\{1,2,\cdots\}\to\{0,1,2\cdots\}$ is a function such that

(1) $h_{1}(t)\leq h_{2}(t)$ , for all $t\geq 1$ .

(2) $\exists t_{0}\geq 1$ such that $h_{1}(t)=h_{2}(t)$ for $t\geq t_{0}$ .

Then $h_{1}$ is a profile function if $h_{1}(u)\leq v+h_{1}(u+v)$ or $h_{1}(v)\leq h_{1}(u+v)$ , for all $u,v\geq 1$ and $u+v\leq t_{0}$ .

Proof: If $u+v>t_{0}$ and $h_{1}(v)>h_{1}(u+v)=h_{2}(u+v)$ then $h_{2}(v)\geq h_{1}(v)>h_{2}(u+v)$ . By assumption, $h_{2}$ is a profile function, thus $h_{2}(u)\leq v+h_{2}(u+v)$ . Therefore $h_{1}(u)\leq h_{2}(u)\leq v+h_{2}(u+v)=v+h_{1}(u+v)$ .

Lemma 2.23.

Suppose $h$ is a profile function, $h_{1}:\{1,2,\cdots\}\to\{0,1,2\cdots\}$ is a function such that

h_{1}(t)=\begin{cases}h(t)&t\neq t_{0}\\ 0&t=t_{0}\end{cases}

Then $h_{1}(t)$ is a profile function if for $1\leq u<t_{0}$ , $h_{1}(t_{0}-u)>0$ implies $h_{1}(u)\leq t_{0}-u$ .

Proof: Since $h_{1}(t_{0})=0$ , $h_{1}(t_{0})\leq h_{1}(t_{0}+u)\leq u+h_{1}(t_{0}+u)$ .

By assumption, for $1\leq u<t_{0}$ , $h_{1}(t_{0}-u)>0$ implies $h_{1}(u)\leq t_{0}-u$ . Therefore, $h_{1}(u)\leq t_{0}-u+h_{1}(t_{0})$ or $h_{1}(t_{0}-u)\leq h_{1}(t_{0})$ .

For the rest situation, namely $1\leq u,v$ such that $u\neq t_{0}$ , $v\neq t_{0}$ and $u+v\neq t_{0}$ , one has $h_{1}(u)=h(u)$ , $h_{1}(v)=h(v)$ and $h_{1}(u+v)=h(u+v)$ . Since $h$ is a profile function, $h_{1}(u)\leq v+h_{1}(u+v)$ or $h_{1}(v)\leq h_{1}(u+v)$ , for all such $u,v$ .

Suppose $A\subset\mathcal{A}_{2}$ is a Hopf subalgebra and $B\subset A$ is normal Hopf subalgebra of $A$ . In order to describe $A//B$ , it’s helpful to know more about $AB^{+}=B^{+}A$ , the ideal generated by $B$ .

Lemma 2.24.

Suppose $A\subset\mathcal{A}_{2}$ is a Hopf subalgebra and $B=C\cap A$ where $C$ is normal Hopf subalgebra of $\mathcal{A}_{2}$ . If $\forall s,t$ such that $0\leq s<h_{C}(t)$ , one has $2^{s}\notin r_{t}$ , then $Sq(r_{1},r_{2},\cdots)$ is not a summand of any $x\in AB^{+}=B^{+}A$ .

Proof: By the proof of Theorem 15.1.6 (c) of Margolis (1983), we have that

\mathcal{A}_{2}C^{+}=C^{+}\mathcal{A}_{2}=span\{Sq(u_{1},u_{2}\cdots)|\exists 0\leq s<h_{C}(t)\text{ such that }2^{s}\in u_{t}\}.

Thus by assumption, $Sq(r_{1},r_{2},\cdots)$ is not a summand of any $x\in\mathcal{A}_{2}C^{+}=C^{+}\mathcal{A}_{2}$ and thus $Sq(r_{1},r_{2},\cdots)$ is not a summand of any $x\in AB^{+}=B^{+}A$ .

We state the following two lemmas here for later argument.

Lemma 2.25.

Given a finite Hopf subalgebra $A$ of $\mathcal{A}_{2}$ . If $B$ is a maximal elementary Hopf subalgebra of $A$ then $B=B_{i}\cap A$ for some $i$ , where

h_{B_{i}}(t)=\begin{cases}0&t<i\\ i&t\geq i\end{cases}

Proof: By example A.6. of Palmieri (1997), $B_{i}$ are all the maximal elementary Hopf subalgebras of $\mathcal{A}_{2}$ . If $B\subset A\subset\mathcal{A}_{2}$ is an elementary subHopf algebra, then $B\subset B_{i}\cap A$ for some $i$ . On the other hand, $B_{i}\cap A$ themselves are elementary Hopf subalgebra, thus if $B$ is maximal, then $B=B_{i}\cap A$ for some $i$ .

Remark 2.26.

The maximal elementary Hopf subalgebras of $A(n)$ are exactly $\{B_{i}\cap A(n)\}$ where $1\leq i\leq[n/2]+1$ , since $B_{i}\cap A(n)\subset B_{[n/2]+1}\cap A(n)$ if $i\geq[n/2]+1$ and $B_{i}\cap A(n)\nsubseteq B_{j}\cap A(n)$ if $i,j\leq[n/2]+1$ and $i\neq j$ .

Let $E(n)\subset A(n)$ be the sub algebra generated by $\{P^{0}_{t}|t\leq n+1\}$ . Then $E(n)$ is an exterior algebra and is a normal Hopf subalgebra of $A(n)$ .

Lemma 2.27 (Proposition 15.3.29 of Margolis (1983)).

There is a map $\eta:A(n-1)\to A(n)//E(n)$ which doubles degree and which is an algebra isomorphism.

