# A generalization of Newton-Maclaurin’s inequalities

Changyu Ren  Changyu Ren, School of Mathematical Science, Jilin University,
Changchun, 130012, Jilin Province, P.R. China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove Newton-Maclaurin type inequalities for functions
obtained by linear combination of two neighboring primary symmetry functions,
which is a generalization of the classical Newton-Maclaurin inequality.

###### Key words and phrases:

Primary symmetry function, Newton-Maclaurin’s inequality, Hessian equations.

###### 1991 Mathematics Subject Classification:

26D05, 26D15

Research of author is supported by an NSFC Grant No. 11871243

##  1\. Introduction

The $k$-th elementary symmetric function of the variables
$x_{1},x_{2},\cdots,x_{n}$ is defined by

$\sigma_{k}(x)=\displaystyle\sum_{1\leq i_{1}<\cdots<i_{k}\leq
n}x_{i_{1}}\cdots x_{i_{k}},\quad 1\leq k\leq n,$

where $x=(x_{1},x_{2},\cdots,x_{n})$. For example, when $n=3$,

$\displaystyle\sigma_{1}(x)=x_{1}+x_{2}+x_{3},\quad\sigma_{2}(x)=x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3},\quad\sigma_{n}(x)=x_{1}x_{2}x_{3}.$

It will be convenient to define $\sigma_{0}(x)=1$, and define
$\sigma_{k}(x)=0$ if $k<0$ or $k>n$. Furthermore, define a $k$-th elementary
symmetric mean as

$E_{k}(x)=\dfrac{\sigma_{k}(x)}{C_{n}^{k}},\quad k=0,1,\cdots,n,$

where $C_{n}^{k}=\dfrac{n!}{k!(n-k)!}$.

######  Theorem 1.1.

(Newton [19] and Maclaurin [17]) Let $x=(x_{1},\cdots,x_{n})$ be an $n$-tuple
non-negative real numbers. Then

(1.1) $E_{k}^{2}(x)\geq E_{k-1}(x)E_{k+1}(x),\quad k=1,2\cdots,n-1,$ (1.2)
$E_{1}(x)\geq E_{2}^{1/2}(x)\geq\cdots\geq E_{n}^{1/n}(x),$

and the inequality is strict unless all entries of $x$ coincide.

The inequalities (1.1) in Theorem 1.1 is a consequence of a rule stated by
Newton [19] which gives a lower bound on the number of nonreal roots of a real
polynomial. Since Newton did not give a proof of his rule, the proof of
Theorem 1.1 is due to MacLaurin [17]. Actually, the Newton’s inequalities
(1.1) work for $n$-tuples of real, not necessarily positive elements. For an
inductive proof in the case where $x_{1},x_{2},\cdots,x_{n}$ are nonnegative,
see [9] $\S$ 2.22. For a proof by differential calculus in the case where
$x_{1},x_{2},\cdots,x_{n}$ are real, see [9] $\S$ 4.3, or [21]. Several
reformulations/generalisations of Newton’s inequalities have been given over
the years, see for instance [18, 21, 20, 2].

The Newton-Maclaurin inequalities play important roles in deriving theoretical
result for fully nonlinear partial differential equations and geometric
analysis. There are many important results that need to use the Newton-
Maclaurin inequalities, such as [3, 4, 5, 6, 7, 11] etc. This is because the
following $k$-Hessian equations and curvature equations

$\displaystyle\sigma_{k}(\lambda(u_{ij}))$ $\displaystyle=f(x,u,\nabla u),$
$\displaystyle\sigma_{k}(\kappa(X))$ $\displaystyle=f(X,\nu)$

are central studies in the field of fully nonlinear partial differential
equations and geometric analysis. Their left-hand side $k$-Hessian operator
$\sigma_{k}$ is the primary symmetric function about the eigenvalues
$\lambda=(\lambda_{1},\cdots,\lambda_{n})$ of the Hessian matrix $(u_{ij})$ or
the principal curvature $\kappa=(\kappa_{1},\cdots,\kappa_{n})$ of the
surface. In recent years, the fully non-linear equations derived from linear
combinations of primary symmetric functions have received increasing
attention. For example, the following special Lagrangian equation

${\rm Imdet}(\delta_{ij}+{\rm
i}u_{ij})=\displaystyle\sum_{k=0}^{[(n-1)/2]}(-1)^{k}\sigma_{2k+1}(\lambda(u_{ij}))=0,$

which derived by Harvey and Lawson in their study of the minimal submanifold
problem [10] . The fully non-linear partial differential equations

$P_{m}(u_{ij})=\displaystyle\sum_{k=0}^{m-1}(l_{k}^{+})^{m-k}(x)\sigma_{k}(u_{ij})=g^{m-1}(x)$

studied by Krylov [12] and Dong [1]. In [13], Li-Ren-Wang studied the
concavity of operators $\displaystyle\sum_{s=0}^{k}\alpha_{s}\sigma_{s}$ and
$\sigma_{k}+\alpha\sigma_{k-1}$, and discussed the curvature estimates for the
convex surface of the corresponding equations

$\displaystyle\sum_{s=0}^{k}\alpha_{s}\sigma_{s}(\kappa(X))=f(X,\nu(X)),\quad
X\in M.$

Guan and Zhang [8] investigated the curvature estimates of the following
curvature equation

$\sigma_{k}(W_{u}(x))+\alpha(x)\sigma_{k-1}(W_{u}(x))=\displaystyle\sum_{l=0}^{k-2}\alpha_{l}(x)\sigma_{l}(W_{u}(x)),\quad
x\in\mathbb{S}^{n}.$

Recently, Liu and Ren [16] discussed the Pogorelov-type $C^{2}$ estimates for
$(k-1)$-convex and $k$-convex solutions of the following Sum Hessian equations

$\sigma_{k}(\lambda(u_{ij}))+\alpha\sigma_{k-1}(\lambda(u_{ij}))=f(x,u,\nabla
u),$

and established a rigidity theorem when the right-hand side of the equation is
constant.

In the further study of the above problem, the Newton-Maclaurin type
inequalities for the operators derived from the left-hand side of the
equations are always needed. A natural question is that whether the Newton-
Maclaurin type inequalities for the operators of linear combinations of these
primary symmetric functions still hold?

Now the main result of this paper is stated as follows.

######  Theorem 1.2.

Let $n\geq 3,1\leq k\leq n-2$. For any real number $\alpha\in\mathbb{R}$ and
any $n$-tuple real numbers $x=(x_{1},x_{2},\cdots,x_{n})\in\mathbb{R}^{n}$, we
have

(1.3) $[\alpha E_{k}(x)+E_{k+1}(x)]^{2}\geq[\alpha E_{k-1}(x)+E_{k}(x)][\alpha
E_{k+1}(x)+E_{k+2}(x)].$

The inequality is strict unless $x_{1}=\cdots=x_{n}$, or

$\dfrac{\alpha E_{k}(x)+E_{k+1}(x)}{\alpha E_{k-1}(x)+E_{k}(x)}=\dfrac{\alpha
E_{k+1}(x)+E_{k+2}(x)}{\alpha E_{k}(x)+E_{k+1}(x)}=-\alpha.$

A special case in which the above equality holds is that there are $n-1$
elements of $x_{1},x_{2},\cdots,x_{n}$ taking the value $-\alpha$.

Furthermore, if $\alpha\geq 0$ and

$\alpha E_{m}(x)+E_{m+1}(x)\geq 0,\quad{\rm for~{}all~{}~{}}m=0,1,\cdots,k,$

then

(1.4) $[\alpha+E_{1}(x)]\geq[\alpha
E_{1}(x)+E_{2}(x)]^{1/2}\geq\cdots\geq[\alpha E_{k}(x)+E_{k+1}(x)]^{1/(k+1)}.$

Theorem 1.2 is the Newton-Maclaurin type inequality for the function obtained
by linear combination of two neighboring terms $E_{k}$ and $E_{k+1}$.
Obviously, it is the classical Newton-Maclaurin inequality when $\alpha=0$.
Furthermore, one may ask if the Newton-Maclaurin type inequality still holds
for the functions which obtained by linear combination of general primary
symmetric functions. That is, for
$\alpha=(\alpha_{1},\cdots,\alpha_{n})\in\mathbb{R}^{n}$,
$x=(x_{1},\cdots,x_{n+1})\in\mathbb{R}^{n+1}$, whether the following
inequalities hold?

(1.5)
$\left(\displaystyle\sum_{k=1}^{n}\alpha_{k}E_{k}(x)\right)^{2}\geq\left(\displaystyle\sum_{k=1}^{n}\alpha_{k}E_{k-1}(x)\right)\left(\displaystyle\sum_{k=1}^{n}\alpha_{k}E_{k+1}(x)\right).$

The answer is no in general. For example, let $n=3$, $\alpha=(1,0,1)$,
$x=(4,4,\dfrac{1}{4},\dfrac{1}{4})$, we have

$\displaystyle E_{1}(x)=$
$\displaystyle\dfrac{1}{4}(4+4+\dfrac{1}{4}+\dfrac{1}{4})=\dfrac{17}{8},$
$\displaystyle E_{2}(x)=$ $\displaystyle\dfrac{1}{6}(4\times 4+4\times
4\times\dfrac{1}{4}+\dfrac{1}{4}\times\dfrac{1}{4})=\dfrac{107}{32},$
$\displaystyle E_{3}(x)=$ $\displaystyle\dfrac{1}{4}(2\times 4\times
4\times\dfrac{1}{4}+2\times
4\times\dfrac{1}{4}\times\dfrac{1}{4})=\dfrac{17}{8},$ $\displaystyle
E_{4}(x)=$ $\displaystyle 4\times 4\times\dfrac{1}{4}\times\dfrac{1}{4}=1,$

therefore,

$\displaystyle[E_{1}(x)+E_{3}(x)]^{2}-[1+E_{2}(x)][E_{2}(x)+E_{4}(x)]=-\dfrac{825}{1024}<0.$

Maybe inequalities (1.5) hold when $\alpha=(\alpha_{1},\cdots,\alpha_{n})$
satisfy some structural conditions.

A straightforward conclusion of Newton’s inequality (1.1) is

(1.6)
$\sigma_{k}^{2}(x)-\sigma_{k-1}(x)\sigma_{k+1}(x)\geq\theta\sigma_{k}^{2}(x),$

where $0<\theta<1$ is a constant depending on $n$ and $k$. The inequality
(1.6) is also a common sense which is widely used in the study of $k$-Hessian
equations and curvature equations, see [4, 5, 6, 7, 11, 13, 14, 15], etc. For
the Sum Hessian operators $\sigma_{k}+\alpha\sigma_{k-1}$, if $\alpha>0$ and
$x=(x_{1},\cdots,x_{n})\in\Gamma_{k}$, Liu and Ren [16] proved

$[\sigma_{k}(x)+\alpha\sigma_{k-1}(x)]^{2}\geq[\sigma_{k-1}(x)+\alpha\sigma_{k-2}(x)][\sigma_{k+1}(x)+\alpha\sigma_{k}(x)]$

by Newton’s inequality (1.6). Here $\Gamma_{k}$ is the Garding’s cone

$\Gamma_{k}=\\{\lambda\in\mathbb{R}^{n}\ |\quad\sigma_{m}(\lambda)>0,\quad
m=1,\cdots,k\\}.$

The requirement $x\in\Gamma_{k}$ of above inequalities restrict their
applications in the study of Sum Hessian equations. In this paper, we remove
the restrictions from $\alpha$ and $x$.

######  Theorem 1.3.

Let $n\geq 3,0\leq k\leq n-1$. For any real number $\alpha\in\mathbb{R}$ and
any $n$-tuple real numbers $x=(x_{1},x_{2},\cdots,x_{n})\in\mathbb{R}^{n}$, we
have

(1.7)
$(1-\theta)[\alpha\sigma_{k}(x)+\sigma_{k+1}(x)]^{2}-[\alpha\sigma_{k-1}(x)+\sigma_{k}(x)][\alpha\sigma_{k+1}(x)+\sigma_{k+2}(x)]\geq
0,$

where $0<\theta<1$ is a constant depending only on $n$ and $k$.

