A correction to the uniqueness of a partial perfect locality over a Frobenius P-category

TL;DR

This paper revises previous claims about the uniqueness of a certain locality in Frobenius P-categories, demonstrating that with the concept of extendable localities, the original results still hold and are simplified.

Contribution

It introduces the notion of extendable perfect localities to correct and uphold the original uniqueness results in Frobenius P-categories.

Findings

Proves the existence of perfect al F^{sc}-locality.

Shows the uniqueness of the extendable perfect locality.

Simplifies previous arguments using the concept of extendability.

Abstract

Let $p$ be a prime, $P$ a finite p-group and $\cal F$ a Frobenius $P$-category. In "Existence, uniqueness and functoriality of the perfect locality over a Frobenius $P$-category", Algebra Colloquium, 23(2016) 541-622, we also claimed the uniqueness of the partial perfect locality $\cal L^{\frak X}$ over any up-closed set $\frak X$ of $\cal F$-selfcentralizing subgroups of $P$, but recently Bob Oliver exhibit some counter-examples, demanding some revision of our arguments. In this Note we show that, up to replacing the perfect localities by the "extendable" perfect localities over any up-closed set $\frak X$ of $\cal F$-selfcentralizing subgroups of $P$, our arguments are correct, still proving the existence and the uniqueness of the perfect $\cal F^{\rm sc}$-locality, since it is "extendable". We take advantage to simplify some of our arguments.

Equations

$$\tau :{\mathcal{T}}_P⟶\mathcal{L}\mathrm{and}\pi :\mathcal{L}⟶\mathcal{F}$$

$${\tau }_{{}_Q}:N_P(Q)⟶\mathcal{P}(Q)\mathrm{and}{\pi }_{{}_Q}:\mathcal{P}(Q)⟶\mathcal{F}(Q)$$

$${\tau }^{{}_X}:{\mathcal{T}}_P^{{}^X}⟶{\mathcal{L}}^{{}^X}\mathrm{and}{\pi }^{{}_X}:{\mathcal{L}}^{{}^X}⟶{\mathcal{F}}^{{}^X}$$

$${\tau }_{{}_Q}^{{}_X}:N_P(Q)⟶{\mathcal{P}}^{{}^X}(Q)\mathrm{and}{\pi }_{{}_Q}^{{}_X}:{\mathcal{P}}^{{}^X}(Q)⟶\mathcal{F}(Q)$$

$$x{\cdot}\tau_{{}_{R}}(v)=\tau_{{}_{Q}}\big{(}\pi_{x}(v)\big{)}{\cdot}x$$

$${\mathcal{P}}^{{}^{X,Q}}\cong {\mathcal{T}}_{{\mathcal{P}}^{{}^X}(Q)}^{{}^{X^{{}^Q}}}$$

$${\tau }_{{}_{R\cdot Q,Q}}^{{}_X}(1)\cdot {\hat{x}}_{{}_Q}=\hat{x}\cdot {\tau }_{{}_{T\cdot Q,Q}}^{{}_X}(1)$$

$$\eqalign{\tau^{{}_{X}}_{{}_{R{\cdot}Q,Q}}(1){\cdot}\hat{x}_{{}_{Q}}{\cdot}\tau^{{}_{X}}_{{}_{Q}}(t)&=\hat{x}{\cdot}\tau^{{}_{X}}_{{}_{T{\cdot}Q,Q}}(t)=\hat{x}{\cdot}\tau^{{}_{X}}_{{}_{T{\cdot}Q}}(t){\cdot}\tau^{{}_{X}}_{{}_{T{\cdot}Q,Q}}(1)\cr&=\tau^{{}_{X}}_{{}_{R{\cdot}Q}}\Big{(}\big{(}\pi^{{}_{X}}_{{}_{R{\cdot}Q,T{\cdot}Q}}(\hat{x})\big{)}(t)\Big{)}{\cdot}\hat{x}{\cdot}\tau^{{}_{X}}_{{}_{T{\cdot}Q,Q}}(1)\cr&=\tau^{{}_{X}}_{{}_{R{\cdot}Q}}\Big{(}\big{(}\pi^{{}_{X}}_{{}_{R{\cdot}Q,T{\cdot}Q}}(\hat{x})\big{)}(t)\Big{)}{\cdot}\tau^{{}_{X}}_{{}_{R{\cdot}Q,Q}}(1){\cdot}\hat{x}_{{}_{Q}}\cr&=\tau^{{}_{X}}_{{}_{R{\cdot}Q,Q}}(1){\cdot}\tau^{{}_{X}}_{{}_{Q}}\Big{(}\big{(}\pi^{{}_{X}}_{{}_{R{\cdot}Q,T{\cdot}Q}}(\hat{x})\big{)}(t)\Big{)}{\cdot}\hat{x}_{{}_{Q}}\cr&=\tau^{{}_{X}}_{{}_{R{\cdot}Q,Q}}(1){\cdot}\tau^{{}_{X}}_{{}_{Q}}\Big{(}\big{(}\pi^{{}_{X}}_{{}_{R,T}}(x)\big{)}(t)\Big{)}{\cdot}\hat{x}_{{}_{Q}}\cr}$$

$$\hat{x}_{{}_{Q}}{\cdot}\tau^{{}_{X}}_{{}_{Q}}(t){\cdot}(\hat{x}_{{}_{Q}})^{-1}=\tau^{{}_{X}}_{{}_{Q}}\Big{(}\big{(}\pi^{{}_{X}}_{{}_{R,T}}(x)\big{)}(t)\Big{)}$$

$${\mathcal{P}}^{{}^{X,W}}\cong {\mathcal{T}}_{L_{\mathcal{F}}^{{}^W}}^{{}^{X^{{}^W}}}$$

$${\mathcal{P}}^{{}^{X,V}}\cong {\mathcal{T}}_{L_{\mathcal{F}}^{{}^V}}^{{}^{X^{{}^V}}}$$

$$N_{{\mathcal{P}}^{{}^{X,V}}}(W)\cong N_{{\mathcal{T}}_{L_{\mathcal{F}}^{{}^V}}^{{}^{X^{{}^V}}}}(W)={\mathcal{T}}_{N_{L_{\mathcal{F}}^{{}^V}}(W)}^{{}^{X^{{}^{V,W}}}}$$

$$N_{P^{{}^V}}(W)⟶N_{L_{\mathcal{F}}^{{}^V}}(W)\mathrm{and}{N}_{L_{\mathcal{F}}^{{}^V}}(W)⟶{\mathcal{F}}^{{}^V}(W)$$

$${\Omega }^{\circ }\cong \Omega \mathrm{and}|\Omega |/|P|\not\equiv 0\mathrm{mod}p$$

$${\mathrm{Res}}_{\phi \times {\mathrm{id}}_P}(\Omega )\cong {\mathrm{Res}}_{{\iota }_Q^P\times {\mathrm{id}}_P}(\Omega )$$

$${\mathcal{T}}_{G^{{}^{\Omega }}}(Q,R)/C_{G^{{}^{\Omega }}}(R)\cong \mathcal{F}(Q,R)$$

$$|{\Omega }^{{\Delta }_{\phi }(Q)}|=|Z(Q)|$$

$${\rm Aut}_{Q\times P}\big{(}(Q\times P)/\Delta_{\eta}(T)\big{)}\cong Z(T)$$

$${\overline{\mathcal{T}}}_{G^{{}^{\Omega }}}^{{}^X}(Q,R)={\mathcal{T}}_{G^{{}^{\Omega }}}^{{}^X}(Q,R)/C_{G^{{}^{\Omega }}}^{{}^{\mathcal{C}X}}(R)$$

$$\matrix{{\cal P}^{{}^{{X}}}&\buildrel\rho^{{}_{{X}}}\over{\cong}&{\cal P}^{\prime^{{X}}}\cr{\atop\lambda^{{}^{\!{X}}}}\searrow&&\hskip-10.0pt\swarrow{\atop\lambda^{\prime^{X}}}\cr&\overline{{\cal T}}^{{}^{{X}}}_{\!G^{{}^{\Omega}}}\cr}$$

$${\Omega }_{{}_W}=\bigcup _{\chi \in \mathcal{F}(W)}{\Omega }^{{\Delta }_{\chi }(W)}$$

$${\overline{g}}_{{}_{\Omega }}^{{}^{X,W}}:N_{{\overline{\mathcal{T}}}_{G^{{}^{\Omega }}}^{{}^X}}(W)⟶{\overline{\mathcal{T}}}_{G^{{}^{{\Omega }_{{}_W}}}}^{{}^{X^{{}^W}}}$$

$${\tau }_{{}_{\mathcal{F}}}^{{}_W}:P^{{}^W}⟶L_{\mathcal{F}}^{{}^W}\mathrm{and}{\pi }_{{}_{\mathcal{F}}}^{{}_W}:L_{\mathcal{F}}^{{}^W}⟶\mathcal{F}(W)$$

$$1\longrightarrow Z(W)\buildrel\tau^{{}_{W}}_{{}_{{\cal F}}}\over{\hbox to20.0pt{\rightarrowfill}}L^{{}^{W}}_{\cal F}\buildrel\pi^{{}_{W}}_{{}_{{\cal F}}}\over{\hbox to20.0pt{\rightarrowfill}}{\cal F}(W)\longrightarrow 1$$

$${\lambda }_{{}_{\mathcal{F}}}^{{}^W}:L_{\mathcal{F}}^{{}^W}⟶N_{G^{{}^{{\Omega }_{{}_W}}}}(W)$$

$$l_{{}_{\mathcal{F}}}^{{}^W}:{\mathcal{T}}_{L_{\mathcal{F}}^{{}^W}}⟶{\mathcal{T}}_{G^{{}^{{\Omega }_{{}_W}}}}$$

$$l_{{}_{\mathcal{F}}}^{{}^{X,W}}:{\mathcal{T}}_{L_{\mathcal{F}}^{{}^W}}^{{}^{X^{{}^W}}}⟶{\overline{\mathcal{T}}}_{G^{{}^{{\Omega }_{{}_W}}}}^{{}^{X^{{}^W}}}$$

$$Y=X-\{\theta (U)\mid \theta \in \mathcal{F}(P,U)\}$$

$${\tau }^{{}_Y}:{\mathcal{T}}_P^{{}^Y}⟶{\mathcal{P}}^{{}^Y}\mathrm{and}{\pi }^{{}_Y}:{\mathcal{P}}^{{}^Y}⟶{\mathcal{F}}^{{}^Y}$$

$$\matrix{{\cal P}^{{}^{{Y}}}&\i&\overline{{\cal T}}^{{}^{Y}}_{\!G^{{}^{\Omega}}}\!\cr\hskip-5.0pt\uparrow&\phantom{\big{\uparrow}}&\hskip-5.0pt\uparrow\cr{\cal M}^{{}^{\Omega,{Y}}}\!&\i&(\overline{{\cal T}}^{{}^{X}}_{\!G^{{}^{\Omega}}}\!)^{{}^{Y}}\cr}$$