In general a map (resp. isomorphism) of algebras induces a functor (resp. isomorphism) of the module categories. Here $\eta$ introduces a small technicality.

Proposition 2.28 (Proposition 15.3.30 of Margolis (1983)).

$\eta$ induces an isomorphism

\eta^{*}:gr(A(n)//E(n))^{ev}\to grA(n-1)

Here $gr(A(n)//E(n))^{ev}$ denotes the full subcategory of $gr(A(n)//E(n))$ of modules concentrated in even degree.

3 Reduction to Hopf subalgebras

In this section we hope to reduce the calculation of $\mathbb{T}(H)$ to the calculation of $\mathbb{T}(E)$ , where $E$ is some proper Hopf subalgebra of $H$ (Theorem 3.14). Then in some sense we can do an induction (see next section). This section is devoted to prove Theorem 3.14.

Recall that by assumption $H$ is a Poincare algebra and thus $H\cong\sigma(n)H^{*}$ . Namely, $H_{0}\cong H_{n}\cong k$ . We denote the unique basis (up to a nonzero scale multiplication) of the $1$ -dimensional vector space $H_{n}$ as $t^{H}_{1}$ .

Definition 3.1.

Let $M$ be an $H$ -module and $Z$ be a normal Hopf subalgebra of $H$ . Define

M^{Z}:=\{x\in M|hx=\epsilon(h)x,\forall h\in Z\},\quad M^{Z}_{1}:=\{t^{Z}_{1}\cdot m|\forall m\in M\}.

$M^{Z}$ is called the (left) invariant $Z$ -submodule of $M$ .

Remark 3.2.

$M^{Z}$ is a well defined $H//Z$ -module where $\alpha\cdot m=h\cdot m,\quad\forall\alpha\in H//Z,m\in M^{Z},$ and $h$ is an arbitrary preimage of $\alpha$ of the quotient map $\pi:H\to H//Z$ .

It’s easy to verify that $Hom_{H}(k,M)\cong\{x\in M|hx=\epsilon(h)x,\forall h\in H\}$ as graded vector spaces. Notice that $H$ itself is an $H$ -module and we can also consider the invariant $H$ -submodule of $H$ .

Lemma 3.3.

Suppose $|H|=n$ , then $H^{H}=H_{n}$

Proof: Let $x=\sum_{i=0}^{n}x_{i}\in H^{H}$ , with $x_{i}\in H_{i}$ . And let $x_{j}$ be the first nonzero summand. Then since $H$ is a Poincare algebra, there is $y_{n-j}\in H_{n-j}$ such that $y_{n-j}x_{j}=t^{H}_{1}\neq 0$ . Therefore, $y_{n-j}x=t^{H}_{1}$ for grading reason. On the other hand, since $x\in H^{H}$ , $y_{n-j}x=\epsilon(y_{n-j})x$ , which leads to $\epsilon(y_{n-j})x=t^{H}_{1}$ . This is true only if $x=at^{H}_{1}$ and $y_{0}=1/a$ for some nonzero $a\in k$ . Thus $H^{H}\subset H_{n}$ . By definition, $\forall h\in H^{+}$ , $|ht^{H}_{1}|>n$ , hence $ht^{H}_{1}=0=\epsilon(h)t^{h}_{1}$ . Therefore, $H_{n}\subset H^{H}$ .

By the same argument, $(\sigma(m)H)^{H}=\sigma(m)H_{n}$ , the top degree of $\sigma(m)H$ . Another observation is that $(\sigma(m)H)^{H}_{1}=\sigma(m)H_{n}$ , since $t^{H}_{1}\cdot\sigma(m)H_{i}\subset\sigma(m)H_{n+i}=0$ for $i>0$ and $t^{H}_{1}\sigma(m)H_{0}=\sigma(m)H_{n}$ .

In general, $M^{H}_{1}$ and $M^{H}$ are not equal. However, this is true when $M$ is free.

Lemma 3.4.

When $M$ is a free $H$ module, $M^{H}=M^{H}_{1}$ .

Proof: Suppose $|H|=n$ . $\forall h\in H,m\in M$ , $h\cdot t^{H}_{1}m=\epsilon(h)t^{H}_{1}m$ by definition, thus $M^{H}_{1}\subset M^{H}$ .

On the other hand, suppose $\{m_{1},\cdots m_{s}\}$ is an $H$ -basis for $M$ as free module, then $\{t^{H}_{1}(m_{1}),\cdots,t^{H}_{1}(m_{s})\}$ is a basis for $M^{H}_{1}$ as vector space, since $t^{H}_{1}\sigma(m)H=\sigma(m)H_{n}$ $M^{H}\cong Hom_{H}(k,M)\cong Hom_{H}(k,\oplus_{i=1}^{s}\sigma(\alpha_{i})H)\cong\oplus_{i=1}^{s}(\sigma(\alpha_{i})H)^{H}\cong\oplus_{i=1}^{s}\sigma(\alpha_{i})H_{n}=\oplus_{i=1}^{s}\sigma(\alpha_{i+n})k.$ Thus $M^{H}=M^{H}_{1}$ since they have same dimension as vector spaces over $k$ .

The element $t^{H}_{1}$ can be used to detect the freeness of an $H$ -module.

Lemma 3.5 (Lemma 12.2.6 of Margolis (1983)).

If $M$ is an $H$ -module, then multiplication by $t^{H}_{1}$ induces a map $f:k\otimes_{H}M\to M$ (of degree $|t^{H}_{1}|$ ). Furthermore, $f$ is a monomorphism if and only if $M$ is free.

Lemma 3.6.

Let $M$ be an $H$ -module. Suppose $t^{H}_{1}m_{1},t^{H}_{1}m_{2},\cdots,t^{H}_{1}m_{n}$ are $k$ -linearly independent. Then $m_{1},m_{2},\cdots,m_{n}$ generate a free $H$ -submodule $L\subset M$ .