######  Remark 1.4.

When $k=0$ or $k=n-1$, (1.3) does not hold, one can take
$x_{1}=x_{2}=\cdots=x_{n}$ for verification. It is a difference between (1.7)
and (1.3).

##  2\. The proof of Theorem 1.2

First, we consider the following inequality in Euclidean space
$\mathbb{R}^{3}$.

######  Lemma 2.1.

For any real number $\alpha\in\mathbb{R}$ and
$z=(z_{1},z_{2},z_{3})\in\mathbb{R}^{3}$, we have

(2.1) $[\alpha E_{1}(z)+E_{2}(z)]^{2}\geq[\alpha E_{0}(z)+E_{1}(z)][\alpha
E_{2}(z)+E_{3}(z)].$

The inequality is strict unless $z_{1}=z_{2}=z_{3}$ or

$\dfrac{\alpha E_{1}(z)+E_{2}(z)}{\alpha E_{0}(z)+E_{1}(z)}=\dfrac{\alpha
E_{2}(z)+E_{3}(z)}{\alpha E_{1}(z)+E_{2}(z)}=-\alpha.$

###### Proof.

A straightforward calculation shows

$\displaystyle 18[\alpha E_{1}(z)+E_{2}(z)]^{2}-18[\alpha
E_{0}(z)+E_{1}(z)][\alpha E_{2}(z)+E_{3}(z)]$ $\displaystyle=$
$\displaystyle[(z_{1}+\alpha)(z_{2}+\alpha)-(z_{1}+\alpha)(z_{3}+\alpha)]^{2}+[(z_{1}+\alpha)(z_{2}+\alpha)-(z_{2}+\alpha)(z_{3}+\alpha)]^{2}$
$\displaystyle+[(z_{1}+\alpha)(z_{3}+\alpha)-(z_{2}+\alpha)(z_{3}+\alpha)]^{2}\geq
0.$

Obviously, the inequality is strict unless $z_{1}=z_{2}=z_{3}$ or any two
elements of $z_{1},z_{2},z_{3}$ valued $-\alpha$. Without loss of generality,
we assume $z_{1}=z_{2}=-\alpha$. Then we obtain

$\displaystyle\alpha E_{0}(z)+E_{1}(z)=\dfrac{z_{3}+\alpha}{3},\alpha
E_{1}(z)+E_{2}(z)=\dfrac{-\alpha(z_{3}+\alpha)}{3},\alpha
E_{2}(z)+E_{3}(z)=\dfrac{\alpha^{2}(z_{3}+\alpha)}{3}.$

Thus,

$\dfrac{\alpha E_{1}(z)+E_{2}(z)}{\alpha E_{0}(z)+E_{1}(z)}=\dfrac{\alpha
E_{2}(z)+E_{3}(z)}{\alpha E_{1}(z)+E_{2}(z)}=-\alpha.$

∎

The following Lemma is a useful tool to prove Newton’s inequalities (1.1) from
Sylvester [22, 20].

######  Lemma 2.2.

If

$F(x,y)=c_{0}x^{m}+c_{1}x^{m-1}y+\cdots+c_{m}y^{m}$

is a homogeneous function of the $n$-th degree in $x$ and $y$ which has all
its roots $x/y$ real, then the same is true for all non-identical 0 equations
$\dfrac{\partial^{i+j}F}{\partial x^{j}\partial y_{j}}=0$, obtained from it by
partial differentiation with respect to $x$ and $y$. Further, if $E$ is one of
these equations, and it has a multiple root $\alpha$, then $\alpha$ is also a
root, of multiplicity one higher, of the equation from which $E$ is derived by
differentiation.

Let us now present the proof of Theorem 1.2.

###### Proof.

We assume $P(t)$ is a polynomial of $n$-degree, with real roots
$x_{1},x_{2},\cdots x_{n}$. Then $P(t)$ is represented as

$P(t)=\prod_{i=1}^{n}(t-x_{i})=E_{0}(x)t^{n}-C_{n}^{1}E_{1}(x)t^{n-1}+C_{n}^{2}E_{2}(x)t^{n-2}-\cdots+(-1)^{n}E_{n}(x),$

and we shall apply Lemma 2.2 to the associated homogeneous polynomial

$F(t,s)=E_{0}(x)t^{n}-C_{n}^{1}E_{1}(x)t^{n-1}s+C_{n}^{2}E_{2}(x)t^{n-2}s^{2}-\cdots+(-1)^{n}E_{n}(x)s^{n}.$

Considering the case of the derivatives$\dfrac{\partial^{n-3}F}{\partial
t^{n-2-k}\partial s^{k-1}}$(for $k=1,\cdots,n-2$), we arrive to the fact that
all the cubic polynomials

(2.2) $E_{k-1}(x)t^{3}-3E_{k}(x)t^{2}s+3E_{k+1}(x)ts^{2}-E_{k+2}(x)s^{3}$

for $k=1,\cdots,n-2$ also have real roots.

If $E_{k-1}(x)=E_{k+2}(x)=0$, it is easy to get

$\displaystyle[\alpha E_{k}(x)+E_{k+1}(x)]^{2}-[\alpha
E_{k-1}(x)+E_{k}(x)][\alpha E_{k+1}(x)+E_{k+2}(x)]$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}[\alpha
E_{k}(x)+E_{k+1}(x)]^{2}+\dfrac{\alpha^{2}}{2}E_{k}^{2}(x)+\dfrac{1}{2}E_{k+1}^{2}(x)\geq
0.$

So we divide it in two cases to deal with (1.3): $E_{k-1}(x)\neq 0$ or
$E_{k+2}(x)\neq 0$.

Case A: When $E_{k-1}(x)\neq 0$, by (2.2), the polynomial

$t^{3}-\dfrac{3E_{k}(x)}{E_{k-1}(x)}t^{2}+\dfrac{3E_{k+1}(x)}{E_{k-1}(x)}t-\dfrac{E_{k+2}(x)}{E_{k-1}(x)}$

has three real roots. We denote the roots by $z_{1},z_{2},z_{3}$ and denote
$z=(z_{1},z_{2},z_{3})$, then

$E_{1}(z)=\dfrac{E_{k}(x)}{E_{k-1}(x)},\quad
E_{2}(z)=\dfrac{E_{k+1}(x)}{E_{k-1}(x)},\quad
E_{3}(z)=\dfrac{E_{k+2}(x)}{E_{k-1}(x)}.$

Using Lemma 2.1 we obtain (1.3).

Case B: When $E_{k+2}(x)\neq 0$, by (2.2), the polynomial

$s^{3}-\dfrac{3E_{k+1}(x)}{E_{k+2}(x)}s^{2}+\dfrac{3E_{k}(x)}{E_{k+2}(x)}s-\dfrac{E_{k-1}(x)}{E_{k+2}(x)}$

has three real roots. Then the proof of (1.3) is similar to Case A.

By Lemma 2.1, the inequalities of (1.3) are strict unless
$x_{1}=x_{2}=\cdots=x_{n}$ or

$\dfrac{\alpha E_{k}(x)+E_{k+1}(x)}{\alpha E_{k-1}(x)+E_{k}(x)}=\dfrac{\alpha
E_{k+1}(x)+E_{k+2}(x)}{\alpha E_{k}(x)+E_{k+1}(x)}=-\alpha.$

A special case in which the above equality holds is that there are $n-1$
elements of $x_{1},x_{2},\cdots,x_{n}$ valued $-\alpha$.

Now, we prove (1.4). When $k=0$, we get from (1.1),

$\displaystyle[\alpha+E_{1}(x)]^{2}-[\alpha E_{1}(x)+E_{2}(x)]=$
$\displaystyle\alpha^{2}+\alpha E_{1}(x)+E_{1}^{2}(x)-E_{2}(x)$
$\displaystyle\geq$ $\displaystyle\alpha(\alpha+E_{1}(x))\geq 0.$

Therefore,

$[\alpha+E_{1}(x)]\geq[\alpha E_{1}(x)+E_{2}(x)]^{1/2}.$

For $1\leq k\leq n-2$, we assume

$[\alpha E_{k-2}(x)+E_{k-1}(x)]^{1/(k-1)}\geq[\alpha
E_{k-1}(x)+E_{k}(x)]^{1/k}.$

Combining (1.3) and the inequalities above, we get

$\displaystyle[\alpha E_{k-1}(x)+E_{k}(x)]^{2}\geq$ $\displaystyle[\alpha
E_{k-2}(x)+E_{k-1}(x)][\alpha E_{k}(x)+E_{k+1}(x)]$ $\displaystyle\geq$
$\displaystyle[\alpha E_{k-1}(x)+E_{k}(x)]^{(k-1)/k}[\alpha
E_{k}(x)+E_{k+1}(x)].$

Thus,

$[\alpha E_{k-1}(x)+E_{k}(x)]^{1/k}\geq[\alpha E_{k}(x)+E_{k+1}(x)]^{1/(k+1)}$

by canceling the common factor.

∎

##  3\. The proof of Theorem 1.3

We divide it in two cases to prove Theorem 1.3:

(i) Special cases: $k=0$, $k=n-1$ or $n=3,k=1$;

(ii) General cases: $n\geq 4,1\leq k\leq n-2$.

###  3.1. Special cases

$\bullet$ When $k=0$, choose $\theta=\dfrac{1}{2}$ in (1.7). Using the
identity $\sigma_{1}^{2}(x)=\displaystyle\sum_{i=1}^{n}x_{i}^{2}+2\sigma_{2}$,
we have

$\displaystyle\dfrac{1}{2}[\alpha+\sigma_{1}(x)]^{2}-[\alpha\sigma_{1}(x)+\sigma_{2}(x)]=$
$\displaystyle\dfrac{\alpha^{2}}{2}+\dfrac{1}{2}\sigma_{1}^{2}(x)-\sigma_{2}(x)\geq
0.$

$\bullet$ When $k=n-1$, choose $\theta=\dfrac{1}{2}$. By using the identity

$\sigma_{n-1}^{2}(x)=\displaystyle\sum_{i=1}^{n}\dfrac{\sigma_{n}^{2}(x)}{x_{i}^{2}}+2\sigma_{n-2}(x)\sigma_{n}(x),$

we get

$\displaystyle\dfrac{1}{2}[\alpha\sigma_{n-1}(x)+\sigma_{n}(x)]^{2}-[\alpha\sigma_{n-2}(x)+\sigma_{n-1}(x)]\alpha\sigma_{n}(x)$
$\displaystyle=$
$\displaystyle\dfrac{\alpha^{2}}{2}\sigma_{n-1}^{2}(x)+\dfrac{1}{2}\sigma_{n}^{2}(x)-\alpha^{2}\sigma_{n-2}(x)\sigma_{n}(x)\geq
0.$

$\bullet$ When $n=3,k=1$, choose $\theta=\dfrac{1}{2}$. Using the identities

$\sigma_{1}(x)^{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+2\sigma_{2}(x),\quad\sigma_{2}^{2}(x)=x_{1}^{2}x_{2}^{2}+x_{1}^{2}x_{3}^{2}+x_{2}^{2}x_{3}^{2}+2\sigma_{1}(x)\sigma_{3}(x),$

we obtain

$\displaystyle\dfrac{1}{2}[\alpha\sigma_{1}(x)+\sigma_{2}(x)]^{2}-[\alpha+\sigma_{1}(x)][\alpha\sigma_{2}(x)+\sigma_{3}(x)]$
$\displaystyle=$
$\displaystyle\dfrac{\alpha^{2}}{2}\sigma_{1}^{2}(x)+\dfrac{1}{2}\sigma_{2}^{2}(x)-\alpha^{2}\sigma_{2}(x)-\alpha\sigma_{3}(x)-\sigma_{1}(x)\sigma_{3}(x)$
$\displaystyle\geq$
$\displaystyle\dfrac{\alpha^{2}}{2}x_{1}^{2}+\dfrac{1}{2}x_{2}^{2}x_{3}^{2}-\alpha
x_{1}x_{2}x_{3}\geq 0.$

###  3.2. General cases

For $n\geq 4,1\leq k\leq n-2$, we denote

(3.1) $a=C_{n}^{k-1},\quad b=C_{n}^{k},\quad c=C_{n}^{k+1},\quad
d=C_{n}^{k+2}.$

Then the inequalities (1.7) can be rewritten as

$(1-\theta)[\alpha bE_{k}(x)+cE_{k+1}(x)]^{2}-[\alpha
aE_{k-1}(x)+bE_{k}(x)][\alpha cE_{k+1}(x)+dE_{k+2}(x)]\geq 0.$

From Lemma 2.2, and by a similar proof to (1.3), we shall complete the proof
Theorem 1.3 via establishing the following inequalities.