Full text

A correction to the uniqueness of a partial

perfect locality over a Frobenius P-category

Lluis Puig

CNRS, Institut de Mathématiques de Jussieu, lluis.puig@imj-prg.fr

6 Av Bizet, 94340 Joinville-le-Pont, France

Abstract: Let p be a prime, P a finite p-group and ℱ a Frobenius P-category. In Existence, uniqueness and functoriality of the perfect locality over a Frobenius P-category, Algebra Colloquium, 23(2016) 541-622, we also claimed the uniqueness of the partial perfect locality ℒ${}^{{}_{{X}}}$ over any up-closed set $\scriptstyle{X}$ of ℱ-selfcentralizing subgroups of P, but recently Bob Oliver exhibit some counter-examples, demanding some revision of our arguments. In this Note we show that, up to replacing the perfect localities by the extendable perfect localities over any up-closed set $\scriptstyle{X}$ of ℱ-selfcentralizing subgroups of P, our arguments are correct, still proving the existence and the uniqueness of the perfect ℱ${}^{{}_{\rm sc}}\!$-locality, since it is extendable. We take advantage to simplify some of our arguments.

**1. Introduction ** 1.1. Let $p$ be a prime, $P$ a finite $p\hbox{-}$group, ${\cal F}$ a Frobenius $P\hbox{-}$category [2] and ${\cal T}_{P}$ the category where the objects are the subgroups of $P\,,$ the morphisms are defined by the $P\hbox{-}$transporters and the composition is defined by the product in $P\,.$ Recall that, according to [3, 17.3], an ${\cal F}\hbox{-}$locality ${\cal L}$ is a finite category where the objects are all the subgroups of $P\,,$ endowed with two functors

[TABLE]

which are the identity on the set of objects, $\pi$ being full, and such that the composition $\pi\circ\tau$ is induced by the conjugation in $P\,;$ we say that ${\cal L}$ is divisible whenever it fulfills the following condition 1.1.2 If $Q\,,$ $R$ and $T$ are subgroups of $P\,,$ for any ${\cal L}\hbox{-}$morphisms $x\,\colon R\to Q$ and $y\,\colon T\to Q$ such that the image of $\pi_{{}_{Q,T}}(y)$ is contained in the image of $\pi_{{}_{Q,R}}(x)\,,$ there is a unique ${\cal L}\hbox{-}$morphism $z\,\colon T\to R$ such that $x{\cdot}z=y\,.$

1.2. Then, it follows from [3, Proposition 18.4 and Theorem 18.6] that a perfect ${\cal F}\hbox{-}$locality, introduced in [3, 17.13], is a divisible ${\cal F}\hbox{-}$locality ${\cal P}$ such that, for any subgroup $Q$ of $P$ fully normalized in ${\cal F}$ [3, 2.6] the finite group ${\cal P}(Q)$ endowed with the group homomorphims

[TABLE]

is the ${\cal F}\hbox{-}$localizer of $Q\,,$ introduced in [3, 18.5]. Actually, as we show in [3, Theorem 20.24] and, more carefully, in [5, Theorem 7.2], ${\cal P}$ is uniquely determined by the full subcategory ${\cal P}^{{}^{\rm sc}}$ over the set of ${\cal F}\hbox{-}$selfcentralizing subgroups of $P\,,$ introduced in [3, 4.8].

1.3. More generally, in order to apply inductive arguments, for any nonempty set ${X}$ of ${\cal F}\hbox{-}$selfcentralizing subgroups of $P$ which contains any

subgroup of $P$ admitting an ${\cal F}\hbox{-}$morphism from some subgroup in ${X}\,,$ we consider the full subcategory ${\cal F}^{{}^{X}}$ of ${\cal F}$ over ${X}$ as the set of objects and, replacing ${\cal T}_{P}$ by its full subcategory ${\cal T}^{{}^{X}}_{P}$ over ${X}\,,$ we may introduce the ${\cal F}^{{}^{X}}\hbox{-}$localities as the finite categories ${\cal L}^{{}^{X}}$ where the objects are the groups in ${X}\,,$ endowed with two functors

[TABLE]

which are the identity on the set of objects, $\pi^{{}_{X}}$ being full, and such that the composition $\pi^{{}_{X}}\circ\tau^{{}_{X}}$ is induced by the conjugation in $P\,.$ In particular, in [5, 2.8] we consider a perfect ${\cal F}^{{}^{X}}\!\hbox{-}$locality ${\cal P}^{{}^{X}}$ and in [5, 6.1] we claimed its existence and uniqueness.

1.4. But, recently, Bob Oliver exhibit some counter-examples to this uniqueness [1]; of course, these counter-examples demand a revision of our arguments in [5]. Our purpose in this Note is to show that, up to restricting the perfect ${\cal F}^{{}^{X}}\!\hbox{-}$localities we consider, our arguments become correct and the uniqueness of these restricted perfect ${\cal F}^{{}^{X}}\!\hbox{-}$localities, called extendable, is true; naturally, our extendable perfect ${\cal F}^{{}^{\rm sc}}\!\hbox{-}$localities include ${\cal P}^{{}^{\rm sc}}$ above. Moreover, we take advantage of this revision to simplify some arguments in [5]. Notations and terminology are the same as in [5] and the main references come from [3].

2. Extendable perfect ${\cal F}^{{}^{X}}\hbox{-}$localities

2.1. With the notation above, let us consider a perfect ${\cal F}^{{}^{X}}\!\hbox{-}$locality ${\cal P}^{{}^{X}}\,;$ that is to say, ${\cal P}^{{}^{X}}$ is a divisible ${\cal F}^{{}^{X}}\!\hbox{-}$locality such that, for any group $Q$ in ${X}$ fully normalized in ${\cal F}$ [3, 2.6], the finite group ${\cal P}^{{}^{X}}\!(Q)$ endowed with the group homomorphims

[TABLE]

is the ${\cal F}\hbox{-}$localizer of $Q\,,$ introduced in [3, 18.5]; in particular, $\pi^{{}_{X}}_{{}_{Q}}$ is surjective and, since $Q$ is ${\cal F}\hbox{-}$selfcentralizing, $\tau^{{}_{X}}_{{}_{Q}}$ is injective [3, Remark 18.7]. Moreover, note that condition 18.6.3 in [3, Theorem 18.6] implies that $Q$ fulfills equality 17.10.1 in [3, Proposition 17.10]; in particular, extending ${\cal P}^{{}^{X}}$ as in [3, 17.4], it follows from [3, Proposition 17.10] that ${\cal P}^{{}^{X}}$ is a coherent ${\cal F}^{{}^{X}}\!\hbox{-}$locality [3, 17.9]; that is to say, we have

[TABLE]

for any pair of subgroups $Q$ and $R$ in ${X}\,,$ any $x\in{\cal P}^{{}^{X}}(Q,R)$ and any $v\in R\,.$

2.2. Actually, considering the normalizer $N_{\cal F}(Q)$ of $Q$ in ${\cal F}$ [3, 2.14] which is a Frobenius $N_{P}(Q)\hbox{-}$category [3, Proposition 2.16], denoting by ${X}^{{}^{Q}}$ the set of subgroups of $N_{P}(Q)$ belonging to ${X}$ and setting $P^{{}^{Q}}=N_{P}(Q)$

and ${\cal F}^{{}^{Q}}\!=N_{\cal F}(Q)\,,$ we can also consider the normalizer $N_{{\cal P}^{{}^{X}}}(Q)$ of $Q$ in ${\cal P}^{{}^{X}}$ [3, 17.4 and 17.5] and, setting ${\cal F}^{{}^{{X},Q}}=({\cal F}^{{}^{Q}})^{{}^{{X}^{{}^{Q}}}}\,,$ it is not difficult to see that ${\cal P}^{{}^{{X},Q}}\!\!=N_{{\cal P}^{{}^{X}}}(Q)$ is actually a perfect ${\cal F}^{{}^{{X},Q}}\!\hbox{-}$locality.

2.3. Moreover, the ${\cal F}^{{}^{{X},Q}}\!\hbox{-}$locality ${\cal P}^{{}^{{X},Q}}\!$ and the group ${\cal P}^{{}^{X}}\!(Q)$ are related throughout the transporter of the $p\hbox{-}$subgroups of ${\cal P}^{{}^{X}}\!(Q)\,;$ explicitly, let us call transporter ${\cal T}_{{\cal P}^{{}^{X}}\!(Q)}$ of ${\cal P}^{{}^{X}}\!(Q)$ the ${\cal F}^{{}^{Q}}\!\hbox{-}$locality formed by the category where the objects are all the subgroups of $P^{{}^{Q}}\,,$ where the morphisms are defined by the elements of the ${\cal P}^{{}^{X}}\!(Q)\hbox{-}$transporters of the corresponding $\tau^{{}_{X}}_{{}_{Q}}\hbox{-}$images, and where the composition is defined by the product in ${\cal P}^{{}^{X}}\!(Q)\,,$ endowed with the obvious functors induced by $\tau^{{}_{X}}_{{}_{Q}}$ and by $\pi^{{}_{X}}_{{}_{Q}}\,.$ Then, denoting by ${\cal T}_{{\cal P}^{{}^{X}}\!(Q)}^{{}^{{X}^{{}^{Q}}}}$ the full subcategory of ${\cal T}_{{\cal P}^{{}^{X}}\!(Q)}$ over ${X}^{{}^{Q}}\,,$ we claim that we have an ${\cal F}^{{}^{Q}}\hbox{-}$locality equivalence [3, 2.9]

[TABLE]

Firstly, we need the following lemma which admits the same proof as in [3, Proposition 24.2].