Proof: By assumption, $k\otimes_{H}L$ is a $k$ vector space spanned by $1\otimes m_{1},1\otimes m_{2},\cdots,1\otimes m_{n}$ . Consider the map $f:k\otimes_{H}L\to L$ defined in Lemma3.5:

f(\sum_{i=1}^{n}a_{i}\otimes m_{i})=\sum_{i=1}^{n}a_{i}t^{H}_{1}m_{i}

where $a_{i}\in k$ . Since $t^{H}_{1}m_{1},\cdots,t^{H}_{1}m_{n}$ are $k$ -linearly independent, $f$ is injective.

By Lemma 3.5, we are done.

In fact, $t^{H}_{1}$ is just the integral of the Hopf algebra $H$ (Montgomery (1993)), and it has some kind of transitivity:

Lemma 3.7.

Suppose $Z\subset H$ is a normal Hopf subalgebra. Denote the generator of $Z_{|Z|}$ and $(H//Z)_{|H//Z|}$ as $t^{Z}_{1}$ and $\overline{t^{H}_{Z}}$ respectively. Then $t^{H}_{Z}t^{Z}_{1}$ is the generator of $H_{|H|}$ where $t^{H}_{Z}$ is a preimage of $\overline{t^{H}_{Z}}$ under the quotient map.

Proof: Show that $t^{H}_{Z}t^{Z}_{1}$ is well-defined first:

\forall a\in HZ^{+},\ \ (t^{H}_{Z}+a)t^{Z}_{1}=t^{H}_{Z}t^{Z}_{1}+hzt^{Z}_{1}=t^{H}_{Z}t^{Z}_{1}

where $a=hz$ for some $h\in H$ and $z\in Z^{+}$ .

Recall that $H$ is a Poincare algebra. By definition, $\exists x\in H_{|H|-|Z|}$ such that $xt^{Z}_{1}\neq 0$ . By Theorem 2.4, $x\in H_{|H//Z|}$ . Since $H_{|H|}=k$ , is a one dimensional vector space generated by $t^{H}_{1}$ , we can assume that

xt^{Z}_{1}=t^{H}_{1}

after a scalar multiplication if necessary.

Now that $xt^{Z}_{1}\neq 0$ , one has $x\notin HZ^{+}$ and hence the quotient image of $x$ in $H//Z$ is nonzero. Therefore $x$ is a preimage of $\overline{t^{H}_{Z}}$ since $|x|=|H//Z|$ and $(H//Z)_{|H//Z|}=k$ , a one dimensional vector space.

Thanks to the transitivity, we have the following two technical lemmas:

Lemma 3.8.

Let $Z\subset H$ be a normal Hopf subalgebra. Suppose $M$ is a finitely generated $H$ -module such that $M^{Z}_{1}$ is a free $H//Z$ module, and $M\downarrow^{H}_{Z}\cong k\oplus(free)$ . Then $M\cong k\oplus(free)$ as $H$ -modules.

Proof: Let $t^{Z}_{1}m_{1},\cdots,t^{Z}_{1}m_{n}$ be a basis of $M^{Z}_{1}$ as free $H//Z$ -module. Then by Lemma 3.7 and Lemma 3.4, we have

t^{H}_{Z}t^{Z}_{1}m_{1}=t^{H}_{1}m_{1},\cdots,t^{H}_{Z}t^{Z}_{1}m_{n}=t^{H}_{1}m_{n}

are $k$ -linearly independent. Therefore $\{m_{i}\}$ generate a free $H$ -submodule of $M$ by Lemma 3.6, named as $L$ . Therefore, $L$ is also a free $Z$ -module, which implies $L^{Z}=L^{Z}_{1}=M^{Z}_{1}$ , generated by $t^{Z}_{1}m_{1},\cdots,t^{Z}_{1}m_{n}$ as free $H//Z$ -module. On the other hand, $M^{Z}_{1}=F^{Z}_{1}=F^{Z}$ , where by assumption $M\downarrow^{H}_{Z}\cong k\oplus F$ for some $F$ free as $Z$ -module. Therefore $L^{Z}=F^{Z}$ , hence $dim(L)=dim(F)$ since both $L$ and $F$ are free $Z$ -modules. Thus $dim(L)=dim(M)-1$ and since $L$ is free, hence injective, as an $H$ -module, we get $M\cong L\oplus k$ , as was to be shown.

Corollary 3.9.

Let $Z$ be a normal Hopf subalgebra. Then

H^{Z}\cong\sigma(|Z|)H//Z

as $H//Z$ -modules.

Proof: By Theorem 2.4, and Lemma 3.4, as graded $k$ -vector spaces,

H^{Z}=t^{Z}_{1}\cdot Z\otimes H//Z=k\{t^{Z}_{1}\}\otimes H//Z\cong\sigma(|Z|)H//Z.

On the other hand,

\overline{t^{H}_{Z}}H^{Z}=t^{H}_{Z}t^{H}_{1}H=t^{H}_{1}H=k\{t^{H}_{1}\}.

Then by Lemma 3.6, $H^{Z}$ has a free $H//Z$ -submodule. Therefore, $H^{Z}\cong\sigma(|Z|)H//Z$ as $H//Z$ - modules for dimensional reason.

Definition 3.10.

Suppose $A$ is an algebra and $M$ is an $A$ -module. Let $\mathcal{B}=\{B_{i}|i\in I\}$ be a collection of nontrivial subalgebras of $A$ . If $M$ is free iff $M$ restricted to every subalgebra $B\in\mathcal{B}$ is free, then we say $A$ has detect property in $grA$ , with detecting set $\mathcal{B}$ .

Here $B$ is nontrivial means $B\neq A$ and $B\neq k$ .