$(1-\theta)[\alpha bE_{1}(z)+cE_{2}(z)]^{2}-[\alpha a+bE_{1}(z)][\alpha
cE_{2}(z)+dE_{3}(z)]\geq 0.$

Here $z=(z_{1},z_{2},z_{3})\in\mathbb{R}^{3}$. Furthermore, it is equivalent
to prove

(3.2) $L(z):=(\alpha b\sigma_{1}+c\sigma_{2})^{2}-(3\alpha
a+b\sigma_{1})(\alpha c\sigma_{2}+3d\sigma_{3})\geq\theta(\alpha
b\sigma_{1}+c\sigma_{2})^{2},$

where

$\sigma_{1}=z_{1}+z_{2}+z_{3},\quad\sigma_{2}=z_{1}z_{2}+z_{1}z_{3}+z_{2}z_{3},\quad\sigma_{3}=z_{1}z_{2}z_{3}.$

Now we prove (3.2). Let us expand the left-hand and right-hand sides of (3.2)
separately. By using identities

$\displaystyle\sigma_{1}^{2}=z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+2\sigma_{2},\quad\sigma_{2}^{2}=z_{1}^{2}z_{2}^{2}+z_{1}^{2}z_{3}^{2}+z_{2}^{2}z_{3}^{2}+2\sigma_{1}\sigma_{3},$

we have

$\displaystyle L(z)=$ $\displaystyle\alpha^{2}b^{2}\sigma_{1}^{2}+\alpha
bc\sigma_{1}\sigma_{2}+c^{2}\sigma_{2}^{2}-3\alpha^{2}ac\sigma_{2}-9\alpha
ad\sigma_{3}-3bd\sigma_{1}\sigma_{3}$ $\displaystyle=$
$\displaystyle\alpha^{2}b^{2}\displaystyle\sum_{i=1}^{3}z_{i}^{2}+\alpha^{2}(2b^{2}-3ac)\sigma_{2}+\alpha
bc\sigma_{1}\sigma_{2}$ (3.3)
$\displaystyle+c^{2}\displaystyle\sum_{p<q}^{3}z_{p}^{2}z_{q}^{2}+(2c^{2}-3bd)\sigma_{1}\sigma_{3}-9\alpha
ad\sigma_{3},$ (3.4) $\displaystyle(\alpha b\sigma_{1}+c\sigma_{2})^{2}=$
$\displaystyle\alpha^{2}b^{2}\displaystyle\sum_{i=1}^{3}z_{i}^{2}+2\alpha^{2}b^{2}\sigma_{2}+2\alpha
bc\sigma_{1}\sigma_{2}+c^{2}\displaystyle\sum_{p<q}^{3}z_{p}^{2}z_{q}^{2}+2c^{2}\sigma_{1}\sigma_{3}.$

For any $z=(z_{1},z_{2},z_{3})\in\mathbb{R}^{3}$, we define a nonnegative
function

$\displaystyle W(z,t)=$
$\displaystyle(z_{1}-z_{2})^{2}(\alpha+tz_{3})^{2}+(z_{1}-z_{3})^{2}(\alpha+tz_{2})^{2}+(z_{2}-z_{3})^{2}(\alpha+tz_{1})^{2}.$

Expanding $W(z,t)$ we get

(3.5) $\displaystyle W(z,t)=$ $\displaystyle
2\alpha^{2}\displaystyle\sum_{i=1}^{3}z_{i}^{2}-2\alpha^{2}\sigma_{2}+2\alpha
t\sigma_{1}\sigma_{2}+2t^{2}\displaystyle\sum_{p<q}^{3}z_{p}^{2}z_{q}^{2}-2t^{2}\sigma_{1}\sigma_{3}-18\alpha
t\sigma_{3}.$

We want to absorb the terms $\sigma_{1}\sigma_{2}$, $\sigma_{1}\sigma_{3}$ and
$\sigma_{2}$ in $L(z)$ by using $(\alpha b\sigma_{1}+c\sigma_{2})^{2}$ and
$W(z,t)$. So let

(3.6) $\displaystyle L(z)=$ $\displaystyle\theta_{1}(\alpha
b\sigma_{1}+c\sigma_{2})^{2}+\theta_{2}W(z,t)+V(z),$

where $\theta_{1}$ and $\theta_{2}$ are constants which will be determined
later, $V(z)$ is the remaining part. By (3.2)-(3.6), comparing the
coefficients of $\sigma_{1}\sigma_{2}$,$\sigma_{1}\sigma_{3}$ and
$\sigma_{2}$, we have

$\left\\{\begin{array}[]{ll}bc=2bc\theta_{1}+2t\theta_{2},&\quad(\sigma_{1}\sigma_{2})\\\
2c^{2}-3bd=2c^{2}\theta_{1}-2t^{2}\theta_{2},&\quad(\sigma_{1}\sigma_{3})\\\
(2b^{2}-3ac)=2b^{2}\theta_{1}-2\theta_{2}.&\quad(\sigma_{2})\end{array}\right.$

We solve the equations above to find

(3.7)
$\displaystyle\theta_{1}=\dfrac{3(2ac^{3}+2b^{3}d-b^{2}c^{2}-3abcd)}{6ac^{3}+6b^{3}d-4b^{2}c^{2}},~{}\theta_{2}=\dfrac{c^{2}(3ac-b^{2})^{2}}{6ac^{3}+6b^{3}d-4b^{2}c^{2}},~{}t=\dfrac{b(3bd-c^{2})}{c(3ac-b^{2})}.$

Using (3.6) again, we obtain

$\displaystyle V(z)=$ $\displaystyle
A_{1}\alpha^{2}\displaystyle\sum_{i=1}^{3}z_{i}^{2}+A_{2}\displaystyle\sum_{p<q}^{3}z_{p}^{2}z_{q}^{2}+A_{3}\alpha\sigma_{3},$

where

$\displaystyle A_{1}=$ $\displaystyle b^{2}-b^{2}\theta_{1}-2\theta_{2},\quad
A_{2}=c^{2}-c^{2}\theta_{1}-2t^{2}\theta_{2},\quad A_{3}=18t\theta_{2}-9ad.$

We claim that For $n\geq 4,1\leq k\leq n-2$,

(3.8) $\displaystyle\theta_{1}>0,\quad\theta_{2}>0,\quad V(z)\geq 0.$

Notice that $W(z,t)$ is nonnegative, so by (3.6) we obtain (3.2). That is,
Theorem 1.3 is proved. Next, we will prove the claim.

######  Lemma 3.1.

For $n\geq 4$, $1\leq k\leq n-2$, we have

(i) $3bd-c^{2}>0$, $3ac-b^{2}>0$;

(ii) $2ac^{3}+2b^{3}d-b^{2}c^{2}-3abcd>0$.

###### Proof.

(i) By (3.1) we have

(3.9)
$\dfrac{a}{b}=\dfrac{C_{n}^{k-1}}{C_{n}^{k}}=\dfrac{k}{n-k+1},\quad\dfrac{b}{c}=\dfrac{C_{n}^{k}}{C_{n}^{k+1}}=\dfrac{k+1}{n-k},\quad\dfrac{c}{d}=\dfrac{C_{n}^{k+1}}{C_{n}^{k+2}}=\dfrac{k+2}{n-k-1}.$

By (3.9), one has

$\displaystyle 3bd-c^{2}=$ $\displaystyle
c^{2}(\dfrac{3bd}{c^{2}}-1)=c^{2}\left(\dfrac{3(k+1)(n-k-1)}{(n-k)(k+2)}-1\right)$
(3.10) $\displaystyle=$ $\displaystyle
c^{2}\cdot\dfrac{-2k^{2}+2(n-2)k+(n-3)}{(n-k)(k+2)}.$

Since the following quadratic function

$\displaystyle f_{1}(k):=-2k^{2}+2(n-2)k+(n-3)$

with respect to variable $k$ is increasing on the interval
$[1,\dfrac{n-2}{2}]$, decreasing on the interval $[\dfrac{n-2}{2},n-2]$, and

$f_{1}(1)=3(n-3)>0,\quad f_{1}(n-2)=n-3>0,$

$f_{1}(k)>0$ on $[1,n-2]$. Combining (3.2), we obtain $3bd-c^{2}>0$.

Similarly,

$3ac-b^{2}=b^{2}\cdot\dfrac{-2k^{2}+2nk-n-1}{(n-k+1)(k+1)},$

and the function $-2k^{2}+2nk-n-1$ respect to variable $k$ is positive on
$[1,n-2]$, so we get $3ac-b^{2}>0$.

(ii) By (3.9), we have

$\displaystyle 2ac^{3}+2b^{3}d-b^{2}c^{2}-3abcd=$ $\displaystyle
b^{2}c^{2}(\dfrac{2ac}{b^{2}}+\dfrac{2bd}{c^{2}}-1-\dfrac{3ad}{bc})$
$\displaystyle=$ $\displaystyle
b^{2}c^{2}\cdot\dfrac{2(n+1)(-k^{2}+kn-k-1)}{(k+1)(k+2)(n-k+1)(n-k)}.$

It is easy to get $-k^{2}+(n-1)k-1>0$ for $1\leq k\leq n-2$.

∎

######  Lemma 3.2.

For $n\geq 4$, $1\leq k\leq n-2$, we have

(i) $A_{1}>0$;

(ii) $A_{2}>0$;

(iii) $A_{1}A_{2}-A_{3}^{2}/36>0$.

###### Proof.

(i) Substituting (3.7) into $A_{1}$, we have

$\displaystyle A_{1}=$ $\displaystyle
b^{2}(1-\theta_{1})-2\theta_{2}=\dfrac{9ab^{3}cd-b^{4}c^{2}}{6ac^{3}+6b^{3}d-4b^{2}c^{2}}-\dfrac{2c^{2}(3ac-b^{2})^{2}}{6ac^{3}+6b^{3}d-4b^{2}c^{2}}$
$\displaystyle=$
$\displaystyle\dfrac{3c(4ab^{2}c^{2}+3ab^{3}d-6a^{2}c^{3}-b^{4}c)}{6ac^{3}+6b^{3}d-4b^{2}c^{2}}$
(3.11) $\displaystyle=$
$\displaystyle\dfrac{3ab^{2}c^{3}}{2c^{2}(3ac-b^{2})+2b^{2}(3bd-c^{2})}\cdot(4+\dfrac{3bd}{c^{2}}-\dfrac{6ac}{b^{2}}-\dfrac{b^{2}}{ac}).$

By (3.9),

$\displaystyle(4+\dfrac{3bd}{c^{2}}-\dfrac{6ac}{b^{2}}-\dfrac{b^{2}}{ac})=$
$\displaystyle\dfrac{2(n+1)[-k^{3}+(n-5)k^{2}+(3n-2)k-n-1]}{k(k+1)(k+2)(n-k+1)(n-k)}.$

Let us show the cubic function

$f_{2}(k):=-k^{3}+(n-5)k^{2}+(3n-2)k-n-1>0.$

Differentiating $f_{2}(k)$, we solve the quadratic equation

$f_{2}^{\prime}(k)=-3k^{2}+2(n-5)k+3n-2=0.$

The quadratic formula gives

$k_{1}=\dfrac{2(n-5)-\sqrt{4n^{2}-4n+76}}{6},\quad
k_{2}=\dfrac{2(n-5)+\sqrt{4n^{2}-4n+76}}{6}.$

It is easy to get

$k_{1}<\dfrac{2(n-5)-(2n-1)}{6}<0,$
$\dfrac{4n-11}{6}=\dfrac{2(n-5)+(2n-1)}{6}<k_{2}<\dfrac{2(n-5)+3n}{6}=\dfrac{5(n-2)}{6},$

and $f_{2}^{\prime}(k)\geq 0$ on $[1,k_{2}]$, $f_{2}^{\prime}(k)\leq 0$ on
$[k_{2},n-2]$. So function $f_{2}(k)$ is increasing on $[1,k_{2}]$ and
decreasing on $[k_{2},n-2]$. On the other hand,
$f_{2}(1)=f_{2}(n-2)=3(n-3)>0$, this implied $f_{2}(k)>0$ on $[1,n-2]$.
Combining (3.2) and (i) of Lemma 3.1, we obtain $A_{1}>0$.