Lemma  2.4. Any ${\cal P}^{{}^{X}}\hbox{-}$morphism is a monomorphism and an epimorphism.

2.5. Now, we already know that any ${\cal P}^{{}^{{X},Q}}\!\hbox{-}$morphism $x\,\colon T\to R$ is induced by a ${\cal P}^{{}^{{X}}}\!\hbox{-}$morphism $\hat{x}\,\colon T{\cdot}Q\to R{\cdot}Q$ which stabilizes $Q$ [3, 2.14.1]; then, it easily follows from the lemma above that $\hat{x}$ is uniquely determined by $x\,,$ and the divisibility of ${\cal P}^{{}^{{X}}}\!$ guarantees the existence of a unique $\hat{x}_{{}_{Q}}\in{\cal P}^{{}^{{X}}}\!(Q)$ fulfilling

[TABLE]

moreover, from the coherence of ${\cal P}^{{}^{{X}}}\!$ (cf.  2.1.2), for any $t\in T\i P^{{}^{Q}}$ we get

[TABLE]

so that from the lemma above we still get

[TABLE]

thus, the element $\hat{x}_{{}_{Q}}$ belongs to the ${\cal P}^{{}^{{X}}}\!(Q)\hbox{-}$transporter {\cal T}_{{\cal P}^{{}^{{X}}}\!(Q)}\big{(}\tau^{{}_{X}}_{{}_{Q}}(R),\tau^{{}_{X}}_{{}_{Q}}(T)\big{)}

and it is not difficult to check that the correspondence sending the ${\cal P}^{{}^{{X},Q}}\!\hbox{-}$mor-phism $x\,\colon T\to R$ to the ${\cal T}_{{\cal P}^{{}^{X}}\!(Q)}\hbox{-}$morphism $\hat{x}_{{}_{Q}}\,\colon T\to R$ defines a faithful ${\cal F}^{{}^{Q}}\hbox{-}$locality functor ${\cal P}^{{}^{{X},Q}}\to{\cal T}_{{\cal P}^{{}^{X}}\!(Q)}^{{}^{{X}^{{}^{Q}}}}$ [3, 2.9]. The “surjectivity” follows again from condition 18.6.3 in [3, Theorem 18.6].

2.6. But, for any ${\cal F}\hbox{-}$selfcentralizing subgroup $W$ of $P$ fully normalized in ${\cal F}\,,$ we still have the normalizer ${\cal F}^{{}^{W}}=N_{\cal F}(W)\,;$ let us set $P^{{}^{W}}=N_{P}(W)\,;$ if $P^{{}^{W}}$ belongs to ${X}\,,$ so that the set ${X}^{{}^{W}}$ of subgroups of $P^{{}^{W}}$ belonging to ${X}$ is not empty, then we also can consider the normalizer ${\cal P}^{{}^{{X},W}}\!=N_{{\cal P}^{{}^{X}}}(W)\,,$ which is again a perfect ${\cal F}^{{}^{{X},W}}\!\hbox{-}$locality, and we always have the existence of the ${\cal F}\hbox{-}$localizer $L^{{}^{W}}_{\cal F}$ of $W$ [3, Theorem 18.6]; thus, we still can consider the transporter ${\cal T}_{L^{{}^{W}}_{\cal F}}$ of $L^{{}^{W}}_{\cal F}$ as an ${\cal F}^{{}^{W}}\!\hbox{-}$locality and the full subcategory ${\cal T}_{L^{{}^{W}}_{\cal F}}^{{}^{{X}^{{}^{W}}}}\!$ of ${\cal T}_{L^{{}^{W}}_{\cal F}}$ over ${X}^{{}^{W}}\!$ as the set of objects. Finally, we say that the perfect ${\cal F}^{{}^{X}}\!\hbox{-}$locality ${\cal P}^{{}^{X}}$ is extendable whenever for any ${\cal F}\hbox{-}$selfcentralizing subgroup $W$ of $P$ fully normalized in ${\cal F}$ such that $P^{{}^{W}}\in{X}$ there exists an ${\cal F}^{{}^{{X},W}}\!\hbox{-}$locality isomorphism††† In [5, 6.18], arguing by induction we claim such an equivalence but, with the notation there, if the group $\scriptstyle U$ is normal in ℱ then the induction argument cannot be applied!

[TABLE]

note that, according to  2.3.1, we may assume that $W$ does not belong to ${X}\,.$

Proposition  2.7. If ${\cal P}^{{}^{X}}\!$ is an extendable perfect ${\cal F}^{{}^{X}}\!\hbox{-}$locality then, for any ${\cal F}\hbox{-}$selfcentralizing subgroup $V$ of $P$ fully normalized in ${\cal F}$ such that $P^{{}^{V}}\in{X}\,,$ ${\cal P}^{{}^{{X},V}}\!\!$ is an extendable perfect ${\cal F}^{{}^{{X},V}}\!\!\hbox{-}$locality.

Proof: From our definition we have an ${\cal F}^{{}^{V}}\hbox{-}$locality isomorphism

[TABLE]

which determines an $N_{{\cal F}^{{}^{V}}}(W)\hbox{-}$locality isomorphism

[TABLE]

where we identify $P^{{}^{V}}$ with its image in $L^{{}^{V}}_{\cal F}$ and, for any ${\cal F}^{{}^{V}}\hbox{-}$selfcentralizing subgroup $W$ of $P^{{}^{V}}\!$ fully normalized in ${\cal F}^{{}^{V}}$ such that $N_{P^{{}^{V}}}(W)\in{X}^{{}^{V}}\,,$ we denote by ${X}^{{}^{V,W}}$ the set of subgroups of $N_{P^{{}^{V}}}(W)$ belonging to ${X}^{{}^{V}}$

But, it is not difficult to check that the normalizer $N_{\!L^{{}^{V}}_{\cal F}}(W)\,,$ endowed with the group homomorphisms

[TABLE]

induced by the structural group homomorphisms of $L^{{}^{V}}_{\cal F}\,,$ is the ${\cal F}^{{}^{V}}\!\hbox{-}$localizer of $W\,.$ We are done.

3. A reduction procedure

3.1. With the notation above, recall that a basic $P\times P\hbox{-}$set [3, 21,4] is a finite nonempty $P\times P\hbox{-}$set $\Omega$ such that $\{1\}\times P$ acts freely on $\Omega\,,$ that we have

[TABLE]

where we denote by $\Omega^{\circ}$ the $P\times P\hbox{-}$set obtained by exchanging both factors, and that, for any subgroup $Q$ of $P$ and any injective group homomorphism $\varphi\,\colon Q\to P$ such that $\Omega$ contains a $P\times P\hbox{-}$subset isomorphic to $(P\times P)/\Delta_{\varphi}(Q)$ where we set $\Delta_{\varphi}(Q)=\{(\varphi(u),u)\}_{u\in Q}\,,$ we have a $Q\times P\hbox{-}$set isomorphism

[TABLE]

3.2. Denoting by $G^{{}^{\Omega}}$ the group of automorphisms of the $\{1\}\times P\hbox{-}$set ${\rm Res}_{\{1\}\times P}(\Omega)\,,$ it is clear that we have an injective map from $P\times\{1\}$ in $G^{{}^{\Omega}}\,;$ we identify its image with the $p\hbox{-}$group $P$ so that, from now on, $P$ is contained in $G^{{}^{\Omega}}$ and acts freely on $\Omega\,.$ Recall that the full subcategory of the $G^{{}^{\Omega}}\hbox{-}$transporter over the set of subgroups of $P$ induces a Frobenius $P\hbox{-}$category [3, Proposition 21.9] and we say that $\Omega$ is an ${\cal F}\hbox{-}$basic $P\times P\hbox{-}$set if, for any pair of subgroups $Q$ and $R$ of $P\,,$ we have

[TABLE]

3.3. Actually, it follows from [3, Proposition 21.12] that an ${\cal F}\hbox{-}$basic $P\times P\hbox{-}$set always exists; more precisely, we say that an ${\cal F}\hbox{-}$basic $P\times P\hbox{-}$set $\Omega$ is natural if it fulfills [5, 3.5]

[TABLE]

for any ${\cal F}\hbox{-}$selfcentralizing subgroup $Q$ of $P$ and any $\varphi\in{\cal F}(P,Q)\,,$ and if it is thick [3, 21.7] outside of the set of ${\cal F}\hbox{-}$selfcentralizing subgroups of $P$ — namely the multiplicity of $(P\times P)/\Delta_{\psi}(R)$ is at least two if $R$ is not ${\cal F}\hbox{-}$selfcentralzing and $\psi$ belongs to ${\cal F}(P,R)\,.$ The existence of natural ${\cal F}\hbox{-}$basic $P\times P\hbox{-}$sets follows from [5, Proposition 3.4] together with [3, Proposition 21.12]; here, we are interested in the following form of [5, Proposition 3.7]

Proposition  3.4 Let $\Omega$ be a natural ${\cal F}\hbox{-}$basic $P\times P\hbox{-}$set, $Q$ and $T$ a pair of ${\cal F}\hbox{-}$selfcentralizing subgroups of $P$ and $\eta$ an element of ${\cal F}(Q,T).$ The multiplicity of $(Q\times P)/\Delta_{\eta}(T)$ in ${\rm Res}_{Q\times P}(\Omega)$ is at most one, and if it is one then we have

[TABLE]

3.5. From now on, $\Omega$ is a natural ${\cal F}\hbox{-}$basic $P\times P\hbox{-}$set. For any subgroup $Q$  of  $P\,,$  it  is  clear  that  $C_{G^{{}^{\Omega}}}(Q)$  is  just  the  group  of  automorphisms  of  the

$Q\times P\hbox{-}$set ${\rm Res}_{Q\times P}(\Omega)$ and it is clear that the correspondence sending $Q$ to $C_{G^{{}^{\Omega}}}(Q)$ induces a contravariant functor $C_{G^{{}^{\Omega}}}$ from ${\cal F}$ to the category ${G}{r}$ of finite groups. Let us denote by $C^{{}^{\rm nsc}}_{G^{{}^{\Omega}}}(Q)$ the subgroup of elements $f\in C_{G^{{}^{\Omega}}}(Q)$ which act trivially on all the $Q\times P\hbox{-}$orbits of $\Omega$ isomorphic to $(Q\times P)/\Delta_{\eta}(T)$ where $T$ is ${\cal F}\hbox{-}$selfcentralizing; in particular, if $Q$ is not ${\cal F}\hbox{-}$selfcentralizing then we have $C^{{}^{\rm nsc}}_{G^{{}^{\Omega}}}(Q)=C_{G^{{}^{\Omega}}}(Q)\,;$ in any case, $C^{{}^{\rm nsc}}_{G^{{}^{\Omega}}}(Q)$ is normal in $C_{G^{{}^{\Omega}}}(Q)$ and, according to Proposition  3.4, the quotient $C_{G^{{}^{\Omega}}}(Q)/C^{{}^{\rm nsc}}_{G^{{}^{\Omega}}}(Q)$ is Abelian.