Theorem 3.11 (Theorem 1.3, Example A.6 of Palmieri (1997)).

Suppose $E$ is a finite Hopf subalgebra of the mod $2$ Steenrod algebra $\mathcal{A}_{2}$ , $M$ is an $E$ -module. Then $M$ is projective if and only if $M$ restricted to $B$ is projective for every elementary Hopf subalgebra $B$ of $E$ .

Remark 3.12.

Here $E$ and $B$ are all graded, and the grading was given by that of $\mathcal{A}_{2}$ .

Remark 3.13.

In particular, if $E$ is not elementary, $E$ has detect property in $grE$ with detecting set $\{B|B\text{ is a maximal elementary Hopf subalgebra }\}$ .

Now we are ready to prove our main theorem:

Theorem 3.14.

Suppose $H$ has a nontrivial normal Hopf subalgebra $Z$ such that $H//Z$ has detect property in $gr(H//Z)$ with detecting set $\mathcal{B}=\{B_{i}|i\in I\}$ . Then

Res:\mathbb{T}(H)\longrightarrow\prod_{E\in\mathcal{E}}\mathbb{T}(E).

is injective, where $\mathcal{E}=\{E_{i}|E_{i}//Z=B_{i},i\in I\}$ .

Proof: By assumption there is a nontrivial normal Hopf subalgebra $Z$ of $H$ such that $H//Z$ has detect property with detecting set $\mathcal{B}$ .

Let $M$ be an endotrivial $H$ -module whose class in $\mathbb{T}(H)$ is in the kernel of the restriction map $Res$ . This means $M\cong k\oplus F$ as $E$ -modules, for $E\in\mathcal{E}$ , where $F$ is a free $E$ -module. Since $Z$ is finite nontrivial connected Hopf algebra, $|t^{Z}_{1}|\neq 0$ and thus $t^{Z}_{1}\in Z^{+}$ . If follows that

M^{Z}=k\oplus F^{Z},\quad M^{Z}_{1}=F^{Z}_{1}.

Since $F$ is a free $E$ -module, $F$ is also a free $Z$ module, thus by Lemma 3.4, $F^{Z}_{1}=F^{Z}$ . Moreover, $F^{Z}$ is free over $E//Z$ since $F$ is free over $E$ , see Corollary 3.9.

Therefore, $M^{Z}_{1}$ is a free $E//Z$ module, for every $E\in\mathcal{E}$ . Using the fact that $H//Z$ has detect property with detecting set $\mathcal{B}=\{B_{i}|i\in I\}=\{E//Z|E\in\mathcal{E}\}$ , we have $M^{Z}_{1}$ is a free $H//Z$ -module. By Lemma 3.8, we are done.

4 Endotrivial group of A(n)

Consider the Hopf algebra $A(n),n\geq 2$ . From now on, let $a=[(n-1)/2]$ , $i$ be an integer with $1\leq i\leq a+1$ and we will work on the field $\mathbb{Z}/2$ .

By Lemma 2.27, there is an isomorphism of algebras $\eta:A(n-1)\to A(n)//E(n)$ which doubles the grading. Use the same symbol as in the proof of Lemma 2.25, the maximal elementary Hopf subalgebras of $A(n-1)$ are exactly $B_{1}\cap A(n-1),\cdots,B_{a+1}\cap A(n-1)$ , with which $A(n-1)$ has detect property. Define

h_{B_{i}^{\prime}(n)}(t)=\begin{cases}h_{B_{i}\cap A(n-1)}(t)+1&1\leq t\leq n+1\\ 0&t>n+1\end{cases}

By Lemma 2.22, $B_{i}^{\prime}(n)$ is a Hopf subalgebra, since $h_{B_{i}^{\prime}(n)}(t)$ is an increasing function when $t\leq i$ and $h_{B_{i}^{\prime}(n)}(t)=h_{A(n)}(t)$ when $t\geq i$ .

Again by Lemma 2.22, for each $B_{i}^{\prime}(n)$ there are Hopf subalgebras $D_{i}(n)$ defined by

h_{D_{i}(n)}(t)=\begin{cases}0&t<i\\ h_{B_{i}^{\prime}(n)}(t)&t\geq i\end{cases}.

Given $n\geq 2$ , recall that $B_{1}^{\prime}(n)=D_{1}(n)$ .

There are injective homomorphisms between the group of endotrivial modules for these Hopf subalgebras:

$\mathbb{T}(A(n))\to\prod_{i=1}^{1+a}\mathbb{T}(B_{i}^{\prime}(n))$ (Proposition 4.1)

$\mathbb{T}(B_{i}^{\prime}(n))\to\mathbb{T}(\ast)\times\mathbb{T}(D_{i}(n)),2\leq i\leq a+1$ (Proposition 4.4).

$\mathbb{T}(D_{i}(n))\to\mathbb{T}(\ast)\times\mathbb{T}(\ast),1\leq i\leq a+1$ (Proposition 4.5 4.6)

All the above maps are the restriction map induced by inclusion of Hopf subalgebras, $\ast$ means certain elementary Hopf subalgebras.

Finally, we define $O_{i}=<P^{0}_{t}|i\leq t\leq n+1>$ .

The rest of this section will be devoted to the proof of Theorem 1.1.

Proposition 4.1.

$\forall n\geq 2$ ,

Res:\mathbb{T}(A(n))\longrightarrow\prod_{i=1}^{a+1}\mathbb{T}(B^{\prime}_{i}(n))

is injective.

Proof: Given a finitely generated $A(n)//E(n)$ -module $M$ , assume that $M|_{B_{i}^{\prime}(n)//E(n)}$ is free in $gr\left(B_{i}^{\prime}(n)//E(n)\right)$ for each $1\leq i\leq a+1$ , we will show that $M$ is a free $A(n)//E(n)$ -module.