(ii) Substituting (3.7) into $A_{2}$, we have

$\displaystyle A_{2}=$ $\displaystyle
c^{2}(1-\theta_{1})-2t^{2}\theta_{2}=\dfrac{9abc^{3}d-b^{2}c^{4}}{6ac^{3}+6b^{3}d-4b^{2}c^{2}}-\dfrac{2b^{2}(3bd-c^{2})^{2}}{6ac^{3}+6b^{3}d-4b^{2}c^{2}}$
$\displaystyle=$
$\displaystyle\dfrac{3b(4b^{2}c^{2}d+3ac^{3}d-6b^{3}d^{2}-bc^{4})}{6ac^{3}+6b^{3}d-4b^{2}c^{2}}$
(3.12) $\displaystyle=$
$\displaystyle\dfrac{3b^{3}c^{2}d}{2c^{2}(3ac-b^{2})+2b^{2}(3bd-c^{2})}\cdot(4+\dfrac{3ac}{b^{2}}-\dfrac{6bd}{c^{2}}-\dfrac{c^{2}}{bd}).$

By using (3.9), we get

$\displaystyle(4+\dfrac{3ac}{b^{2}}-\dfrac{6bd}{c^{2}}-\dfrac{c^{2}}{bd})=$
$\displaystyle\dfrac{2(n+1)[k^{3}-(2n+2)k^{2}+(n^{2}+3n-5)k-n^{2}+2n-3]}{(k+1)(k+2)(n-k+1)(n-k)(n-k-1)}.$

Denote

$f_{3}(k):=k^{3}-(2n+2)k^{2}+(n^{2}+3n-5)k-n^{2}+2n-3.$

Differentiating $f_{3}(k)$ and solving the quadratic equation

$f_{3}^{\prime}(k)=3k^{2}-2(2n+2)k+n^{2}+3n-5=0,$

we get

$k_{1}=\dfrac{2(2n+2)-\sqrt{4n^{2}-4n+76}}{6},\quad
k_{2}=\dfrac{2(2n+2)+\sqrt{4n^{2}-4n+76}}{6}.$

Obviously,

$\dfrac{n+4}{6}=\dfrac{2(2n+2)-3n}{6}<k_{1}<\dfrac{2(2n+2)-(2n-1)}{6}=\dfrac{2n+5}{6},$
$k_{2}>\dfrac{2(2n+2)+2n-1}{6}>n,$

and $f_{3}^{\prime}(k)\geq 0$ on $[1,k_{1}]$, $f_{3}^{\prime}(k)\leq 0$ on
$[k_{1},n-2]$. So function $f_{3}(k)$ is increasing on $[1,k_{1}]$ and
decreasing on $[k_{1},n-2]$. Since $f_{3}(1)=f_{3}(n-2)=3(n-3)>0$, we have
$f_{3}(k)>0$ on $[1,n-2]$. Combining (3.2) and (i) of Lemma 3.1, we obtain
$A_{2}>0$.

(iii) Substituting (3.7) into $A_{3}$, we have

$\displaystyle A_{3}=$ $\displaystyle
18t\theta_{2}-9ad=\dfrac{9bc(3ac-b^{2})(3bd-c^{2})}{3ac^{3}+3b^{3}d-2b^{2}c^{2}}-9ad$
$\displaystyle=$
$\displaystyle\dfrac{9(b^{3}c^{3}-3abc^{4}-3b^{4}cd+11ab^{2}c^{2}d-3a^{2}c^{3}d-3ab^{3}d^{2})}{3ac^{3}+3b^{3}d-2b^{2}c^{2}}.$

By using the expressions $A_{1}$ and $A_{2}$ in (3.2) and (3.2) respectively,
we have

$\displaystyle A_{1}A_{2}-A_{3}^{2}/36$ $\displaystyle=$
$\displaystyle\dfrac{9(12ab^{3}c^{3}d+20a^{2}b^{2}c^{2}d^{2}-ab^{2}c^{5}-b^{5}c^{2}d-12a^{2}bc^{4}d-12ab^{4}cd^{2}-3a^{3}c^{3}d^{2}-3a^{2}b^{3}d^{3})}{4(3ac^{3}+3b^{3}d-2b^{2}c^{2})}$
$\displaystyle=$
$\displaystyle\dfrac{9a^{2}b^{2}c^{2}d^{2}}{4(3ac^{3}+3b^{3}d-2b^{2}c^{2})}\cdot\left(\dfrac{12bc}{ad}+20-\dfrac{c^{3}}{ad^{2}}-\dfrac{b^{3}}{a^{2}d}-\dfrac{12c^{2}}{bd}-\dfrac{12b^{2}}{ac}-\dfrac{3ac}{b^{2}}-\dfrac{3bd}{c^{2}}\right).$

From (3.9), we get

$\displaystyle(\dfrac{12bc}{ad}+20-\dfrac{c^{3}}{ad^{2}}-\dfrac{b^{3}}{a^{2}d}-\dfrac{12c^{2}}{bd}-\dfrac{12b^{2}}{ac}-\dfrac{3ac}{b^{2}}-\dfrac{3bd}{c^{2}})$
$\displaystyle=$
$\displaystyle\dfrac{4(n+1)^{2}[-(n+5)k^{2}+(n^{2}+4n-5)k-n^{2}+1]}{k^{2}(k+1)(k+2)(n-k+1)(n-k)(n-k-1)^{2}}.$

Denote

$f_{4}(k):=-(n+5)k^{2}+(n^{2}+4n-5)k-n^{2}+1.$

It is easy to get that function $f_{4}(k)$ is increasing on
$\Big{[}1,\dfrac{n-1}{2}\Big{]}$ and decreasing on
$\Big{[}\dfrac{n-1}{2},n-2\Big{]}$, and

$f_{4}(1)=f_{4}(n-2)=3(n-3)>0,$

so we have $f_{4}(k)>0$ for $1\leq k\leq n-2$. Combining formulas above we
obtain

$A_{1}A_{2}-A_{3}^{2}/36>0.$

∎

$\bullet$ Proof of the Claim (3.8).

Lemma 3.1 implies $\theta_{1}>0,\theta_{2}>0$. By Lemma 3.2,

$\displaystyle A_{1}\alpha^{2}z_{1}^{2}+A_{2}z_{2}^{2}z_{3}^{2}\geq$
$\displaystyle
2\sqrt{A_{1}A_{2}}|\alpha\sigma_{3}|>\dfrac{1}{3}|A_{3}\alpha\sigma_{3}|,$
$\displaystyle A_{1}\alpha^{2}z_{2}^{2}+A_{2}z_{1}^{2}z_{3}^{2}\geq$
$\displaystyle
2\sqrt{A_{1}A_{2}}|\alpha\sigma_{3}|>\dfrac{1}{3}|A_{3}\alpha\sigma_{3}|,$
$\displaystyle A_{1}\alpha^{2}z_{3}^{2}+A_{2}z_{1}^{2}z_{2}^{2}\geq$
$\displaystyle
2\sqrt{A_{1}A_{2}}|\alpha\sigma_{3}|>\dfrac{1}{3}|A_{3}\alpha\sigma_{3}|.$

Combing the inequalities above, we get

$\displaystyle V(z)=$ $\displaystyle
A_{1}\alpha^{2}\displaystyle\sum_{i=1}^{3}z_{i}^{2}+A_{2}\displaystyle\sum_{p<q}^{3}z_{p}^{2}z_{q}^{2}+A_{3}\alpha\sigma_{3}\geq
0.$

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$\mathrm{Mor}_{C_{{\mathcal{L}}}(E)}(P,P^{\prime})$ are the set of morphisms
$\varphi\in\mathrm{Mor}_{{\mathcal{L}}}(PE,P^{\prime}E)$ whose underlying
homomorphisms are the identity on $E$ and send $P$ into $P^{\prime}$ (see [6,
Def 2.4]). By [6, Prop 2.5], the category $C_{{\mathcal{L}}}(E)$ is the
centric linking system associated to $C_{{\mathcal{F}}}(E)$.

For every $E\in\mathrm{Ob}({\mathcal{F}}^{e})$, let
$\widetilde{C}_{{\mathcal{L}}}(E)$ denote the category whose objects are the
pairs $(P,\alpha)$ with $P\in{\mathcal{F}}^{c}$ and
$\alpha\in\mathrm{Mor}_{{\mathcal{F}}}(E,Z(P))$. The morphisms in
$\widetilde{C}_{{\mathcal{L}}}(E)$ are defined by

$\mathrm{Mor}_{\widetilde{C}_{{\mathcal{L}}}(E)}((P,\alpha),(Q,\beta)):=\\{\varphi\in\mathrm{Mor}_{{\mathcal{L}}}(P,Q)\
|\ \pi(\varphi)\circ\alpha=\beta\\}$

where $\pi:{\mathcal{L}}\to{\mathcal{F}}^{c}$ denotes the canonical projection
functor. There is a natural map

$f:\operatorname*{hocolim}_{E\in({\mathcal{F}}^{e})^{op}}|\widetilde{C}_{{\mathcal{L}}}(E)|\smash{\,\mathop{\longrightarrow}\limits\,}|{\mathcal{L}}|$

induced by the forgetful functors $(P,\alpha)\to P$. It is proved by Broto,
Levi, and Oliver [6, Thm 2.6] that $f$ is a homotopy equivalence, and that
there is a homotopy equivalence
$|C_{{\mathcal{L}}}(E)|\to|\widetilde{C}_{{\mathcal{L}}}(E)|$ induced by the
functor $P\to(P,incl)$. This homotopy equivalence given by $f$ is called _the
centralizer decomposition_ for $(S,{\mathcal{F}},{\mathcal{L}})$ over the
collection of all nontrivial elementary abelian subgroups of $S$.

Associated to this decomposition, there is a Bousfield-Kan spectral sequence

$E_{2}^{s,t}=\underset{({\mathcal{F}}^{e})^{op}}{\lim{}^{s}}H^{t}(|C_{{\mathcal{L}}}(-)|;{\mathbb{F}}_{p})\Rightarrow
H^{s+t}(|{\mathcal{L}}|;{\mathbb{F}}_{p}).$

The centralizer decomposition is _sharp_ if $E_{2}^{s,t}=0$ for $s\geq 1$.
When the centralizer decomposition is sharp, then there is an isomorphism

$H^{*}(|{\mathcal{L}}|;{\mathbb{F}}_{p})\cong\underset{E\in({\mathcal{F}}^{e})^{op}}{\lim}H^{*}(|C_{{\mathcal{L}}}(E);{\mathbb{F}}_{p}).$

Broto, Levi, and Oliver [6] proved that the centralizer decomposition for
every $p$-local finite group is sharp. We can also state their result in terms
of the cohomology of the centralizer fusion systems.

######  Theorem 9.4 (Broto-Levi-Oliver [6], p. 821).

Let ${\mathcal{F}}$ be a saturated fusion system over $S$. Then, for every
$n\geq 0$, and every $i\geq 1$,

$\underset{({\mathcal{F}}^{e})^{op}}{\lim{}^{i}}\
H^{n}(C_{{\mathcal{F}}}(-);{\mathbb{F}}_{p})=0.$

Recall that a subgroup $Q\leq S$ is _central_ in ${\mathcal{F}}$ if
$C_{{\mathcal{F}}}(Q)={\mathcal{F}}$. The product of all central subgroups in
${\mathcal{F}}$ is called the center of ${\mathcal{F}}$ and denoted by
$Z({\mathcal{F}})$. The main result of this section is the following:

######  Theorem 9.5.

If the subgroup decomposition over ${\mathcal{F}}$-centric subgroups is sharp
for every $p$-local finite group $(S,{\mathcal{F}},{\mathcal{L}})$ with
$Z({\mathcal{F}})\neq 1$, then it is sharp for every $p$-local finite group.