3.6. More generally, for any $Q\in{X}$ denote by $C^{{}^{{\cal C}{X}}}_{G^{{}^{\Omega}}}(Q)$ the subgroup of elements $f\in C_{G^{{}^{\Omega}}}(Q)$ which act trivially on all the $Q\times P\hbox{-}$orbits of $\Omega$ isomorphic to $(Q\times P)/\Delta_{\eta}(T)$ where $T$ belongs to ${X}\,;$ it is easily checked that the correspondence sending $Q\in{X}$ to $C^{{}^{{\cal C}{X}}}_{G^{{}^{\Omega}}}(Q)$ defines a subfunctor $C^{{}^{{\cal C}{X}}}_{\!G^{{}^{\Omega}}}\,\colon{\cal F}^{{}^{X}}\to{G}{r}$ of the restriction of $C_{G^{{}^{\Omega}}}$ to ${\cal F}^{{}^{X}}\,,$ and we consider the quotient ${\cal F}^{{}^{X}}\!\hbox{-}$locality $\overline{{\cal T}}_{\!G^{{}^{\Omega}}}^{{}^{X}}={\cal T}_{G^{{}^{\Omega}}}^{{}^{X}}/C^{{}^{{\cal C}{X}}}_{\!G^{{}^{\Omega}}}$ — noted $\bar{\cal L}^{{}^{\rm n,{X}}}$ in [5, 5.1.2] — sending any pair of groups $Q$ and $R$ in ${X}$ to

[TABLE]

here we are interested in the following form of [5, Corollaries 5.20 and 5.21].

Proposition  3.7. For any perfect ${\cal F}^{{}^{{X}}}\!\hbox{-}$locality ${\cal P}^{{}^{X}}$ there is a unique naturally ${\cal F}^{{}^{{X}}}\!\hbox{-}$isomorphic class of faithful ${\cal F}^{{}^{X}}\hbox{-}$locality functors $\lambda\!^{{}^{X}}\,\colon{\cal P}^{{}^{X}}\to\overline{{\cal T}}^{{}^{{X}}}_{G^{{}^{\Omega}}}\,.$ Moreover, if ${\cal P}^{\prime^{{X}}}\!$ is a perfect ${\cal F}^{{}^{{X}}}\!\hbox{-}$locality which is ${\cal F}^{{}^{{X}}}\!\hbox{-}$locality isomorphic to ${\cal P}^{{}^{X}}$ then there is a commutative diagram of ${\cal F}^{{}^{{X}}}\!\hbox{-}$locality functors

[TABLE]

3.8. With the notation in  2.2 above, for any ${\cal F}\hbox{-}$selfcentralizing subgroup $W$ of $P$ fully normalized in ${\cal F}$ such that $P^{{}^{W}}\!\in{X}\,,$ it follows from [3, Proposition 21.11] that the subset of $\Omega$

[TABLE]

is actually an ${\cal F}^{{}^{W}}\!\hbox{-}$basic $P^{{}^{W}}\!\!\times P^{{}^{W}}\!\!\hbox{-}$set; mutatis mutandi, denote by $G^{{}^{\Omega_{{}_{W}}}}$ the group of $\{1\}\times P^{{}^{W}}\!\!\hbox{-}$set automorphisms of ${\rm Res}_{\{1\}\times P^{{}^{W}}\!}(\Omega_{{}_{W}})$ and identify $P^{{}^{W}}\!$ with $P^{{}^{W}}\!\!\times\{1\}\,;$ since the quotient $N_{G^{{}^{\Omega}}}(W)/C_{G^{{}^{\Omega}}}(W)$ is isomorphic to ${\cal F}(W)$ (cf.  3.2.1), it is clear that $N_{G^{{}^{\Omega}}}(W)$ stabilizes $\Omega_{{}_{W}}$ and therefore we have a canonical group homomorphism from $N_{G^{{}^{\Omega}}}(W)$ to $G^{{}^{\Omega_{{}_{W}}}}\,;$ again, we are interested in the following form of [5, Proposition 6.15].

Proposition  3.9. With the notation above, for any pair of subgroups $Q$ and $R$ of $P^{{}^{W}}\!$ containing $W$ and any element $\varphi$ in ${\cal F}^{{}^{W}}\!(Q,R)\,,$ there exists at most one $Q\times P^{{}^{W}}\!\!\hbox{-}$orbit in $\Omega_{{}_{W}}$ isomorphic to $(Q\times P^{{}^{W}})/\Delta_{\varphi}(R)\,,$ $\Omega_{{}_{W}}$ is a natural ${\cal F}^{{}^{W}}\!\hbox{-}$basic $P^{{}^{W}}\!\times P^{{}^{W}}\!\hbox{-}$set and, in particular, $C_{G^{{}^{\Omega_{{}_{W}}}}}(Q)$ is an Abelian $p\hbox{-}$group.

3.10. It follows from this proposition that, as in  3.6 above, if $P^{{}^{W}}$ belongs to ${X}$ then we get the quotient ${\cal F}^{{}^{{X},W}}\!\hbox{-}$locality $\overline{{\cal T}}_{\!G^{{}^{\Omega_{{}_{W}}}}}^{{}^{{X}^{{}^{W}}}}\,;$ actually, it follows from Propositions  3.4 and  3.9 above that, with the notation in  2.2 and  2.6 above, the canonical group homomorphism from $N_{G^{{}^{\Omega}}}(W)$ to $G^{{}^{\Omega_{{}_{W}}}}$ induces an ${\cal F}^{{}^{{X},W}}\!\hbox{-}$locality functor

[TABLE]

note that, according to Proposition  3.7 above, we have faithful ${\cal F}^{{}^{{X},W}}\!\hbox{-}$locality functors from ${\cal P}^{{}^{{X},W}}=N_{{\cal P}^{{}^{X}}}(W)$ to both ${\cal F}^{{}^{{X},W}}\!\hbox{-}$localities $N_{\overline{{\cal T}}^{{}^{X}}_{\!G^{{}^{\Omega}}}\!}(W)$ and $\overline{{\cal T}}^{{}^{{X}^{{}^{W}}}}_{\!G^{{}^{\Omega_{{}_{W}}}}}$ and we may assume that they agree with $\overline{{g}}^{{}^{{X},W}}_{{}_{\Omega}}\,.$

3.11. On the other hand, let $L^{{}^{W}}_{\cal F}$ be the ${\cal F}\hbox{-}$localizer of $W$ [3, Theorem 18.6]; that is to say, $L^{{}^{W}}_{\cal F}$ is a finite group endowed with an injective and a surjective group homomorphisms

[TABLE]

$\tau^{{}_{W}}_{{}_{{\cal F}}}(P^{{}^{W}})$ is a Sylow $p\hbox{-}$subgroup of $L^{{}^{W}}_{\cal F}\,,$ the composition $\pi^{{}_{W}}_{{}_{{\cal F}}}\!\circ\tau^{{}_{W}}_{{}_{{\cal F}}}\!$ is defined by the conjugation in ${\cal F}(W)$ and we also have the exact sequence

[TABLE]

Below, we restate [5, Proposition 6.19].

Proposition  3.12. With the notation above, there is a unique $C_{\!G^{{}^{\Omega_{{}_{W}}}}}(W)\hbox{-}$ conjugacy class of group homomorphisms

[TABLE]

compatible with the structural group homomorphisms from $P^{{}^{W}}$ and to ${\cal F}(W)\,.$

3.13. As in  2.6 above, denote by ${\cal T}_{\!L^{{}^{W}}_{\cal F}}$ the ${\cal F}^{{}^{W}}\!\hbox{-}$locality determined by $\tau^{{}_{W}}_{{}_{{\cal F}}}$ and by the transporter of the group $L^{{}^{W}}_{\cal F}\,;$ it is clear that any group homomorphism $\lambda^{{}^{W}}_{{}_{{\cal F}}}\,\colon L^{{}^{W}}_{\cal F}\to N_{\!G^{{}^{\Omega_{{}_{W}}}}}(W)$ in  3.12.1 above determines an ${\cal F}^{{}^{W}}\!\hbox{-}$locality functor

[TABLE]

and two of them are naturally ${\cal F}^{{}^{W}}\!\hbox{-}$isomorphic [5, 2.9]; moreover, if $P^{{}^{W}}\in{X}\,,$ it is not difficult to see that the full subcategory ${\cal T}_{\!L^{{}^{W}}_{\cal F}}^{{}^{{X}^{{}^{W}}}}$ of ${\cal T}_{\!L^{{}^{W}}_{\cal F}}$ over ${X}^{{}^{W}}$ as the set of objects is a perfect ${\cal F}^{{}^{{X},W}}\!\hbox{-}$locality, and from  3.13.1 we get an ${\cal F}^{{}^{{X},W}}\!\hbox{-}$locality functor

[TABLE]

4. Existence and uniqueness of an extendable perfect ${\cal F}^{{}^{{X}}}\!\hbox{-}$locality

4.1. With the notation in  1.3 above, our main purpose is to prove that

Theorem. There exists an extendable perfect ${\cal F}^{{}^{{X}}}\!\hbox{-}$locality ${\cal P}^{{}^{X}}\!$, which is unique up to ${\cal F}^{{}^{{X}}}\!\hbox{-}$locality isomorphisms. The existence and the uniqueness of the ${\cal F}\hbox{-}$localizer $L^{{}^{P}}_{\cal F}$ of $P$ [3, Theorem 18.6] proves the existence and the uniqueness of the extendable perfect ${\cal F}^{{}^{X}}\hbox{-}$locality whenever ${X}=\{P\}\,;$ indeed, $L^{{}^{P}}_{\cal F}$ is actually a semidirect product $P\!\mathchar 9583\relax\!K$ where $K\cong{\cal F}(P)/{\cal F}_{P}(P)$ is a $p^{\prime}\hbox{-}$group and, for any ${\cal F}\hbox{-}$selfcentralizing normal subgroup $W$ of $P\,,$ the ${\cal F}^{{}^{{X},W}}\!\hbox{-}$locality equivalence  2.6.1 is obvious.