The trick is that there is a natural splitting $M=M_{1}\oplus\sigma(1)M_{2}$ where $M_{1}$ and $M_{2}$ are all concentrated in even degree. This is because that $A(n)//E(n)$ is concentrated in even degree. Proposition 2.28 guarantees that $(\eta^{*}M_{j})|_{B_{i}\cap A(n-1)}$ is free for $j=1,2$ ; and then the detect property of $A(n-1)$ with detecting set $\{B_{i}\cap A(n-1)|1\leq i\leq a+1\}$ implies $\eta^{*}M_{j}$ is a free $A(n-1)$ -module. Again, since $\eta^{*}$ is an isomorphism between the two module categories, $M_{j}$ is a free $A(n)//E(n)$ -module, $j=1,2$ . Namely, $M=M_{1}\oplus\sigma(1)M_{2}$ is a free $A(n)//E(n)$ -module.

The above argument shows that $A(n)//E(n)$ has detect property with detecting sets $\{B_{i}^{\prime}(n)//E(n)|1\leq i\leq a+1\}$ . By Theorem 3.14, we are done.

Lemma 4.2.

Given two Hopf subalgebras $E_{1}$ , $E_{2}$ of $H$ and suppose $M$ is a finitely generated $H$ -module such that $M|_{E_{j}}$ is a free $E_{j}$ -module for $j=1,2$ .

M is a free H-module, if

(1) $\exists t_{1}\neq 0\in E_{1},t_{2}\neq 0\in E_{2}$ such that $|t_{1}t_{2}|=|H|$ , and

(2)for all homogeneous $y\neq 0\in E^{+}_{2},|y|>|t_{1}|$ or $y=z$ with $z^{2}=0,0\neq zt_{1}\in E^{+}_{1}$ .

Proof: Suppose $f:\oplus_{j=1}^{q}H\to M$ with $f(h_{1},h_{2},\cdots,h_{q})=\sum_{j=1}^{q}h_{j}m_{j}$ is a surjective $H$ -module homomorphism with $q$ is minimal. Such $f$ and $q$ exist since $M$ is finitely generated. Furthermore, we assume that $|m_{j}|\leq|m_{j+1}|$ . We show that $m_{1}$ generates a free $H$ -submodule first.

Suppose $m_{1}=\sum_{l}x_{l}b_{l}$ for some homogeneous $x_{l}\in E_{1}^{+},b_{l}\in M$ . Then for each $l$ , one has $|x_{l}b_{l}|=|x_{l}|+|b_{l}|>|m_{1}|$ , contradiction. Hence $xw$ is not a summand of $m_{1}$ , $\forall x\in E_{1}^{+},w\in M$ . By assumption $M$ is a free $E_{1}$ module, thus $xm_{1}\neq 0$ . In particular, $t_{1}m_{1}\neq 0$ .

Suppose $t_{1}m_{1}=\sum_{l}y_{l}c_{l}$ for some homogeneous $y_{l}\in E_{2}^{+},c_{l}\in M$ and $\exists y_{l_{0}}\neq z$ . Then $|y_{l_{0}}c_{l_{0}}|=|y_{l_{0}}|+|c_{l_{0}}|>|t_{1}|+|c_{l_{0}}|\geq|t_{1}m_{1}|$ , which is impossible. Thus $t_{1}m_{1}=z\cdot c$ where $c=\sum_{l}c_{l}$ , then $z\cdot t_{1}m_{1}=(z)^{2}c=0$ . On the other hand, by condition (2), $0\neq zt_{1}\in E_{1}$ , which is a contradiction since we showed $xm_{1}\neq 0$ , for all nonzero $x\in E_{1}$ . In conclusion, $yw$ is not a summand of $t_{1}m_{1}$ for all $y\in E_{2}^{+}$ and $w\in M$ . Again since $M$ is a free $E_{2}$ module, $yt_{1}m_{1}\neq 0$ .

In summary, $yt_{1}m_{1}\neq 0,\forall y\in E_{2}^{+}$ . In particular, $t_{2}t_{1}m_{1}\neq 0$ , which implies that $t_{2}t_{1}\neq 0$ . By assumption, $|t_{2}t_{1}|=|H|$ . Therefore, $t_{2}t_{1}=t^{H}_{1}$ since $H$ is a finite connected Hopf algebra and $t_{2}t_{1}\neq 0$ . By Lemma 3.6, $m_{1}$ generates a free $H$ -submodule $L_{1}\subset M$ .

Since $q$ is minimal, $m_{j}\notin L_{1}$ for $j\geq 2$ . Define $M_{2}=M/L_{1}$ , then $f_{2}:\oplus_{j=1}^{q-1}H\to M_{2}$ with $f(h_{1},\cdots,h_{q-1})=\sum_{j=1}^{q-1}h_{j}\overline{m_{j+1}}$ is a surjective $H$ -module homomorphism with $q-1$ is minimal , where $|\overline{m_{j}}|\leq|\overline{m_{j+1}}|$ . By an induction on the number of generators, one may assume that $M_{2}$ is free. Therefore, $M=L_{1}\oplus M_{2}$ is a free $H$ -module. The proof of Proposition 4.4, Proposition 4.5, and Proposition 4.6 rely heavily on the above Lemma. To apply this Lemma, one must know well about the degree of the related algebraic generators $P^{s}_{t}$ .

Lemma 4.3.

For $s,u>0$ and $t,v\geq 1$ ,

(1) If $s+t=u+v$ and $t\geq v$ then $|P^{s}_{t}|\geq|P^{u}_{v}|$ ;

(2) If $s+t>u+v$ , then $|P^{s}_{t}|>|P^{u}_{v}|$ .