###### Proof.

Let $n\geq 0$ be a fixed integer, and let $H^{n}(-;{\mathbb{F}}_{p})$ denote
the group cohomology functor considered as a module over
${\mathcal{O}}^{c}({\mathcal{F}})$. By Theorem 9.3, there is a spectral
sequence

$E_{2}^{s,t}=\underset{({\mathcal{F}}^{e})^{op}}{\lim{}^{s}}\
H^{t}({\mathcal{O}}^{c}(C_{{\mathcal{F}}}(-));H^{n}(-;{\mathbb{F}}_{p}))\Rightarrow
H^{s+t}({\mathcal{O}}^{c}({\mathcal{F}});H^{n}(-;{\mathbb{F}}_{p})).$

For every $E\in\mathrm{Ob}({\mathcal{F}}^{e})$, the centralizer fusion system
$C_{{\mathcal{F}}}(E)$ has a nontrivial center, hence by assumption, the
subgroup decomposition for
$(C_{S}(E),C_{{\mathcal{F}}}(E),C_{{\mathcal{L}}}(L))$ is sharp. This gives
that $E_{2}^{s,t}=0$ for all $t\geq 1$, and for $t=0$, we have

$E_{2}^{s,0}=\underset{({\mathcal{F}}^{e})^{op}}{\lim{}^{s}}H^{0}({\mathcal{O}}^{c}(C_{{\mathcal{F}}}(-));H^{n}(-;{\mathbb{F}}_{p}))\cong\underset{({\mathcal{F}}^{e})^{op}}{\lim{}^{s}}H^{n}(C_{{\mathcal{F}}}(-);{\mathbb{F}}_{p}).$

By Theorem 9.4, these higher limits vanish for $s\geq 1$, hence
$H^{i}({\mathcal{O}}^{c}({\mathcal{F}});H^{n}(-;{\mathbb{F}}_{p}))=0$ for all
$i\geq 1$. ∎

Another consequence of the spectral sequence argument given above is the
following: Given a $p$-local group $(S,{\mathcal{F}},{\mathcal{L}})$, if we
know that

  1. (1)

the subgroup decomposition for $(S,{\mathcal{F}},{\mathcal{L}})$ is sharp, and

  2. (2)

for every $E\in\mathrm{Ob}({\mathcal{F}}^{e})$, the subgroup decomposition for
$(C_{S}(E),C_{{\mathcal{F}}}(E),C_{{\mathcal{L}}}(L))$ is sharp,

then we can conclude that the centralizer decomposition for
$(S,{\mathcal{F}},{\mathcal{L}})$ over the nontrivial elementary abelian
$p$-subgroups is sharp. We can consider this as a propagation result between
two different types of homology decompositions in the sense given by Grodal
and Smith in [15].

##  10\. The normalizer decomposition for ${\mathcal{O}}^{c}({\mathcal{F}})$

Let ${\mathcal{F}}$ be a saturated fusion system over $S$, and let
$sd({\mathcal{F}}^{c})$ denote the poset category of all chains
$\sigma=(P_{0}<P_{1}<\cdots<P_{n})$ with
$P_{i}\in\mathrm{Ob}({\mathcal{F}}^{c})$ where there is a unique morphism
$\sigma\to\tau$ in $sd({\mathcal{F}}^{c})$ if $\tau$ is a subchain of
$\sigma$. Two chains $\sigma=(P_{0}<P_{1}<\cdots<P_{n})$ and
$\tau=(Q_{0}<Q_{1}<\cdots<Q_{n})$ are ${\mathcal{F}}$-conjugate if there is an
isomorphism $\varphi:P_{n}\to Q_{n}$ in ${\mathcal{F}}$ such that
$\varphi(P_{i})=Q_{i}$ for all $i$. In this case we write
$\sigma\sim_{{\mathcal{F}}}\tau$. The ${\mathcal{F}}$-conjugacy class of a
chain $\sigma$ in $sd({\mathcal{F}}^{c})$ is denoted by $[\sigma]$.

######  Definition 10.1.

The category of _${\mathcal{F}}$ -conjugacy classes of chains in
${\mathcal{F}}^{c}$_, denoted by ${\overline{s}}d({\mathcal{F}}^{c})$, is the
poset category whose objects are ${\mathcal{F}}$-conjugacy classes $[\sigma]$
of chains of subgroups in $\mathrm{Ob}({\mathcal{F}}^{c})$ where there is a
unique morphism $[\sigma]\to[\tau]$ in ${\overline{s}}d({\mathcal{F}}^{c})$ if
$\tau$ is ${\mathcal{F}}$-conjugate to a subchain of $\sigma$.

Let $G$ be a discrete group with a Sylow $p$-subgroup $S$ realizing the fusion
system ${\mathcal{F}}$. If ${\mathcal{C}}$ is the collection of all
$p$-centric subgroups in $G$, then we have
${\mathcal{F}}^{c}\cong{\mathcal{F}}_{{\mathcal{C}}}(G)$. We defined the
category of orbit simplices of $G$, denoted by $Sd_{{\mathcal{C}}}(G)/G$, to
be the category whose objects are the $G$-orbits $[\sigma]$ of the chains of
subgroups in ${\mathcal{C}}$ and such that there is a unique morphism
$[\sigma]\to[\tau]$ in $Sd_{{\mathcal{C}}}(G)/G$ if there is an element $g\in
G$ such that $g\tau$ is a face of $\sigma$ (see Definition 6.3). It is easy to
see that since ${\mathcal{F}}^{c}\cong{\mathcal{F}}_{{\mathcal{C}}}(G)$, the
poset categories $Sd_{{\mathcal{C}}}(G)/G$ and
$\overline{s}d({\mathcal{F}}^{c})$ are isomorphic.

If $\sigma=(P_{0}<\cdots<P_{n})$ is a chain of subgroups of a discrete group
$G$, then the stabilizer of $\sigma$ under the $G$-action defined by
conjugation is the subgroup $N_{G}(\sigma)=\bigcap_{i}N_{G}(P_{i})$. Note that
$C_{G}(P_{n})\leq N_{G}(\sigma)$ is a normal subgroup and the quotient group
$N_{G}(\sigma)/C_{G}(P_{n})$ is a subgroup of $N_{G}(P_{n})/C_{G}(P_{n})={\rm
Aut}_{G}(P_{n})\leq{\rm Aut}(P_{n})$. We define the $G$-automorphism group of
$\sigma$ to be the subgroup

${\rm Aut}_{G}(\sigma):=N_{G}(\sigma)/C_{G}(P_{n})\leq{\rm Aut}(P_{n}).$

For a chain of $p$-groups $\sigma=(P_{0}<\cdots<P_{n})$ in $S$, the
automorphism group of $\sigma$ is defined by

${\rm Aut}(\sigma):=\\{\alpha\in{\rm Aut}(P_{n})\,|\,\alpha(P_{i})=P_{i}\text{
for all }i\\}\leq{\rm Aut}(P_{n}).$

In Section 8, for every subgroup $K\leq{\rm Aut}(Q)$, we defined the
$K$-normalizer group $N_{G}^{K}(Q)$ and the $K$-normalizer fusion system
$N_{{\mathcal{F}}}^{K}(Q)$ (see Definition 8.1). It is easy to see that if we
take $K={\rm Aut}(\sigma)\leq{\rm Aut}(P_{n})$, then
$N_{G}^{K}(P_{n})=N_{G}(\sigma)$. For the $K$-normalizer fusion system
$N_{{\mathcal{F}}}^{K}(P_{n})$, we have the following alternative description.

######  Definition 10.2.

Let ${\mathcal{F}}$ be a saturated fusion system over $S$, and let
$\sigma=(P_{0}<\cdots<P_{n})$ be a chain of subgroups in $S$. The _normalizer
fusion system_ $N_{{\mathcal{F}}}(\sigma)$ is the fusion system over
$N_{S}(\sigma)$ whose morphisms
$\mathrm{Mor}_{N_{{\mathcal{F}}}(\sigma)}(P,R)$ are all the morphisms
$\varphi\in\mathrm{Mor}_{{\mathcal{F}}}(P,R)$ which extend to a morphism
$\widetilde{\varphi}:P_{n}P\to P_{n}R$ in ${\mathcal{F}}$ in such a way that
$\widetilde{\varphi}(P_{i})=P_{i}$ for all $i\in\\{1,\cdots,n\\}$.

We say a chain $\sigma=(P_{0}<\cdots<P_{n})$ in $S$ is _fully
${\mathcal{F}}$-normalized_ if $|N_{S}(\sigma)|\geq|N_{S}(\tau)|$ for every
$\tau\sim_{{\mathcal{F}}}\sigma$. Note that this coincides with the definition
of fully $K$-normalized in Section 8 for the subgroup $K={\rm
Aut}(\sigma)\leq{\rm Aut}(P_{n})$. Using the results of Section 8 we conclude
the following proposition which was also proved in [22, Prop 3.9] using a
different approach.

######  Proposition 10.3.

Let ${\mathcal{F}}$ be a saturated fusion system over $S$, and let
$\sigma=(P_{0}<\cdots<P_{n})$ be a chain of subgroups in $S$ which is fully
${\mathcal{F}}$-normalized. Then

  1. (1)

the normalizer fusion system $N_{{\mathcal{F}}}(\sigma)$ is saturated, and

  2. (2)

if ${\mathcal{F}}={\mathcal{F}}_{S}(G)$ for a discrete group $G$ with a Sylow
$p$-subgroup $S$, then $N_{S}(\sigma)$ is a Sylow $p$-subgroup of
$N_{G}(\sigma)$ and
${\mathcal{F}}_{N_{S}(\sigma)}(N_{G}(\sigma))=N_{{\mathcal{F}}}(\sigma)$.

###### Proof.

The first statement follows from Theorem 8.2 and the second part follows from
Propositions 8.4 and 8.6. ∎

Let $\sigma$ be a chain of subgroups in ${\mathcal{F}}^{c}$ which is fully
${\mathcal{F}}$-normalized. Let ${\mathcal{N}}:=N_{{\mathcal{F}}}(\sigma)$,
and let ${\mathcal{N}}^{cn}$ denote the full subcategory of ${\mathcal{N}}$
generated by subgroups of $N_{S}(\sigma)$ which are ${\mathcal{F}}$-centric.
By Lemma 8.10, if $P\leq S$ is ${\mathcal{F}}$-centric then it is also
${\mathcal{N}}$-centric, hence ${\mathcal{N}}^{cn}$ is a full subcategory of
${\mathcal{N}}^{c}$. By Lemma 8.13, if $P$ is an ${\mathcal{N}}$-centric-
radical subgroup in $N_{S}(\sigma)$, then it is ${\mathcal{F}}$-centric. Let
${\mathcal{N}}^{cr}$ denote the full subcategory of ${\mathcal{N}}$ generated
by ${\mathcal{N}}$-centric-radical subgroups of $N_{S}(\sigma)$. Then we have

${\mathcal{N}}^{cr}\subseteq{\mathcal{N}}^{cn}\subseteq{\mathcal{N}}^{c}.$

For higher limits over different collections of subgroups, we have the
following result.

######  Proposition 10.4 ([6, Cor 3.6]).

Let ${\mathcal{F}}$ be a saturated fusion system over $S$. Let ${\mathcal{C}}$
be the collection of subgroups of $S$ closed under taking overgroups such that
$\mathrm{Ob}({\mathcal{F}}^{cr})\subseteq{\mathcal{C}}\subseteq\mathrm{Ob}({\mathcal{F}}^{c})$.
Then for every ${\mathbb{Z}}_{(p)}{\mathcal{O}}^{c}({\mathcal{F}})$-module
$M$, and for every $i\geq 0$, there is an isomorphism

$\underset{{\mathcal{O}}({\mathcal{F}}_{{\mathcal{C}}})}{\lim{}^{i}}M\cong\underset{{\mathcal{O}}^{c}({\mathcal{F}})}{\lim{}^{i}}M.$

###### Proof.

We can filter the module $M$ with atomic functors, i.e. with functors that
vanish except on a single ${\mathcal{F}}$-conjugacy class, and prove the
result by induction using long exact sequences. Hence it is enough to prove
the above isomorphism when $M$ is an atomic functor.