4.2. Thus, we may assume that ${X}\not=\{P\}$ and will argue by induction on $|{X}|\,.$ Choose a minimal element $U$ in ${X}$ fully normalized in ${\cal F}$ and set

[TABLE]

then, by the induction hypothesis, we may assume that there exists an extendable perfect ${\cal F}^{{}^{{Y}}}\hbox{-}$locality ${\cal P}^{{}^{{Y}}}\,,$ endowed with the structural functors

[TABLE]

which is unique up to ${\cal F}^{{}^{Y}}\hbox{-}$locality isomorphisms. At this point, according to Proposition  3.7 above, we may assume that ${\cal P}^{{}^{{Y}}}$ is an ${\cal F}^{{}^{{Y}}}\!\hbox{-}$sublocality of the ${\cal F}^{{}^{{Y}}}\!\hbox{-}$locality $\overline{{\cal T}}^{{}^{Y}}_{\!G^{{}^{\Omega}}}$ introduced in  3.6 above; then, denoting by $(\overline{{\cal T}}^{{}^{X}}_{\!G^{{}^{\Omega}}}\!)^{{}^{Y}}$ the full subcategory of $\overline{{\cal T}}^{{}^{X}}_{\!G^{{}^{\Omega}}}\!$ over ${Y}$ as the set of objects, we have an obvious functor $(\overline{{\cal T}}^{{}^{X}}_{\!G^{{}^{\Omega}}}\!)^{{}^{Y}}\longrightarrow\overline{{\cal T}}^{{}^{Y}}_{\!G^{{}^{\Omega}}}$ and we look to the pull-back

[TABLE]

which defines a coherent ${\cal F}^{{}^{{Y}}}\!\hbox{-}$locality ${\cal M}^{{}^{\Omega,{Y}}}\!$ [3, 17.9] endowed with obvious structural functors

[TABLE]

4.3. We extend ${\cal M}^{{}^{\Omega,{Y}}}\!$ to a coherent ${\cal F}^{{}^{{X}}}\!\hbox{-}$sublocality ${\cal M}^{{}^{\Omega,{X}}}\!$ of $\overline{{\cal T}}^{{}^{X}}_{\!G^{{}^{\Omega}}}\!$ which contains ${\cal M}^{{}^{\Omega,{Y}}}\!$ as a full subcategory over ${Y}$ and fulfills

[TABLE]

for any $Q\in{X}$ and any $V\in{X}-{Y}\,,$ and denote by

[TABLE]

the corresponding structural functors; finally, we consider the quotient ${\cal F}^{{}^{{X}}}\hbox{-}$lo-cality $\bar{\cal M}^{{}^{\Omega,{X}}}\!$ of ${\cal M}^{{}^{\Omega,{X}}}\!$ defined by

[TABLE]

for any $Q,R\in{X}\,,$ together with the induced natural maps — denoted by $\bar{\upsilon}^{{}^{\Omega,{X}}}$ and $\bar{\rho}^{{}^{\Omega,{X}}}\,.$ Then, the proof of the Theorem above can be reduced to the proof of the following fact, that we prove in the next section 4.3.4. The structural functor $\bar{\rho}^{{}^{\Omega,{X}}}\!$ admits an ${\cal F}^{{}^{{X}}}\!\hbox{-}$locality functorial section.

4.4. Let us first prove this reduction. Choose an ${\cal F}^{{}^{{X}}}\!\hbox{-}$locality functorial section $\bar{\sigma}^{{}_{\Omega,{X}}}\,\colon{\cal F}^{{}^{{X}}}\to\bar{\cal M}^{{}^{\Omega,{X}}}\,;$ for any pair of groups $Q$ and $R$ in ${Y}\,,$ we know that (cf.  2.1)

[TABLE]

and therefore, denoting by ${\cal P}^{{}^{\Omega,{Y}}}\!(Q,R)$ the converse image of \bar{\sigma}^{{}_{\Omega,{X}}}_{{}_{Q,R}}\big{(}{\cal F}^{{}^{{X}}}\!(Q,R)\big{)} in ${\cal M}^{{}^{\Omega,{X}}}(Q,R)\,,$ it is clear that the canonical map ${\cal M}^{{}^{\Omega,{X}}}\!(Q,R)\to{\cal P}^{{}^{Y}}\!(Q,R)$ induces a bijection ${\cal P}^{{}^{\Omega,{Y}}}\!(Q,R)\cong{\cal P}^{{}^{Y}}\!(Q,R)\,;$ that is to say, looking to the pull-back

[TABLE]

— which defines a coherent ${\cal F}^{{}^{{X}}}\!\hbox{-}$locality ${\cal P}^{{}^{\Omega,{X}}}\!$ [3, 17.9] endowed with obvious structural functors

[TABLE]

— and denoting by $({\cal P}^{{}^{\Omega,{X}}}\!)^{{}^{Y}}$ the full subcategory of ${\cal P}^{{}^{\Omega,{X}}}\!$ over ${Y}$ as the set of objects, it follows from those bijections above that we have an ${\cal F}^{{}^{Y}}\!\hbox{-}$locality isomorphism $({\cal P}^{{}^{\Omega,{X}}}\!)^{{}^{Y}}\cong{\cal P}^{{}^{Y}}\,.$

4.5. That is to say, for any $Q\in{Y}$ fully normalized in ${\cal F}\,,$ we already know that ${\cal P}^{{}^{\Omega,{X}}}\!(Q)$ is an ${\cal F}\hbox{-}$localizer of $Q$ and, for any $V\in{X}-{Y}\,,$ it follows from the pull-back  4.4.2 above that we have the exact sequence

[TABLE]

and it is easily checked that the group ${\cal P}^{{}^{\Omega,{X}}}\!(V)\,,$ endowed with the group homomorphisms

[TABLE]

determined by the functors $\tau^{{}_{\Omega,{X}}}$ and $\pi^{{}_{\Omega,{X}}}\,,$ is actually an ${\cal F}^{{}^{{X}}}\hbox{-}$localizer of $V$ whenever $V$ is fully normalized in ${\cal F}\,;$ consequently, it follows from 2.1 above that ${\cal P}^{{}^{\Omega,{X}}}\!$ is a perfect ${\cal F}^{{}^{{X}}}\!\hbox{-}$locality.

4.6. We claim that ${\cal P}^{{}^{\Omega,{X}}}\!$ is actually an extendable perfect ${\cal F}^{{}^{{X}}}\!\hbox{-}$locality; indeed, let $W$ be an ${\cal F}\hbox{-}$selfcentralizing subgroup of $P$ fully normalized in ${\cal F}$ such that $P^{{}^{W}}=N_{P}(W)$ belongs to ${X}\,;$ thus, if $P^{{}^{W}}$ does not belong to ${Y}$ then we have ${X}^{{}^{W}}=\{P^{{}^{W}}\}$ and $P^{{}^{W}}$ is the unique object in both ${\cal F}^{{}^{{X},W}}\!\hbox{-}$localities $N_{{\cal P}^{{}^{\Omega,{X}}}}(W)$ and ${\cal T}_{L^{{}^{W}}_{\cal F}}^{{}^{{X}^{{}^{W}}}}\,;$ in this case, since

[TABLE]

where $K\cong{\cal F}^{{}^{{X},W}}\!\!(P^{{}^{W}})/{\cal F}_{\!P^{{}^{W}}}(P^{{}^{W}})\,,$ it is clear that we get the equivalence  2.6.1. Otherwise ${Y}^{{}^{W}}$ is not empty and, setting ${\cal P}^{{}^{\Omega,{X},W}}=N_{{\cal P}^{{}^{\Omega,{X}}}}(W)$ and denoting by ${\cal P}^{{}^{\Omega,{Y},W}}\!$ the full subcategory of ${\cal P}^{{}^{\Omega,{X},W}}\!$ over ${Y}^{{}^{W}}\,,$ from  4.4 above we get an ${\cal F}^{{}^{{Y},W}}\!\hbox{-}$locality isomorphism

[TABLE]

but, since ${\cal P}^{{}^{Y}}$ is extendable, it follows from our definition in  2.6 above that we still get an ${\cal F}^{{}^{{Y},W}}\!\hbox{-}$locality isomorphism

[TABLE]