Proof: (1) By definition, $|P^{s}_{t}|=2^{s}(2^{t}-1)=2^{s+t}-2^{s}$ and $|P^{u}_{v}|=2^{u}(2^{v}-1)=2^{u+v}-2^{u}$ . By assumption $u\geq s$ and hence $|P^{s}_{t}|\geq|P^{u}_{v}|$ .

(2) By assumption, $s+t>1$ . By (1), since $P^{0}_{s+t}=2^{s+t}-1>2^{s+t-1}>2^{u+v-1}=P^{u+v-1}_{1}$ , we are done.

Proposition 4.4.

For each $2\leq i\leq a+1$ , $n\geq 2$ ,

Res:\mathbb{T}(B_{i}^{\prime}(n))\longrightarrow\mathbb{T}(E(n))\times\mathbb{T}(D_{i}(n))

is injective.

Proof: By Lemma 2.12, $P^{0}_{n+1}$ commutes with all the $P^{s}_{t}\in B_{i}^{\prime}(n)$ , thus $<P^{0}_{n+1}>$ is a normal Hopf subalgebra and $B_{i}^{\prime}(n)//<P^{0}_{n+1}>$ is a connected Hopf algebra, denote it as $H$ . Denote $E(n)//<P^{0}_{n+1}>$ and $D_{i}(n)//<P^{0}_{n+1}>$ as $E_{1}$ and $E_{2}$ respectively, then $E_{1}$ and $E_{2}$ are connected Hopf subalgebra of $H$ . We will show that $H$ has detect property in $grH$ with detecting set $\{E_{1},E_{2}\}$ .

As an algebra, $E_{1}$ is generated by $\mathcal{F}=\{\overline{P^{0}_{t}}|1\leq t\leq n\}$ , $E_{2}$ is generated by $\mathcal{G}=\{\overline{P^{s}_{t}}|i\leq t\leq n,0\leq s\leq h_{D_{i}(n)}(t)\}$ and $H$ is generated by $\mathcal{F}\cup\mathcal{G}$ .

Consider the element $t_{1}=\prod_{j=1}^{i-1}\overline{P^{0}_{j}}$ and $t_{2}=\prod_{t=i}^{n}\left(\prod_{s=0}^{h_{B_{i}^{\prime}(n)}(t)-1}\overline{P^{s}_{t}}\right)$ where $t_{1}\in E_{1}$ and $t_{2}\in E_{2}$ . By Corollary 2.21 and Lemma 2.24, they are nonzero. Next, we show they satisfy the assumption of Lemma 4.2.

Since $|t_{1}|=2^{i}-i$ and $|\overline{P^{0}_{i}}|=2^{i}-1$ , by Lemma 4.3, $\forall y\in E^{+}_{2}$ , $|y|>|t_{1}|$ .

Notice that the Milnor basis of highest degree in $B_{i}^{\prime}(n)$ is $Sq(r_{1},r_{2},\cdots,r_{n+1})$ with $r_{t}=2^{h_{B_{i}^{\prime}(n)}(t)}-1$ , one has

|B_{i}^{\prime}(n)|=\sum_{t=1}^{n+1}\left(2^{h_{B_{i}^{\prime}(n)}(t)}-1\right)\left(2^{t}-1\right).

Since $|P^{0}_{n+1}|=2^{n+1}-1$ and $h_{B_{i}^{\prime}(n)}(n+1)=1$ , by Theorem 2.4,

|H|=|B_{i}^{\prime}(n)|-|<P^{0}_{n+1}>|=\sum_{t=1}^{n}\left(2^{h_{B_{i}^{\prime}(n)}(t)}-1\right)\left(2^{t}-1\right).

On the other hand

|t_{1}|+|t_{2}|=\sum_{t=1}^{n}\left(2^{h_{B_{i}^{\prime}(n)}(t)}-1\right)\left(2^{t}-1\right),

Therefore, we have $|H|=|t_{1}|+|t_{2}|$ . By Lemma 4.2(1), $H$ has detect property in $grH$ with detecting set $\{E_{1},E_{2}\}$ . By Theorem 3.14, we are done.

Proposition 4.5.

For each $2\leq i\leq a+1$ , $n\geq 2$

Res:\mathbb{T}(D_{i}(n))\longrightarrow\mathbb{T}(B_{i}\cap D_{i}(n))\times\mathbb{T}(B_{i+1}\cap D_{i}(n))

is injective.

Proof: Consider $Y_{i}(n)\subset D_{i}(n)$ with profile function

h_{Y_{i}(n)}(t)=\begin{cases}2i+1-t&i+1\leq t\leq 2i-1\\ 0&otherwise\end{cases}.

Since $n\geq 2$ and $i\leq a+1$ , $Y_{i}(n)\subset A(2i-1)$ is a Hopf subalgebra of $D_{i}(n)$ by Lemma 2.22 and Lemma 2.23. Define $X_{i}(n)\subset D_{i}(n)$ by the function

h_{X_{i}(n)}(t)=\begin{cases}i+1&t=i\\ 1&t=2i\\ 0&otherwise\end{cases}.

Directly by Theorem 2.16, $X_{i}(n)$ is a Hopf subalgebra of $D_{i}(n)$ .

By Lemma 2.12 (3) $[x,y]=0$ , $\forall x\in D_{i}(n),y\in Y_{i}(n)$ . Namely, $Y_{i}(n)$ is a normal Hopf subalgebra of $D_{i}(n)$ , and $D_{i}(n)//Y_{i}(n)$ is a connected Hopf algebra, denoted as $H$ . Define $E_{1}^{\prime}=\left(B_{i}\cap D_{i}(n)\right)//Y_{i}(n)$ and $E_{2}=\left(B_{i+1}\cap D_{i}(n)\right)//Y_{i}(n)$ . We will show that $H$ has detect property in $grH$ with detecting sets $\{E_{1}^{\prime},E_{2}\}$ .