Let $Q\in\mathrm{Ob}({\mathcal{F}}^{c})$ be such that $M$ vanishes on
subgroups outside of the ${\mathcal{F}}$-conjugacy class of $Q$. If
$Q\in{\mathcal{C}}$, then by [26, Lemma 1.5(a)] we can conclude that the above
isomorphism holds. So, assume that $Q\not\in{\mathcal{C}}$. In this case the
restriction of $M$ to the subcategory
${\mathcal{O}}({\mathcal{F}}_{{\mathcal{C}}})$ is zero. So in this case we
need to show that
$\underset{{\mathcal{O}}^{c}({\mathcal{F}})}{\lim{}^{i}}M=0$.

Let $\Gamma$ be a finite group and $N$ be a ${\mathbb{Z}}_{(p)}G$-module. Let
$F_{N}$ denote the ${\mathcal{O}}_{p}(\Gamma)$-module such that $F_{N}(1)=N$
and $F(P)=0$ for all $p$-subgroups $1\neq P\leq\Gamma$. We define

$\Lambda^{*}(\Gamma;N):=\underset{{\mathcal{O}}_{p}(\Gamma)}{\lim{}^{*}}F_{N}.$

By [6, Prop 3.2], there is an isomorphism

$\underset{{\mathcal{O}}^{c}({\mathcal{F}})}{\lim{}^{*}}M\cong\Lambda^{*}(\mathrm{Out}_{{\mathcal{F}}}(Q);M(Q)).$

Since $Q$ is not ${\mathcal{F}}$-radical, the group
$\mathrm{Out}_{{\mathcal{F}}}(Q)$ has a normal $p$-subgroup. Then by [19, Prop
6.1(ii)] we have $\Lambda^{*}(\mathrm{Out}_{{\mathcal{F}}}(Q);M(Q))=0$. Hence
the proof is complete. ∎

Now, we are ready to prove the following.

######  Proposition 10.5.

Let $\sigma=(P_{0}<\cdots<P_{n})$ be a fully ${\mathcal{F}}$-normalized chain
of subgroups of $S$ with $P_{i}\in\mathrm{Ob}({\mathcal{F}}^{c})$. Let $G$ be
a discrete group realizing ${\mathcal{F}}$, and ${\mathcal{C}}$ be the
collection of all $p$-centric subgroups in $G$. Then for every
${\mathbb{F}}_{p}\overline{\mathcal{F}}_{{\mathcal{C}}}(G)$-module $M$, there
is an isomorphism

$H^{*}(\overline{\mathcal{F}}_{{\mathcal{C}}}(N_{G}(\sigma));M)\cong
H^{*}({\mathcal{O}}^{c}(N_{{\mathcal{F}}}(\sigma));M).$

###### Proof.

By Proposition 10.3, the fusion orbit category
$\overline{\mathcal{F}}_{{\mathcal{C}}}(N_{G}(\sigma))$ is equivalent to the
orbit category ${\mathcal{O}}^{cn}({\mathcal{N}})$ where
${\mathcal{N}}=N_{{\mathcal{F}}}(\sigma)$. Since
${\mathcal{N}}^{cr}\subseteq{\mathcal{N}}^{cn}\subseteq{\mathcal{N}}^{c}$,
applying Proposition 10.4 to the fusion system ${\mathcal{N}}$, we obtain the
desired isomorphism. ∎

As a consequence we obtain the following theorem (stated as Theorem 1.7 in the
introduction).

######  Theorem 10.6.

Let ${\mathcal{F}}$ be a saturated fusion system over $S$. Then for every
${\mathbb{F}}_{p}{\mathcal{O}}^{c}({\mathcal{F}})$-module $M$, there is a
spectral sequence

$E_{2}^{s,t}\cong\underset{\overline{s}d({\mathcal{F}}^{c})}{\lim{}^{s}}H^{t}(O^{c}(N_{{\mathcal{F}}}(\sigma));M)\Rightarrow
H^{s+t}({\mathcal{O}}^{c}({\mathcal{F}});M).$

###### Proof.

Let $G$ be a discrete group with a Sylow $p$-subgroup $S$ such that
${\mathcal{F}}\cong{\mathcal{F}}_{S}(G)$. Let ${\mathcal{C}}$ be the
collection of all $p$-centric subgroups in $G$. The collection ${\mathcal{C}}$
is closed under taking overgroups, hence it is closed under taking products.
By Proposition 7.8, there is a spectral sequence

$E_{2}^{s,t}=\underset{Sd_{{\mathcal{C}}}(G)/G}{\lim{}^{s}}H^{t}(\overline{\mathcal{F}}_{{\mathcal{C}}}(N_{G}(\sigma));M)\Rightarrow
H^{s+t}(\overline{\mathcal{F}}_{{\mathcal{C}}}(G);M).$

Since ${\mathcal{F}}^{c}\cong{\mathcal{F}}_{{\mathcal{C}}}(G)$, the poset
categories $Sd_{{\mathcal{C}}}(G)/G$ and $\overline{s}d({\mathcal{F}}^{c})$
are isomorphic. In each ${\mathcal{F}}$-conjugacy class
$[\sigma]\in\overline{s}d({\mathcal{F}}^{c})$, we can choose a chain $\sigma$
which is fully ${\mathcal{F}}$-normalized, so we can apply Proposition 10.5 to
replace $H^{t}(\overline{\mathcal{F}}_{{\mathcal{C}}}(N_{G}(\sigma));M)$ with
$H^{t}({\mathcal{O}}^{c}(N_{{\mathcal{F}}}(\sigma));M)$ for every
$[\sigma]\in\overline{s}d({\mathcal{F}}^{c})$. We also have
$\overline{\mathcal{F}}_{{\mathcal{C}}}(G)\cong{\mathcal{O}}^{c}({\mathcal{F}})$.
Hence applying Proposition 3.7, we obtain the desired spectral sequence. ∎

There is a normalizer decomposition for a $p$-local finite group introduced by
Libman [21]. For each chain $\sigma=(P_{0}<\dots<P_{n})$, let ${\rm
Aut}_{{\mathcal{L}}}(\sigma)$ denote the subgroup of $\prod_{i=1}^{n}{\rm
Aut}_{{\mathcal{L}}}(P_{i})$ formed by tuples $(\alpha_{0},\dots,\alpha_{n})$
of automorphisms $\alpha_{i}\in{\rm Aut}_{{\mathcal{L}}}(P_{i})$ such that
$\alpha_{i}|_{P_{i-1}}=\alpha_{i-1}$ for every $i\in\\{1,\dots,n\\}$. In
${\mathcal{L}}$, every morphism is a monomorphism and an epimorphism in the
categorical sense (see [21, 2.10 and Prop 2.11]). As a consequence, the map
$\pi_{0}:{\rm Aut}_{{\mathcal{L}}}(\sigma)\to{\rm Aut}_{{\mathcal{L}}}(P_{0})$
defined by $(\alpha_{0},\dots,\alpha_{n})\to\alpha_{0}$ is a monomorphism, so
the automorphism group ${\rm Aut}_{{\mathcal{L}}}(\sigma)$ can be considered
as a subgroup of ${\rm Aut}_{{\mathcal{L}}}(P_{0})$ via the projection map
$\pi_{0}$. This allows us to identify ${\mathcal{B}}({\rm
Aut}_{{\mathcal{L}}}(\sigma))$ with a subcategory of ${\mathcal{L}}$. Note
that if $\tau$ is a subcomplex of $\sigma$, then there is group homomorphism
${\rm Aut}_{{\mathcal{L}}}(\sigma)\to{\rm Aut}_{{\mathcal{L}}}(\tau)$ defined
by restriction.

######  Theorem 10.7 ([21, Thm A], [1, Thm 5.36]).

Let $(S,{\mathcal{F}},{\mathcal{L}})$ be a $p$-local finite group. Then there
is a functor $\delta:\overline{s}d({\mathcal{F}}^{c})\to Top$ such that the
following hold:

  1. (1)

There is a homotopy equivalence

$\operatorname*{hocolim}_{[\sigma]\in\overline{s}d({\mathcal{F}}^{c})}\delta([\sigma])\smash{\,\mathop{\longrightarrow}\limits^{\simeq}\,}|{\mathcal{L}}|.$

  2. (2)

For each $\sigma\in sd({\mathcal{F}}^{c})$, there is a natural homotopy
equivalence

$B{\rm
Aut}_{{\mathcal{L}}}(\sigma)\smash{\,\mathop{\longrightarrow}\limits^{\simeq}\,}\delta([\sigma]).$

  3. (3)

For each $\sigma\in sd({\mathcal{F}}^{c})$, the map from $B{\rm
Aut}_{{\mathcal{L}}}(\sigma)$ to $|{\mathcal{L}}|$ induced by the equivalences
in (1) and (2) is equal to the map induced by the inclusion of
${\mathcal{B}}({\rm Aut}_{{\mathcal{L}}}(\sigma))$ into ${\mathcal{L}}$.

Associated to the normalizer decomposition there is a Bousfield-Kan spectral
sequence

$E_{2}^{s,t}=\underset{\overline{s}d({\mathcal{F}}^{c})}{\lim{}^{s}}H^{t}({\rm
Aut}_{{\mathcal{L}}}(-);{\mathbb{F}}_{p}))\Rightarrow
H^{s+t}(|{\mathcal{L}}|;{\mathbb{F}}_{p}).$

We say the normalizer decomposition for $(S,{\mathcal{F}},{\mathcal{L}})$ is
_sharp_ if $E_{2}^{s,t}=0$ for all $s>0$ and $t\geq 0$. We prove below
(Theorem 10.9) that the sharpness of the normalizer decomposition is
equivalent to the sharpness of the subgroup decomposition. For this we need
the following lemma:

######  Lemma 10.8.

Let $(S,{\mathcal{F}},{\mathcal{L}})$ be a $p$-local finite group and let
$\sigma$ be a chain of ${\mathcal{F}}$-centric subgroups in $S$ which is fully
${\mathcal{F}}$-normalized. Let ${\mathcal{N}}:=N_{{\mathcal{F}}}(\sigma)$
denote the normalizer fusion system of $\sigma$. Then for every $n\geq 0$, we
have

$H^{t}({\mathcal{O}}^{c}({\mathcal{N}});H^{n}(-;{\mathbb{F}}_{p}))=0$

for $t>0$, and for $t=0$, we have

$H^{0}({\mathcal{O}}^{c}({\mathcal{N}});H^{n}(-;{\mathbb{F}}_{p}))=H^{n}({\mathcal{N}};{\mathbb{F}}_{p})\cong
H^{n}({\rm Aut}_{{\mathcal{L}}}(\sigma);{\mathbb{F}}_{p}).$

###### Proof.

Let $\sigma=(P_{0}<\cdots<P_{n})$. The subgroup $Q:=P_{n}$ is
${\mathcal{F}}$-centric and normal in
${\mathcal{N}}=N_{{\mathcal{F}}}(\sigma)$. Hence by [5, Prop 4.3], there is a
finite group $H$ with a Sylow $p$-subgroup $T\cong N_{S}(\sigma)$ such that
${\mathcal{N}}\cong{\mathcal{F}}_{T}(H)$. Furthermore, by [5, Prop 4.3 (a),
(b)], if ${\mathcal{L}}^{\prime}$ is the centric linking system associated to
${\mathcal{N}}$, then $H\cong{\rm Aut}_{{\mathcal{L}}^{\prime}}(Q)$ and
${\mathcal{L}}^{\prime}\cong{\mathcal{L}}_{T}^{c}(H)$. The normalizer linking
system $N_{{\mathcal{L}}}(\sigma)$ of a chain $\sigma$ can be defined in a
similar way to the normalizer linking system of a subgroup (see [6, Def 6.1]).
For a fully ${\mathcal{F}}$-normalized chain $\sigma=(P_{0}<\cdots<P_{n})$, we
define the normalizer linking system $N_{{\mathcal{L}}}(\sigma)$ to be the
category whose objects are $N_{{\mathcal{F}}}(\sigma)$-centric subgroups in
$N_{S}(\sigma)$, and whose morphisms are defined by