4.7. Always assuming that ${Y}^{{}^{W}}$ is not empty, note that in  3.10 above $\overline{{g}}^{{}^{{Y},W}}_{{}_{\Omega}}$ sends $N_{{\cal P}^{{}^{{Y}}}\!}(W)\!$ isomorphically to its image in $\overline{{\cal T}}^{{}^{{Y}^{{}^{W}}}}_{G^{{}^{\Omega_{{}_{W}}}}}$ — still noted ${\cal P}^{{}^{\Omega,{Y},W}}\,;$ then, from this inclusion, mutatis mutandi we can define a coherent ${\cal F}^{{}^{{Y},W}}\!\hbox{-}$ locality ${\cal M}^{{}^{\Omega,{Y},W}}\!$ as in  4.2.3, and coherent ${\cal F}^{{}^{{X},W}}\!\hbox{-}$localities ${\cal M}^{{}^{\Omega,{X},W}}\!\i\overline{{\cal T}}^{{}^{{X}^{{}^{W}}}}_{\!G^{{}^{\Omega_{{}_{W}}}}}$ and $\bar{\cal M}^{{}^{\Omega,{X},W}}\!$ as in  4.3; moreover, it is clear that $\bar{\sigma}^{{}_{\Omega,{X}}}$ induces an ${\cal F}^{{}^{{X},W}}\!\hbox{-}$locality functorial section $\bar{\sigma}^{{}_{\Omega,{X},W}}\,\colon{\cal F}^{{}^{{X},W}}\to\bar{\cal M}^{{}^{\Omega,{X},W}}$ and that we can define a coherent ${\cal F}^{{}^{{X},W}}\!\hbox{-}$locality ${\cal P}^{{}^{\Omega,{X},W}}$ as in  4.4.2 above which still fulfills

[TABLE]

we denote by $\tau^{{}_{\Omega,{X},W}}\,\colon{\cal T}_{P^{{}^{W}}}^{{}^{{X}^{{}^{W}}}}\to{\cal P}^{{}^{\Omega,{X},W}}$ and by $\pi^{{}_{\Omega,{X},W}}\,\colon{\cal P}^{{}^{\Omega,{X},W}}\to{\cal F}^{{}^{{X},W}}$ the structural functors.

4.8. On the other hand, since ${\cal T}_{\!L^{{}^{W}}_{\cal F}}^{{}^{{X}^{{}^{W}}}}$ is a perfect ${\cal F}^{{}^{{X},W}}\!\hbox{-}$locality (cf.  3.13), it follows from Proposition  3.7 (or from  3.13.2) that ${\cal T}_{\!L^{{}^{W}}_{\cal F}}^{{}^{{X}^{{}^{W}}}}\!$ is actually an ${\cal F}^{{}^{{X},W}}\!\!\hbox{-}$sublocality of $\overline{{\cal T}}^{{}^{{X}^{{}^{W}}}}_{\!G^{{}^{\Omega_{{}_{W}}}}}\,;$ in particular, denoting by $(\overline{{\cal T}}^{{}^{{X}^{{}^{W}}}}_{\!G^{{}^{\Omega_{{}_{W}}}}})^{{}^{{Y}^{{}^{W}}}}$ and by $({\cal T}_{\!L^{{}^{W}}_{\cal F}}^{{}^{{X}^{{}^{W}}}})^{{}^{{Y}^{{}^{W}}}}$ the respective full subcategories of $\overline{{\cal T}}^{{}^{{X}^{{}^{W}}}}_{\!G^{{}^{\Omega_{{}_{W}}}}}$ and of ${\cal T}_{\!L^{{}^{W}}_{\cal F}}^{{}^{{X}^{{}^{W}}}}$ over ${Y}^{{}^{W}}\,,$ it is easily checked that the canonical functor

[TABLE]

sends $({\cal T}_{\!L^{{}^{W}}_{\cal F}}^{{}^{{X}^{{}^{W}}}})^{{}^{{Y}^{{}^{W}}}}$ isomorphically onto ${\cal T}_{\!L^{{}^{W}}_{\cal F}}^{{}^{{Y}^{{}^{W}}}}\i\overline{{\cal T}}^{{}^{{Y}^{{}^{W}}}}_{\!G^{{}^{\Omega_{{}_{W}}}}}\,.$

4.9. Moreover, from 4.4 we know that the canonical functor

[TABLE]

sends $({\cal P}^{{}^{\Omega,{X}}}\!)^{{}^{Y}}$ isomorphically onto ${\cal P}^{{}^{Y}}\,;$ but, it follows from our definition in  3.10 that, denoting by $(\overline{{g}}^{{}^{{X},W}}_{{}_{\Omega}})^{{}^{Y}}$ the restriction of $\overline{{g}}^{{}^{{X},W}}_{{}_{\Omega}}$ to the normalizer in $(\overline{{\cal T}}^{{}^{{X}}}_{G^{{}^{\Omega}}})^{{}^{{Y}}}\!$ of $W\,,$ we have a commutative diagram of functors

[TABLE]

where the vertical arrows are defined by the functors  4.8.1 and  4.9.1; hence, since the functor  4.9.1 sends $({\cal P}^{{}^{\Omega,{X}}}\!)^{{}^{Y}}$ isomorphically onto ${\cal P}^{{}^{Y}}$ (cf.  4.4), this functor sends $N_{({\cal P}^{{}^{\Omega,{X}}}\!)^{{}^{Y}}}(W)$ isomorphically onto $N_{{\cal P}^{{}^{Y}}}(W)$ and we already know that $\overline{{g}}^{{}^{{Y},W}}_{{}_{\Omega}}$ sends $N_{{\cal P}^{{}^{Y}}}(W)$ isomorphically onto ${\cal P}^{{}^{\Omega,{Y},W}}\!\!$ (cf.  4.7), which is isomorphic to ${\cal T}_{\!L^{{}^{W}}_{\cal F}}^{{}^{{Y}^{{}^{W}}}}$ (cf.  4.6.2).

4.10. At this point, it follows from Proposition  3.7 that there exist an ${\cal F}^{{}^{{Y},W}}\!\hbox{-}$locality functor ${l}_{{}_{{\cal F}}}^{{}^{{Y},W}}\,\colon{\cal T}_{\!L^{{}^{W}}_{\cal F}}^{{}^{{Y}^{{}^{W}}}}\to\overline{{\cal T}}^{{}^{{Y}^{{}^{W}}}}_{\!G^{{}^{\Omega_{{}_{W}}}}}$ which sends ${\cal T}_{\!L^{{}^{W}}_{\cal F}}^{{}^{{Y}^{{}^{W}}}}$ isomorphically to ${\cal P}^{{}^{\Omega,{Y},W}}\,,$ and that this functor is naturally ${\cal F}^{{}^{{Y},W}}\!\hbox{-}$isomorphic to the inclusion ${\cal T}_{\!L^{{}^{W}}_{\cal F}}^{{}^{{Y}^{{}^{W}}}}\i\overline{{\cal T}}^{{}^{{Y}^{{}^{W}}}}_{\!G^{{}^{\Omega_{{}_{W}}}}}$ in  4.8 above; that is to say, according to our definition in [5, 2.9] and since the kernel of the structural group homomorphism from

$\overline{{\cal T}}^{{}^{{Y}^{{}^{W}}}}_{\!G^{{}^{\Omega_{{}_{W}}}}}(P^{{}^{W}})$ to ${\cal F}^{{}^{{Y},W}}\!(P^{{}^{W}})$ is the image of $C_{\!G^{{}^{\Omega_{{}_{W}}}}}(P^{{}^{W}})\i{\cal T}_{\!G^{{}^{\Omega_{{}_{W}}}}}(P^{{}^{W}})\,,$ there is $z\in C_{\!G^{{}^{\Omega_{{}_{W}}}}}(P^{{}^{W}})$ such that, denoting by $\overline{z}^{{}^{{Y}^{{}^{W}}}}_{{}_{Q}}$ the image of $z$ in $\overline{{\cal T}}^{{}^{{Y}^{{}^{W}}}}_{\!G^{{}^{\Omega_{{}_{W}}}}}(Q)$ for any $Q\in{Y}^{{}^{W}}\,,$ we get

[TABLE]

in $\overline{{\cal T}}^{{}^{{Y}^{{}^{W}}}}_{\!G^{{}^{\Omega_{{}_{W}}}}}(Q,R)\,,$ for any pair of groups $Q$ and $R$ in ${Y}^{{}^{W}}\,.$

4.11. But, we also can consider the images $\overline{z}^{{}^{{X}^{{}^{W}}}}_{{}_{Q}}$ of $z$ in $\overline{{\cal T}}^{{}^{{X}^{{}^{W}}}}_{\!G^{{}^{\Omega_{{}_{W}}}}}(Q)$ for any $Q\in{X}^{{}^{W}}\,.$ Hence, up to replacing our choice of ${\cal T}_{\!L^{{}^{W}}_{\cal F}}^{{}^{{X}^{{}^{W}}}}\!$ as a ${\cal F}^{{}^{{X},W}}\!\!\hbox{-}$sublocality of $\overline{{\cal T}}^{{}^{{X}^{{}^{W}}}}_{\!G^{{}^{\Omega_{{}_{W}}}}}$ by the choice of $\overline{z}^{{}^{{X}^{{}^{W}}}}_{{}_{Q}}\!\!{\cdot}{\cal T}_{\!L^{{}^{W}}_{\cal F}}^{{}^{{X}^{{}^{W}}}}(Q,R){\cdot}(\overline{z}^{{}^{{X}^{{}^{W}}}}_{{}_{R}}\!)^{-1}$ in $\overline{{\cal T}}^{{}^{{X}^{{}^{W}}}}_{\!G^{{}^{\Omega_{{}_{W}}}}}(Q,R)\,,$ for any pair of groups $Q$ and $R$ in ${X}^{{}^{W}}\!\,,$ in $\overline{{\cal T}}^{{}^{{Y}^{{}^{W}}}}_{\!G^{{}^{\Omega_{{}_{W}}}}}$ we actually may assume that we get

[TABLE]

In this situation, it follows from our definitions in  4.7 above that in $\overline{{\cal T}}^{{}^{{X}^{{}^{W}}}}_{\!G^{{}^{\Omega_{{}_{W}}}}}$ the coherent ${\cal F}^{{}^{{X},W}}\!\hbox{-}$sublocality ${\cal M}^{{}^{\Omega,{X},W}}$ contains ${\cal T}_{\!L^{{}^{W}}_{\cal F}}^{{}^{{X}^{{}^{W}}}}\!\,.$