Suppose $M$ is a finitely generated $H$ -module, and $M|_{E_{1}^{\prime}}$ , $M|_{E_{2}}$ are free. We want to show that $M$ is a free $H$ -module.

By Lemma 2.24, $X_{i}(n)\cap\left(Y_{i}(n)^{+}D_{i}(n)\right)=0$ , namely $\pi|_{X_{i}(n)}$ is an isomorphism of graded algebras from $X_{i}(n)$ to $\pi X_{i}(n)$ , where $\pi:D_{i}(n)\to H$ is the quotient map. By Lemma 3.11 and Lemma 2.25, $X_{i}(n)$ has detect property in $grX_{i}(n)$ with detecting set $\{B_{i}\cap X_{i}(n)\}$ , hence $\pi X_{i}(n)$ has detect property in $gr\left(\pi X_{i}(n)\right)$ with detecting set $\{\pi\left(B_{i}\cap X_{i}(n)\right)\}$ . Now that $M$ is a free $E_{1}^{\prime}$ -module and $\pi(B_{i}\cap X_{i}(n))$ is a Hopf subalgebra of $E_{1}^{\prime}$ , it is free over $\pi(B_{i}\cap X_{i}(n))$ , and thus $M$ is a free $\pi X_{i}(n)$ -module. Now, denote $E_{1}=\pi X_{i}(n)$ .

Consider the element $t_{1}=\prod_{j=0}^{i}\overline{P^{j}_{i}}$ and $t_{2}=\prod_{t=i+1}^{n+1}\left(\prod_{s=h_{Y_{i}(n)}(t)}^{h_{D_{i}(n)}(t)-1}\overline{P^{s}_{t}}\right)$ where $t_{1}\in E_{1}$ and $t_{2}\in E_{2}$ . By Corollary 2.21 and Lemma 2.24, they are nonzero. Next, we show they satisfy the assumption (2) of Lemma 4.2.

Take $z=\overline{P^{0}_{2i}}$ , we know $z^{2}=0$ . By definition of the profile functions, $z\in E_{1}\cap E_{2}$ . Since $|t_{1}|=(2^{i+1}-1)(2^{i}-1)$ and $|\overline{P^{i}_{i+1}}|=(2^{i}-1)2^{i}$ , by Lemma 4.3, $\forall y\in E^{+}_{2}$ , $|y|>|t_{1}|$ as long as $y\neq z$ . Moreover, $zt_{1}\in E_{1}$ because $t_{1},z\in E_{1}$ and by Corollary 2.21 and 2.24, $zt_{1}\neq 0$ .

By definition,

|t_{1}t_{2}|=\sum_{t=i}^{n+1}\left(2^{h_{D_{i}(n)}(t)}-2^{h_{Y_{i}(n)}(t)}\right)\left(2^{t}-1\right).

By the definition of profile function,

|D_{i}(n)|=\sum_{t=i}^{n+1}\left(2^{h_{D_{i}(n)}(t)}-1\right)\left(2^{t}-1\right).

and

|Y_{i}(n)|=\sum_{t=i}^{n+1}\left(2^{h_{Y_{i}(n)}(t)}-1\right)\left(2^{t}-1\right).

Therefore, we have

|H|=|D_{i}(n)|-|Y_{i}(n)|=|t_{1}|+|t_{2}|.

By Lemma 4.2, $M$ is a free $H$ -module, namely, $H$ has detect property in $grH$ with detecting set $\{E_{1}^{\prime},E_{2}\}$ . By Theorem 3.14, we are done.

Proposition 4.6.

For each $n\geq 2$

Res:\mathbb{T}(D_{1}(n))\longrightarrow\mathbb{T}(E(n))\times\mathbb{T}(B_{2}\cap D_{1}(n))

is injective.

Proof: By Lemma 2.12, $P^{0}_{n+1}$ commutes with all the $P^{s}_{t}\in D_{1}(n)$ , thus $<P^{0}_{n+1}>$ is a normal Hopf subalgebra and $D_{1}(n)//<P^{0}_{n+1}>$ is a connected Hopf algebra, denote it as $H$ . Moreover, denote $E(n)//<P^{0}_{n+1}>$ and $B_{2}\cap D_{1}(n)//<P^{0}_{n+1}>$ as $E_{1}^{\prime}$ and $E_{2}$ respectively, then $E_{1}^{\prime}$ and $E_{2}$ are connected Hopf subalgebra of $H$ , and we will show that $H$ has detect property in $grH$ with detecting set $\{E_{1}^{\prime},E_{2}\}$ .

Suppose $M$ is a finitely generated $H$ -module, and $M|_{E^{\prime}_{1}}$ , $M|_{E_{2}}$ are free. We want to show that $M$ is a free $H$ -module.

By assumption $n\geq 2$ , thus $|A(1)|=6<|P^{0}_{n+1}|=2^{n+1}-1$ . Therefore $A(1)\cap<P^{0}_{n+1}>^{+}D_{1}(n)=0$ , which implies $\pi|_{A(1)}$ is an isomorphism of graded algebras from $A(1)$ to $\pi A(1)$ , where $\pi:D_{1}(n)\to D_{1}(n)//<P^{0}_{n+1}>$ is the quotient map. That is, $\pi A(1)$ has detect property in $gr(\pi A(1))$ with detecting set $\{\pi E(1)\}$ . Since $M$ is free on $E_{1}^{\prime}=\pi E(n)$ and $\pi E(1)$ is a normal Hopf subalgebra of $\pi E(n)$ , $M$ is a free $\pi E(1)$ -module by Theorem 2.4. It follows that $M$ is a free graded $\pi A(1)$ -module. Now we denote $E_{1}=\pi A(1)$ .