$\mathrm{Mor}_{N_{{\mathcal{L}}}(\sigma)}(P,P^{\prime})=\\{\varphi\in\mathrm{Mor}_{{\mathcal{L}}}(P_{n}P,P_{n}P^{\prime})|\pi(\varphi)(P)\leq
P^{\prime},\pi(\varphi)(P_{i})\leq P_{i}\text{ for all }i\\}.$

The projection functor $\pi^{N}:N_{{\mathcal{L}}}(\sigma)\to
N_{{\mathcal{F}}}(\sigma)$ is defined to be inclusion on objects and it sends
a morphism $\varphi\in\mathrm{Mor}_{N_{{\mathcal{L}}}(\sigma)}(P,P^{\prime})$
to $\pi(\varphi)|_{P}$. By using Lemma 8.10 and repeating the argument in [6,
Lemma 6.2], we can conclude that $N_{{\mathcal{L}}}(\sigma)$ is the linking
system associated to $N_{{\mathcal{F}}}(\sigma)$. By the uniqueness of
associated linking systems, we obtain that ${\mathcal{L}}^{\prime}\cong
N_{{\mathcal{L}}}(\sigma)$. This gives

$H\cong{\rm Aut}_{{\mathcal{L}}^{\prime}}(Q)\cong{\rm
Aut}_{N_{{\mathcal{L}}}(\sigma)}(P_{n})=\\{\varphi\in{\rm
Aut}_{{\mathcal{L}}}(P_{n})|\pi(\varphi)(P_{i})\leq P_{i}\text{ for all
}i\\}={\rm Aut}_{{\mathcal{L}}}(\sigma).$

If a fusion system is realized by a finite group then by [11, Thm B], the
subgroup decomposition for the associated $p$-local finite group is sharp.
Hence we have

$H^{t}({\mathcal{O}}^{c}({\mathcal{N}});H^{n}(-;{\mathbb{F}}_{p}))=0$

for every $t>0$. For $t=0$, we have

$H^{0}({\mathcal{O}}^{c}({\mathcal{N}});H^{n}(-))\cong\underset{{\mathcal{O}}^{c}({\mathcal{N}})}{\lim}H^{n}(-;{\mathbb{F}}_{p})=H^{n}({\mathcal{N}};{\mathbb{F}}_{p}).$

Since ${\mathcal{N}}\cong{\mathcal{F}}_{T}(H)$ and $H$ is finite, we have

$H^{n}({\mathcal{N}};{\mathbb{F}}_{p})\cong H^{n}(H;{\mathbb{F}}_{p})\cong
H^{*}({\rm Aut}_{{\mathcal{L}}}(\sigma);{\mathbb{F}}_{p})$

so the proof is complete. ∎

Now we prove our main theorem in this section.

######  Theorem 10.9.

Let ${\mathcal{F}}$ be a saturated fusion system over $S$. For every $n\geq
0$, and for every $i\geq 0$, there is an isomorphism

$\underset{{\mathcal{O}}^{c}({\mathcal{F}})}{\lim{}^{i}}H^{n}(-;{\mathbb{F}}_{p})\cong\underset{\overline{s}d({\mathcal{F}}^{c})}{\lim{}^{i}}H^{n}({\rm
Aut}_{{\mathcal{L}}}(-);{\mathbb{F}}_{p}).$

###### Proof.

By Theorem 10.6, there is a spectral sequence

$E_{2}^{s,t}\cong\underset{\overline{s}d({\mathcal{F}}^{c})}{\lim{}^{s}}H^{t}(O^{c}(N_{{\mathcal{F}}}(-));H^{n}(-))\Rightarrow
H^{s+t}({\mathcal{O}}^{c}({\mathcal{F}});H^{n}(-)).$

By Lemma 10.8, $E_{2}^{s,t}=0$ for every $t>0$, and for $t=0$, we have

$E_{2}^{s,0}\cong\underset{\overline{s}d({\mathcal{F}}^{c})}{\lim{}^{s}}H^{0}({\mathcal{O}}^{c}(N_{{\mathcal{F}}}(\sigma));H^{n}(-;{\mathbb{F}}_{p}))\cong\underset{\overline{s}d({\mathcal{F}}^{c})}{\lim{}^{s}}H^{n}({\rm
Aut}_{{\mathcal{L}}}(\sigma);{\mathbb{F}}_{p}).$

Hence we obtain

$\underset{{\mathcal{O}}^{c}({\mathcal{F}})}{\lim{}^{i}}H^{n}(-;{\mathbb{F}}_{p}))\cong
E_{2}^{i,0}\cong\underset{\overline{s}d({\mathcal{F}}^{c})}{\lim{}^{i}}H^{n}({\rm
Aut}_{{\mathcal{L}}}(-);{\mathbb{F}}_{p})$

for every $i\geq 0$. ∎

As an immediate corollary of Theorem 10.9, we conclude the following.

######  Corollary 10.10.

For every $p$-local finite group $(S,{\mathcal{F}},{\mathcal{L}})$, the
subgroup decomposition is sharp if and only if the normalizer decomposition is
sharp.

This reduces the problem of showing the sharpness of subgroup decomposition to
a problem of showing the sharpness of the normalizer decomposition. Of course
this could be an equivalently difficult problem to solve.

##  11\. Fusion systems realized by finite groups

Let $G$ be a discrete group and ${\mathcal{C}}$ be a collection of subgroups
of $G$. Let ${\mathcal{O}}_{{\mathcal{C}}}G$ denote the orbit category of $G$
over the collection ${\mathcal{C}}$ and let
$\widetilde{\beta}:{\mathcal{O}}_{{\mathcal{C}}}(G)\to G\text{-Sets}$ denote
the functor which sends each $H\in{\mathcal{C}}$ to the transitive $G$-set
$G/H$. The $G$-space

$X_{{\mathcal{C}}}^{\beta}:=\operatorname*{hocolim}_{{\mathcal{O}}_{{\mathcal{C}}}(G)}\widetilde{\beta}$

is called the Dwyer space for the subgroup decomposition. If
$\beta:{\mathcal{O}}_{{\mathcal{C}}}(G)\to Space$ is the functor defined by
$\beta=EG\times_{G}\widetilde{\beta}$, then there is a mod-$p$ homology
isomorphism

$\operatorname*{hocolim}_{{\mathcal{O}}_{{\mathcal{C}}}(G)}\beta\smash{\,\mathop{\longrightarrow}\limits^{\simeq}\,}BG$

when ${\mathcal{C}}$ is a $p$-ample collection (see [12] for a definition).
This decomposition is called the mod-$p$ subgroup decomposition for $BG$. The
Dwyer space $X_{{\mathcal{C}}}^{\beta}$ is often denoted by
$E_{{\mathcal{C}}}G$ or $E{\mathbf{O}}_{{\mathcal{C}}}$ and called the
universal space for the collection ${\mathcal{C}}$ (see Grodal [13]).

For every $H\in{\mathcal{C}}$, the fixed point subspace
$(E{\mathbf{O}}_{{\mathcal{C}}})^{H}$ is homotopy equivalent to the
realization of the subposet ${\mathcal{C}}_{\geq
H}=\\{K\in{\mathcal{C}}\,|\,K\geq H\\}$ of ${\mathcal{C}}$, hence it is
contractible because ${\mathcal{C}}_{\geq H}$ has a minimal element (see
[15]). We show below that under certain conditions, for a finite group $G$,
the orbit space $C_{G}(P)\backslash(E{\mathbf{O}}_{{\mathcal{C}}})^{P}$ is
${\mathbb{Z}}_{(p)}$-acyclic. We start with a lemma.

######  Lemma 11.1.

Let $G$ be a finite group and $X$ be a $G$-CW-complex such that for every
$p$-subgroup $P\leq G$, the fixed point subspace $X^{P}$ is contractible. Then
the orbit space $X/G$ is ${\mathbb{Z}}_{(p)}$-acyclic.

###### Proof.

Let $S\leq G$ be a Sylow $p$-subgroup of $G$. By a transfer argument it is
enough to show that $X/S$ is ${\mathbb{Z}}_{(p)}$-acyclic (see [7, Sect
III.2]). Consider the fixed point map $f:X\to pt$. Since for every $P\leq S$,
the map $X^{P}\to pt$ is a homotopy equivalence, we can conclude that $f$ is
an $S$-homotopy equivalence (see [4, Prop 2.5.8]). This gives that the orbit
space $X/S$ is contractible, hence it is ${\mathbb{Z}}_{(p)}$-acyclic. ∎

As a consequence of the above lemma we obtain the following:

######  Proposition 11.2.

Let $G$ be a finite group and ${\mathcal{C}}$ be a collection of $p$-subgroups
in $G$ closed under taking $p$-overgroups. Then for every $P\in{\mathcal{C}}$,
the orbit space $C_{G}(P)\backslash(E{\mathbf{O}}_{{\mathcal{C}}})^{P}$ is
${\mathbb{Z}}_{(p)}$-acyclic.

###### Proof.

Let $X=E{\mathbf{O}}_{{\mathcal{C}}}$ and $P$ be a subgroup in
${\mathcal{C}}$. The fixed point set $X^{P}$ has a natural $N_{G}(P)$-action
on it, hence it has an action of $C_{G}(P)$ via restriction. For every
$p$-subgroup $Q\leq C_{G}(P)$, we have $(X^{P})^{Q}=X^{PQ}$. Since
${\mathcal{C}}$ is closed under taking $p$-overgroups, we have
$PQ\in{\mathcal{C}}$, hence $X^{PQ}$ is contractible. This allows us to apply
Lemma 11.1 and conclude that $C_{G}(P)\backslash X^{P}$ is
${\mathbb{Z}}_{(p)}$-acyclic. ∎

An interesting consequence of this proposition is the following:

######  Theorem 11.3.

Let $G$ be a finite group and ${\mathcal{C}}$ be a collection of $p$-subgroups
of $G$ closed under taking $p$-overgroups. Then for every
${\mathbb{Z}}_{(p)}\overline{\mathcal{F}}_{{\mathcal{C}}}(G)$-module $M$,
there is an isomorphism

$H^{*}(\overline{\mathcal{F}}_{{\mathcal{C}}}(G);M)\cong
H^{*}({\mathcal{O}}_{{\mathcal{C}}}(G);\mathrm{Res}_{pr}M)$

induced by the projection functor
$pr:{\mathcal{O}}_{{\mathcal{C}}}(G)\to\overline{\mathcal{F}}_{{\mathcal{C}}}(G)$.

###### Proof.

Let $X=E{\mathbf{O}}_{{\mathcal{C}}}$ and $R={\mathbb{Z}}_{(p)}$. The isotropy
subgroups of $X$ are in ${\mathcal{C}}$ and for every $P\in{\mathcal{C}}$, the
fixed point set $X^{P}$ is $R$-acyclic. This implies that the chain complex
$C_{*}(X^{?};R)$ is a projective resolution of the constant functor
$\underline{R}$ as an $R{\mathcal{O}}_{{\mathcal{C}}}(G)$-module. By
Proposition 11.2, the chain complex of
$R\overline{\mathcal{F}}_{{\mathcal{C}}}(G)$-modules $C_{*}(C_{G}(?)\backslash
X^{?};R)$ is also $R$-acyclic, hence it gives a projective resolution of
$\underline{R}$ as an $R\overline{\mathcal{F}}_{{\mathcal{C}}}(G)$-module.
Combining these we obtain

$\displaystyle H^{*}(\overline{\mathcal{F}}_{{\mathcal{C}}}(G);M)$
$\displaystyle\cong
H^{*}(\mathrm{Hom}_{R\overline{\mathcal{F}}_{{\mathcal{C}}}(G)}(C_{*}(C_{G}(?)\backslash
X^{?};R),M))$ $\displaystyle\cong
H^{*}(\mathrm{Hom}_{R{\mathcal{O}}_{{\mathcal{C}}}(G)}(C_{*}(X^{?};R),\mathrm{Res}_{pr}M))$
$\displaystyle\cong
H^{*}({\mathcal{O}}_{{\mathcal{C}}}(G);\mathrm{Res}_{pr}M).$

The isomorphism in the middle follows from Lemma 4.7. ∎

For a finite group $G$, let $O_{p}(G)$ denote the (unique) maximal normal
subgroup in $G$. A $p$-subgroup $P\leq G$ is called a _principal $p$-radical
subgroup_ if it is $p$-centric and $O_{p}(N_{G}(P)/PC_{G}(P))$ is trivial. The
collection of principal $p$-radical subgroups is denoted by
${\mathcal{D}}_{p}(G)$. Now as a corollary of Theorem 11.3, a theorem of
Grodal [13, Thm 1.2] gives the following.