4.12. In particular, if ${X}^{{}^{W}}\!={Y}^{{}^{W}}\!$ then we have

[TABLE]

so that we are done. Assume that ${X}^{{}^{W}}\!\not={Y}^{{}^{W}}\!\,;$ then, by the very definition of $\overline{{\cal T}}^{{}^{{X}^{{}^{W}}}}_{\!G^{{}^{\Omega_{{}_{W}}}}}$ (cf.  3.6.1 and  4.3), for any $V\in{X}^{{}^{W}}\!-{Y}^{{}^{W}}\!$ we have

[TABLE]

and therefore, since $p$ does not divide $|\tilde{\cal F}^{{}^{W}}\!(P^{{}^{W}}\!,V)|$ [3, Proposition 6.7], we have a surjective group homomorphism

[TABLE]

mapping $z=(z_{\tilde{\theta}})_{\tilde{\theta}\in\tilde{\cal F}^{{}^{W}}\!(P^{{}^{W}}\!,V)}$ on

[TABLE]

4.13. At this point, considering the contravariant functor

[TABLE]

mapping any $Q\in{Y}^{{}^{W}}$ on $\{0\}$ and any $V\in{X}^{{}^{W}}-{Y}^{{}^{W}}$ on ${\rm Ker}(\nabla^{{}^{\Omega,{X},W}}_{V})\,,$ and the quotient ${\cal F}^{{}^{{X},W}}\!\hbox{-}$locality \overline{{\cal T}}^{{}^{{X}^{{}^{W}}}}_{\!G^{{}^{\Omega_{{}_{W}}}}}\big{/}{d}^{{}^{\Omega,{X},W}} [5, 2.10], it is easily checked that the coherent ${\cal F}^{{}^{{X},W}}\!\hbox{-}$localities ${\cal P}^{{}^{\Omega,{X},W}}$ (cf.  4.7) and ${\cal T}_{\!L^{{}^{W}}_{\cal F}}^{{}^{{X}^{{}^{W}}}}$ have the same image in this quotient; indeed, it follows from equalities  4.11.1 above that their images coincide over ${Y}^{{}^{W}}$ and, since for any $V\in{X}^{{}^{W}}-{Y}^{{}^{W}}$ we have

[TABLE]

${\cal P}^{{}^{\Omega,{X},W}}(V)$ and ${\cal T}_{\!L^{{}^{W}}_{\cal F}}^{{}^{{X}^{{}^{W}}}}(V)$ map both isomorphically onto \big{(}\overline{{\cal T}}^{{}^{{X}^{{}^{W}}}}_{\!G^{{}^{\Omega_{{}_{W}}}}}\big{/}{d}^{{}^{\Omega,{X},W}}\big{)}(V)\,. In particular, we get ${\cal P}^{{}^{\Omega,{X},W}}\cong{\cal T}_{\!L^{{}^{W}}_{\cal F}}^{{}^{{X}^{{}^{W}}}}$ since the functors from ${\cal P}^{{}^{\Omega,{X},W}}$ and ${\cal T}_{\!L^{{}^{W}}_{\cal F}}^{{}^{{X}^{{}^{W}}}}$ to the quotient \overline{{\cal T}}^{{}^{{X}^{{}^{W}}}}_{\!G^{{}^{\Omega_{{}_{W}}}}}\big{/}{d}^{{}^{\Omega,{X},W}} are faithful. This proves our claim in  4.6.

4.14. It remains to prove the uniqueness; thus, assume that ${\cal P}^{{}^{X}}$ and ${\cal P}^{\prime^{X}}$ are two extendable perfect ${\cal F}^{{}^{X}}\hbox{-}$localities; it follows from Proposition  3.7 that we may assume that both are ${\cal F}^{{}^{X}}\hbox{-}$sublocalities of the ${\cal F}^{{}^{X}}\hbox{-}$locality $\overline{{\cal T}}_{\!G^{{}^{\Omega}}}^{{}^{X}}$ introduced in  3.6 above. On the other hand, since the respective full sub-categories ${\cal P}^{{}^{Y}}$ of ${\cal P}^{{}^{X}}$ and ${\cal P}^{\prime^{Y}}$ of ${\cal P}^{\prime^{X}}$ over ${Y}$ as the set of objects are still two extendable perfect ${\cal F}^{{}^{Y}}\hbox{-}$localities, it follows from our induction hypothesis that they are ${\cal F}^{{}^{Y}}\hbox{-}$locality isomorphic. Consequently, considering the inclusions of ${\cal P}^{{}^{Y}}$ and ${\cal P}^{\prime^{Y}}$ in $\overline{{\cal T}}_{\!G^{{}^{\Omega}}}^{{}^{Y}}$ induced by the inclusions

[TABLE]

and by the canonical functor $(\overline{{\cal T}}^{{}^{{X}}}_{\!G^{{}^{\Omega}}})^{{}^{{Y}}}\!\!\to\overline{{\cal T}}^{{}^{{Y}}}_{\!G^{{}^{\Omega}}}$ (cf.  4.9.1), the existence of an ${\cal F}^{{}^{Y}}\hbox{-}$locality isomorphism ${\cal P}^{{}^{Y}}\cong{\cal P}^{\prime^{Y}}$ determines two ${\cal F}^{{}^{Y}}\hbox{-}$locality functors from ${\cal P}^{{}^{Y}}$ to $(\overline{{\cal T}}_{\!G^{{}^{\Omega}}}^{{}^{X}})^{{}^{Y}}\,;$ then, it follows again from Proposition  3.7 that the functors ainsi obtained are naturally ${\cal F}^{{}^{Y}}\hbox{-}$isomorphic.

4.15. That is to say, as in  4.10 above, since the kernel of the structural group homomorphism from $\overline{{\cal T}}^{{}^{{Y}}}_{\!G^{{}^{\Omega}}}(P)$ to ${\cal F}^{{}^{{Y}}}\!(P)$ is the image of $C_{\!G^{{}^{\Omega}}}(P)$ in ${\cal T}^{{}^{{Y}}}_{\!G^{{}^{\Omega}}}(P)\,,$ there is $z\in C_{\!G^{{}^{\Omega}}}(P)$ such that, denoting by $\overline{z}^{{}^{{Y}}}_{{}_{Q}}$ the image of $z$ in $\overline{{\cal T}}^{{}^{{Y}}}_{\!G^{{}^{\Omega}}}(Q)$ for any $Q\in{Y}\,,$ in $\overline{{\cal T}}^{{}^{{Y}}}_{\!G^{{}^{\Omega}}}(Q,R)$ we get

[TABLE]

for any pair of groups $Q$ and $R$ in ${Y}\,.$ As above, considering the images $\overline{z}^{{}^{{X}}}_{{}_{Q}}$ of $z$ in $\overline{{\cal T}}^{{}^{{X}}}_{\!G^{{}^{\Omega}}}(Q)$ for any $Q\in{X}$ and modifying our choice of ${\cal P}^{\prime^{{X}}}\!$ as a ${\cal F}^{{}^{{X}}}\!\!\hbox{-}$sublocality of $\overline{{\cal T}}^{{}^{{X}}}_{\!G^{{}^{\Omega}}}$ by the choice of $\overline{z}^{{}^{{X}}}_{{}_{Q}}{\cdot}{\cal P}^{\prime^{{X}}}(Q,R){\cdot}(\overline{z}^{{}^{{X}}}_{{}_{R}}\!)^{-1}$ in $\overline{{\cal T}}^{{}^{{X}}}_{\!G^{{}^{\Omega}}}(Q,R)$ for any pair of groups $Q$ and $R$ in ${X}\,,$ we actually may assume that in $\overline{{\cal T}}^{{}^{{Y}^{{}^{W}}}}_{\!G^{{}^{\Omega_{{}_{W}}}}}$ we have ${\cal P}^{{}^{{Y}}}={\cal P}^{\prime^{{Y}}}\,.$

4.16. Moreover, as in  4.12 above, by the very definition of $\overline{{\cal T}}^{{}^{{X}}}_{\!G^{{}^{\Omega}}}$ (cf.  3.6.1 and  4.3), for any $V\in{X}-{Y}$ we have

[TABLE]

and therefore, since $p$ does not divide $|\tilde{\cal F}(P,V)|$ [3, Proposition 6.7], we have a surjective group homomorphism

[TABLE]

mapping $z=(z_{\tilde{\theta}})_{\tilde{\theta}\in\tilde{\cal F}(P,V)}$ on

[TABLE]

4.17. At this point, considering the contravariant Dirac functor

[TABLE]

mapping any $Q\in{Y}$ on $\{0\}$ and any $V\in{X}-{Y}$ on ${\rm Ker}(\nabla^{{}^{\Omega,{X}}}_{V})\,,$ and the quotient ${\cal F}^{{}^{{X}}}\hbox{-}$locality \overline{{\cal T}}^{{}^{{X}}}_{\!G^{{}^{\Omega}}}\big{/}{d}^{{}^{\Omega,{X}}} [5, 2.10], it is easily checked that the coherent ${\cal F}^{{}^{{X}}}\hbox{-}$localities ${\cal P}^{{}^{{X}}}$ and ${\cal P}^{\prime^{{X}}}$ have the same image in this quotient; indeed, it follows from  4.15 above that their images coincide over ${Y}$ and, since for any $V\in{X}-{Y}$ we have

[TABLE]

${\cal P}^{{}^{{X}}}(V)$ and ${\cal P}^{\prime^{{X}}}(V)$ map both isomorphically onto \big{(}\overline{{\cal T}}^{{}^{{X}}}_{\!G^{{}^{\Omega}}}\big{/}{d}^{{}^{\Omega,{X}}}\big{)}(V)\,. In particular, we get ${\cal P}^{{}^{{X}}}\cong{\cal P}^{\prime^{{X}}}$ since the functors from ${\cal P}^{{}^{{X}}}$ and ${\cal P}^{\prime^{{X}}}$ to the quotient \overline{{\cal T}}^{{}^{{X}}}_{\!G^{{}^{\Omega}}}\big{/}{d}^{{}^{\Omega,{X}}} are faithful. This proves the uniqueness.