Consider the element $t_{1}=\overline{P^{0}_{1}P^{1}_{1}}\in E_{1}$ and $t_{2}=\prod_{t=2}^{n}\overline{P^{0}_{t}P^{1}_{t}}\in E_{2}$ . By Corollary 2.21 and Lemma 2.24, they are nonzero. Next, we show they satisfy the assumption (2) of Lemma 4.2.

Take $z=\overline{P^{0}_{2}}\in E_{1}\cap E_{2}$ we know $z^{2}=0$ . Since $|t_{1}|=3$ and $|\overline{P^{1}_{2}}|=6$ , by Lemma 4.3, $\forall y\in E^{+}_{2}$ , $|y|>|t_{1}|$ as long as $y\neq z$ . Moreover, by Corollary 2.21 and 2.24, $zt_{1}\neq 0$ .

By definition,

|t_{1}t_{2}|=\sum_{t=1}^{n}3\left(2^{t}-1\right).

Notice that the Milnor basis of highest degree in $D_{1}(n)$ is that $Sq(r_{1},r_{2},\cdots r_{n+1})$ with $r_{t}=3$ for $t\leq n$ , and $r_{n+1}=1$ , one has

|D_{1}(n)|=2^{n+1}-1+\sum_{t=1}^{n}3\left(2^{t}-1\right).

Since $|P^{0}_{n+1}|=2^{n+1}-1$ , by Theorem 2.4,

|H|=|D_{1}(n)|-|<P^{0}_{n+1}>|=\sum_{t=1}^{n}3\left(2^{t}-1\right).

Therefore, we have $|H|=|t_{1}|+|t_{2}|.$ By Lemma 4.2, $M$ is a free $H$ -module, namely, $H$ has detect property in $grH$ with detecting set $\{E_{1}^{\prime},E_{2}\}$ . By Theorem 3.14, we are done.

Theorem 4.7.

Given $n\geq 2$ , the morphism of groups

f:\mathbb{Z}\oplus\mathbb{Z}\to\mathbb{T}(A(n))

which sends $(m,l)$ to $[\sigma(m)\Omega_{A(n)}^{l}(\mathbb{Z}/2)]$ is an isomorphism of groups.

Proof: By Proposition 4.1 4.4 4.5 4.6, one gets

Res:\mathbb{T}(A(n))\longrightarrow\prod_{L}\mathbb{T}(L)

is an injective group homomorphism, where each $L$ is elementary and contains $O_{a+1}$ . By Proposition 2.8, $Res^{L}_{O_{a+1}}:\mathbb{T}(L)\to\mathbb{T}(O_{a+1})$ is an isomorphism. Compose $Res$ with $Res^{L}_{O_{a+1}}$ on each component of $\prod_{L}\mathbb{T}(L)$ , one gets an injection

Res:\mathbb{T}(A(n))\longrightarrow\mathbb{T}(O_{a+1}).

Recall that $\sigma(m)\Omega_{A(n)}^{l}(\mathbb{Z}/2)\in\mathbb{T}(A(n))$ and it maps to $\sigma(m)\Omega_{O_{a+1}}^{l}(\mathbb{Z}/2)$ under the restriction $Res^{A(n)}_{O_{a+1}}$ , the above injection is in fact an isomorphism.

Recall that the group of endotrivial modules are the same as the stable Picard group, thus one has the following Corollary:

Corollary 4.8.

Given $n\geq 2$ , the morphism of groups

f:\mathbb{Z}\oplus\mathbb{Z}\to Pic(A(n))

which sends $(m,l)$ to $[\sigma(m)\Omega_{A(n)}^{l}(\mathbb{Z}/2)]$ is an isomorphism of groups.

References

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The Stable Picard Group of A​(n)A(n)

Abstract

keywords:

1 Introduction

Theorem 1.1 (Corollary 4.8).

2 Preliminaries

2.1 Finite connected graded Hopf algebra

Definition 2.1.

Theorem 2.2 (Theorem 12.2.5 & 12.2.9 of Margolis (1983)).

Theorem 2.3 (Proposition 12.2.8 of Margolis (1983)).

Theorem 2.4 (Theorem 1.3 of Aguiar and Lauve (2013)).

Remark 2.5.

2.2 Endotrivial modules

Lemma 2.6.

Definition 2.7 (Definition 2.15 of Gheorghe et al. (2018)).

Proposition 2.8 (Prop3.2 of Bhattacharya and Ricka (2017)).

Remark 2.9.

Remark 2.10.

2.3 Mod 2 Steenrod algebra

Lemma 2.11.

Lemma 2.12.

Lemma 2.13.

Lemma 2.14.

Definition 2.15.

Theorem 2.16 (Theorem 15.1.6 of Margolis (1983)).

Remark 2.17.

Remark 2.18.

Definition 2.19.

Remark 2.20.

Corollary 2.21.

Lemma 2.22.

Lemma 2.23.

Lemma 2.24.

Lemma 2.25.

Remark 2.26.

Lemma 2.27 (Proposition 15.3.29 of Margolis (1983)).

Proposition 2.28 (Proposition 15.3.30 of Margolis (1983)).

3 Reduction to Hopf subalgebras

Definition 3.1.

Remark 3.2.

Lemma 3.3.

Lemma 3.4.

Lemma 3.5 (Lemma 12.2.6 of Margolis (1983)).

Lemma 3.6.

Lemma 3.7.

Lemma 3.8.

Corollary 3.9.

Definition 3.10.

Theorem 3.11 (Theorem 1.3, Example A.6 of Palmieri (1997)).

Remark 3.12.

Remark 3.13.

Theorem 3.14.

4 Endotrivial group of A(n)

Proposition 4.1.

Lemma 4.2.

Lemma 4.3.

Proposition 4.4.

Proposition 4.5.

Proposition 4.6.

Theorem 4.7.

Corollary 4.8.

References

The Stable Picard Group of $A(n)$