######  Corollary 11.4.

Let $G$ be a finite group, and ${\mathcal{C}}$ and ${\mathcal{C}}^{\prime}$ be
two collections of $p$-subgroups of $G$ closed under taking $p$-overgroups
satisfying ${\mathcal{C}}^{\prime}\cap
D_{p}(G)\subseteq{\mathcal{C}}\subseteq{\mathcal{C}}^{\prime}$. Then for any
${\mathbb{Z}}_{(p)}\overline{\mathcal{F}}_{{\mathcal{C}}}(G)$-module $M$,
there is an isomorphism

$H^{*}(\overline{\mathcal{F}}_{{\mathcal{C}}}(G);M)\cong
H^{*}(\overline{\mathcal{F}}_{{\mathcal{C}}^{\prime}}(G);\mathrm{Res}_{{\mathcal{C}}^{\prime}}M).$

###### Proof.

By Theorem 11.3, it is enough to prove that

$H^{*}({\mathcal{O}}_{{\mathcal{C}}}(G);\mathrm{Res}_{pr}M)\cong
H^{*}({\mathcal{O}}_{{\mathcal{C}}^{\prime}}(G);\mathrm{Res}_{pr}\mathrm{Res}_{{\mathcal{C}}^{\prime}}M).$

Since $C_{G}(P)$ acts trivially on $(\mathrm{Res}_{pr}M)(P)$ for every
$P\in{\mathcal{C}}$, the result follows from [13, Thm 1.2]. ∎

As a consequence we obtain a generalization of the vanishing result due to
Diaz and Park [11].

######  Theorem 11.5.

Let ${\mathcal{F}}={\mathcal{F}}_{S}(G)$ for a finite group $G$ with a Sylow
$p$-subgroup $S$. Let ${\mathcal{C}}$ be a collection of subgroups of $S$
closed under taking $p$-overgroups such that ${\mathcal{C}}$ includes all
${\mathcal{F}}$-centric-radical subgroups in $S$. Then for every $n\geq 0$ and
every $i\geq 1$,

$\underset{{\mathcal{O}}({\mathcal{F}}_{{\mathcal{C}}})}{\lim{}^{i}}H^{n}(-;{\mathbb{F}}_{p})=0.$

###### Proof.

Let ${\mathcal{F}}={\mathcal{F}}_{S}(G)$. By Lemma 8.9, a subgroup $P\leq S$
is ${\mathcal{F}}$-centric if and only if it is $p$-centric in $G$. For $P\leq
S$, we have

${\rm Out}_{{\mathcal{F}}}(P):={\rm Aut}_{{\mathcal{F}}}(P)/{\rm Inn}(P)\cong
N_{G}(P)/PC_{G}(P).$

This means $P\leq S$ is ${\mathcal{F}}$-radical if and only if
$O_{p}(N_{G}(P)/PC_{G}(P))=1$. Hence $P\leq S$ is ${\mathcal{F}}$-centric-
radical if and only if $P$ is a principal $p$-radical subgroup.

Let ${\mathcal{C}}^{\prime}$ denote the collection of all $p$-subgroups in
$G$. We denote the orbit category over the collection ${\mathcal{C}}^{\prime}$
by ${\mathcal{O}}_{p}(G)$. By the assumption on ${\mathcal{C}}$ we have

${\mathcal{C}}^{\prime}\cap{\mathcal{D}}_{p}(G)={\mathcal{D}}_{p}(G)\subseteq{\mathcal{C}}\subseteq{\mathcal{C}}^{\prime}.$

Applying Theorem 11.3 and Corollary 11.4 to these collections, we obtain

$\underset{{\mathcal{O}}({\mathcal{F}}_{{\mathcal{C}}})}{\lim{}^{i}}H^{n}(-;{\mathbb{F}}_{p})\cong\underset{{\mathcal{O}}({\mathcal{F}}_{{\mathcal{C}}^{\prime}})}{\lim{}^{i}}H^{n}(-;{\mathbb{F}}_{p})\cong\underset{{\mathcal{O}}_{p}(G)}{\lim{}^{i}}H^{n}(-;{\mathbb{F}}_{p}).$

Since $H^{n}(-;{\mathbb{F}}_{p})$ is a cohomological Mackey functor, by [18,
Prop 5.14],

$\underset{{\mathcal{O}}_{p}(G)}{\lim{}^{i}}H^{n}(-;{\mathbb{F}}_{p})=0.$

Hence the proof is complete. ∎

It is interesting to ask whether Proposition 11.2 and Theorem 11.3 still hold
when $G$ is an infinite group. The following example illustrates that these
results do not hold for infinite groups in general.

######  Example 11.6.

Let ${\mathcal{F}}$ be the fusion system of the symmetric group $S_{3}$ at
prime $3$. If we apply the Leary-Stancu construction to ${\mathcal{F}}$, we
find the infinite group

$G=\langle b,a\,|\,b^{3}=1,aba^{-1}=b^{2}\rangle\cong
C_{3}\rtimes{\mathbb{Z}}$

realizing ${\mathcal{F}}$ (see [16, Ex 4.3]). The subgroup $\langle
a^{2}\rangle\cong{\mathbb{Z}}$ is central in $G$ and $G/\langle a^{2}\rangle$
is isomorphic to the symmetric group $S_{3}$. Hence we have an extension of
groups $1\to{\mathbb{Z}}\to G\to S_{3}\to 1$ whose extension class in
${\mathbb{F}}_{3}$-coefficients is zero. This gives that for every $n\geq 0$,

$H^{n}(G;{\mathbb{F}}_{3})\cong H^{n}(S_{3};{\mathbb{F}}_{3})\oplus
H^{n-1}(S_{3};{\mathbb{F}}_{3}).$

Note that

$H^{*}(S_{3};{\mathbb{F}}_{3})\cong
H^{*}(C_{3};{\mathbb{F}}_{3})^{C_{2}}\cong(\Lambda(y)\otimes{\mathbb{F}}_{3}[x])^{C_{2}}$

where $|y|=1$, $|x|=2$, and $C_{2}$ acts on $x$ and $y$ via $x\to-x$ and
$y\to-y$. Hence $H^{i}(S_{3};{\mathbb{F}}_{3})\cong{\mathbb{F}}_{3}$ for
$i=0,3$ mod $4$, and zero otherwise. We have

$H^{*}({\mathcal{F}};{\mathbb{F}}_{3})=\lim_{P\in{\mathcal{O}}({\mathcal{F}})}H^{*}(P;{\mathbb{F}}_{3})\cong
H^{*}(S_{3};{\mathbb{F}}_{3})\neq H^{*}(G;{\mathbb{F}}_{3}),$

since $H^{n-1}(S_{3};{\mathbb{F}}_{3})\neq 0$ for $n=0,1$ mod $4$. Hence this
is an example of the situation where the cohomology of the infinite group $G$
realizing the fusion system ${\mathcal{F}}$ is not isomorphic to the
cohomology of ${\mathcal{F}}$ (see [16]).

Let ${\mathcal{C}}$ be the collection of all finite $3$-subgroups in $G$. We
can take $X=E_{{\mathcal{C}}}G$ to be the real line with $G$-action via the
homomorphism $G\to{\mathbb{Z}}$. For any commutative ring $R$, the chain
complex $C_{*}(X^{?};R)$ gives a sequence

$0\to
R[G/C_{3}^{?}]\smash{\,\mathop{\longrightarrow}\limits^{1-a}\,}R[G/C_{3}^{?}]\to\underline{R}\to
0$

which is a projective resolution of $\underline{R}$ as an
$R{\mathcal{O}}_{{\mathcal{C}}}(G)$-module. For every
$R{\mathcal{O}}_{{\mathcal{C}}}(G)$-module $M$, the higher limits
$\lim^{*}_{{\mathcal{O}}_{{\mathcal{C}}}(G)}M$ are the cohomology modules of
the cochain complex

$0\to M(C_{3})\smash{\,\mathop{\longrightarrow}\limits^{1-a}\,}M(C_{3})\to 0.$

Hence $\lim^{0}_{{\mathcal{O}}_{{\mathcal{C}}}(G)}M\cong\ker(1-a)$ and
$\lim^{1}_{{\mathcal{O}}_{{\mathcal{C}}}(G)}M\cong\mathrm{coker}(1-a)$. If we
take $M=H^{n}(-;{\mathbb{F}}_{p})$ with $n=0,3$ mod $4$, then the action of
the element $a\in G$ on $M$ is trivial, hence we have

$\underset{{\mathcal{O}}_{{\mathcal{C}}}(G)}{\lim{}^{i}}H^{n}(-;{\mathbb{F}}_{p})\cong{\mathbb{F}}_{3}\neq
0$

for $i=0,1$ and $n=0,3$ mod $4$.

The orbit space $C_{G}(C_{3})\backslash
X^{C_{3}}\cong{\mathbb{Z}}\backslash{\mathbb{R}}\cong S^{1}$ is not
${\mathbb{Z}}_{(3)}$-acyclic. This shows that Proposition 11.2 does not hold
for this group. If we take the orbit spaces of $C_{G}(P)$-actions on $X^{P}$
for all $P\in{\mathcal{C}}$, then we obtain a chain complex

$0\to H_{1}(?)\to R\mathrm{Mor}_{{\mathcal{F}}_{{\mathcal{C}}}(G)}(?,C_{3})\to
R\mathrm{Mor}_{{\mathcal{F}}_{{\mathcal{C}}}(G)}(?,C_{3})\to\underline{R}\to
0$

of $R\overline{\mathcal{F}}_{{\mathcal{C}}}(G)$-modules. Note that
$H_{1}(P)\cong R$ for $P=1$ and $P=C_{3}$, and the restriction map
$H_{1}(C_{3})\to H_{1}(1)$ is given by multiplication by 2. Thus if we take
$R={\mathbb{Z}}_{(3)}$, then $H_{1}(?)\cong\underline{R}$, and we can splice
the sequences to get a projective resolution for $\underline{R}$ as an
$R\overline{\mathcal{F}}_{{\mathcal{C}}}(G)$-module. Using this projective
resolution, we obtain that for every
${\mathbb{Z}}_{(3)}\overline{\mathcal{F}}_{{\mathcal{C}}}(G)$-module $M$,

$\underset{\overline{\mathcal{F}}_{{\mathcal{C}}}(G)}{\lim{}^{i}}M\cong
H^{i}(C_{2};M(C_{3}))=0$

for every $i\geq 1$. In particular,

$\lim{}^{i}_{\overline{\mathcal{F}}_{{\mathcal{C}}}(G)}H^{n}(-;{\mathbb{F}}_{3})=0$

for all $i\geq 1$. Hence Conjecture 1.1 holds for the fusion system
${\mathcal{F}}={\mathcal{F}}_{C_{3}}(S_{3})$.

To summarize, for the infinite group $G=C_{3}\rtimes{\mathbb{Z}}$ realizing
the fusion system ${\mathcal{F}}={\mathcal{F}}_{C_{3}}(S_{3})$, we conclude
that

  1. (1)

the statements of Proposition 11.2 and Theorem 11.3 do not hold for $G$ when
we take ${\mathcal{C}}$ as the collection of all $3$-subgroups in $G$,

  2. (2)

the mod-$3$ subgroup decomposition for $BG$ over the collection of all finite
$3$-subgroups is not sharp for $G$, and

  3. (3)

Conjecture 1.1 holds for the fusion system ${\mathcal{F}}$.

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