5. Existence and uniqueness of the sections from ${\cal F}^{{}^{X}}$ to $\bar{\cal M}^{{}^{\Omega,{X}}}$

5.1. With the hypothesis and notation in  4.3 above, our purpose in this section is to prove that

Theorem. The structural functor $\bar{\rho}^{{}^{\Omega,{X}}}\!\,\colon\bar{\cal M}^{{}^{\Omega,{X}}}\!\to{\cal F}^{{}^{{X}}}$ admits an ${\cal F}^{{}^{{X}}}\!\hbox{-}$locality functorial section $\bar{\sigma}^{{}_{\Omega,{X}}}\,\colon{\cal F}^{{}^{{X}}}\to\bar{\cal M}^{{}^{\Omega,{X}}}\!\,.$

Actually, since we assume that $U\not=P\,,$ we also have $U\not=N_{P}(U)=P^{{}^{U}}$ and therefore $P^{{}^{U}}$ belongs to ${Y}^{{}^{U}}\,;$ thus, this theorem is just the existence part of [5, Theorem 6.22] but we restate the proof in our new context; indeed, here we assume that ${\cal P}^{{}^{Y}}$ is an extendable perfect ${\cal F}^{{}^{Y}}\!\hbox{-}$locality and therefore the ${\cal F}^{{}^{{Y},U}}\!\hbox{-}$locality isomorphism in [5, 6.18]

[TABLE]

follows from our definition in  2.6; in particular, as in  4.11 above, in $\overline{{\cal T}}^{{}^{{Y}^{{}^{U}}}}_{\!G^{{}^{\!\Omega_{{}_{U}}}}}\!$ we may assume that ${\cal P}^{{}^{{Y},U}}={\cal T}_{\!L^{{}^{U}}_{\cal F}}^{{}^{{Y}^{{}^{U}}}}\,.$

5.2. Since ${Y}^{{}^{U}}$ is not empty, as in  4.7 above we can define the coherent ${\cal F}^{{}^{{Y},U}}\!\hbox{-}$locality ${\cal M}^{{}^{\Omega,{Y},U}}\!$ via the pull-back (cf.  4.2.3)

[TABLE]

and the coherent ${\cal F}^{{}^{{X},U}}\!\hbox{-}$localities ${\cal M}^{{}^{\Omega,{X},U}}\!\i\overline{{\cal T}}^{{}^{{X}^{{}^{U}}}}_{\!G^{{}^{\Omega_{{}_{W}}}}}$ and $\bar{\cal M}^{{}^{\Omega,{X},U}}\!$ as in  4.3, with the second structural functors

[TABLE]

Now recall that, denoting by $\tilde{\cal F}^{{}^{{X}}}$ and $\tilde{\cal F}^{{}^{{X},U}}$ the respective exterior quotients of ${\cal F}^{{}^{{X}}}$ and ${\cal F}^{{}^{{X},U}}$ [3, 1.3], the coherency of $\bar{\cal M}^{{}^{\Omega,{X}}}\!$ and $\bar{\cal M}^{{}^{\Omega,{X},U}}\!$ determines contravariant functors [5, 2.8.3]

[TABLE]

as usual, the existence of $\bar{\sigma}^{{}_{\Omega,{X}}}\!$ depends on the vanishing of the cohomology class of a suitable ${K}{e}{r}(\bar{\rho}^{{}^{\Omega,{X}}})\hbox{-}$valued $2\hbox{-}$cocycle and, from the reduction procedure developed in section  3, we will move to the corresponding ${K}{e}{r}(\bar{\rho}^{{}^{\Omega,{X},U}})\hbox{-}$va-lued $2\hbox{-}$cocycle.

5.3. From the commutative diagram  4.2.3 we get the following commutative diagram of the normalizers of $U$

[TABLE]

moreover, we are setting $N_{{\cal P}^{{}^{{Y}}}}(U)={\cal P}^{{}^{{Y},U}}$ and we have the commutative diagram  4.9.2 for $W=U\,.$ Consequently, the ${\cal F}^{{}^{{Y},U}}\!\hbox{-}$ and ${\cal F}^{{}^{{X},U}}\!\hbox{-}$locality functors (cf.  3.10.1)

[TABLE]

successively induce the new ${\cal F}^{{}^{{Y},U}}\hbox{-}$locality functor (cf.  5.2.1)

[TABLE]

and, moreover, the ${\cal F}^{{}^{{X},U}}\hbox{-}$locality functors (cf.  4.3)

[TABLE]

Similarly, since we are assuming that ${\cal P}^{{}^{{Y},U}}={\cal T}_{\!L^{{}^{U}}_{\cal F}}^{{}^{{Y}^{{}^{U}}}}$ (cf.  4.11.1), the ${\cal F}^{{}^{{X},U}}\!\hbox{-}$lo-cality functor (cf. 3.13.2)

[TABLE]

and the pull-back  5.2.1 above determine new ${\cal F}^{{}^{{X},U}}\!\hbox{-}$locality functors (cf.  4.3)

[TABLE]

5.4. At this point, denoting by $\tilde{\iota}_{{}_{{X},U}}\,\colon\tilde{\cal F}^{{}^{{X},U}}\!\to\tilde{\cal F}^{{}^{{X}}}$ the canonical functor, it is well-known that, for any $n\in{N}\,,$ the restriction induces a group homomorphism (cf.  5.1.2)

[TABLE]

moreover, $\bar{h}^{{}^{{X},U}}_{{}_{\Omega}}$ induces a natural map [5, 2.10.1]

[TABLE]

and therefore, for any $n\in{N}\,,$ we also get a group homomorphism

[TABLE]

In [5, Proposition 6.9, 6.12.3 and 6.21.7] we prove that, for any $n\in{N}\,,$ the composition of the homomorphisms  5.4.1 and  5.4.3 determines an isomorphism

[TABLE]

5.5. Let us explicit the announced ${K}{e}{r}(\bar{\rho}^{{}^{\Omega,{X}}})\hbox{-}$valued $2\hbox{-}$cocycle. For any ${\cal F}^{{}^{{X}}}\!\hbox{-}$morphism $\varphi\,\colon R\to Q\,,$ choose a lifting $x_{\varphi}$ in ${\cal M}^{{}^{\Omega,{X}}}\!(Q,R)$ (cf.  4.3) and denote by $\bar{x}_{\varphi}$ the image of $x_{\varphi}$ in $\bar{\cal M}^{{}^{\Omega,{X}}}\!(Q,R)\,;$ actually, we can do our choice in such a way that we have (cf.  4.3)

[TABLE]

for any $u\in Q\,,$ where $\kappa^{{}_{{X}}}_{{}_{Q}}(u)\in{\cal F}^{{}^{{X}}}\!(Q)$ denotes the conjugation by the image of $u\,;$ indeed, if we have $\kappa^{{}_{{X}}}_{{}_{Q}}(u)\circ\varphi=\varphi$ then we get $u=\varphi(z)$ for a suitable $z\in Z(R)\,;$ since $\bar{\cal M}^{{}^{\Omega,{X}}}$ is coherent, in this case we obtain

[TABLE]

More precisely, if $Q$ and $R$ are contained in $P^{{}^{U}}$ and $\varphi\,\colon R\to Q$ comes from an ${\cal F}^{{}^{{X},U}}\!\hbox{-}$morphism, it is quite clear that we may assume that $x_{\varphi}$ belongs to \big{(}N_{\!{\cal M}^{{}^{\Omega,{X}}}}(U)\big{)}(Q,R) and then that ${h}^{{}^{{X},U}}_{{}_{\Omega}}(x_{\varphi})$ belongs to the image of ${\cal T}_{\!L^{{}^{U}}_{\cal F}}^{{}^{{X}^{{}^{U}}}}\!(Q,R)$ via ${l}_{{}_{{\cal F}}}^{{}^{{X},U}}\,,$ so that actually we have (cf.  5.3.6)

[TABLE]

5.6. Then, for any triple of subgroups $Q\,,$ $R$ and $T$ in ${X}\,,$ and any pair of ${\cal F}\hbox{-}$morphisms $\psi\,\colon T\to R$ and $\varphi\,\colon R\to Q\,,$ since $x_{\varphi}{\cdot}x_{\psi}$ and $x_{\varphi\circ\psi}$ have the same image $\varphi\circ\psi$ in ${\cal F}(Q,T)\,,$ the divisibility of ${\cal M}^{{}^{\Omega,{X}}}$ guarantees the existence and the uniqueness of $k_{\varphi,\psi}\in{\rm Ker}(\rho^{{}_{\Omega,{X}}}_{T})$ fulfilling

[TABLE]

Denote by $\bar{k}_{\varphi,\psi}$ the image of $k_{\varphi,\psi}$ in ${\rm Ker}(\bar{\rho}^{{}_{\Omega,{X}}}_{T})\,;$ since $\bar{\cal M}^{{}^{\Omega,{X}}}\!$ is coherent, on the one hand for any $u\in Q$ and any $v\in R$ we get (cf.  5.5.1)

[TABLE]

hence, from the divisibility of $\bar{\cal M}^{{}^{\Omega,{X}}}$ we obtain

[TABLE]

That is to say, for any $n\in{N}\,,$ setting [3, 1.5]

[TABLE]

we have obtained an element $\bar{k}=\{\bar{k}_{\tilde{q}}\}_{\tilde{q}\in{F}{c}{t}(\Delta_{2},\tilde{\cal F}^{{}^{{X}}})}$ in {C}^{2}\big{(}\tilde{\cal F}^{{}^{\Omega,{X}}}\!,{K}{e}{r}(\bar{\rho}^{{}_{\Omega,{X}}})\big{)} where we set $\bar{k}_{\tilde{q}}=\bar{k}_{\tilde{q}(1\bullet 2),\tilde{q}(0\bullet 1)}=\bar{k}_{{q}(1\bullet 2),{q}(0\bullet 1)}$ for some representative ${q}\,\colon\Delta_{2}\to{\cal F}^{{}^{{X}}}$ of $\tilde{q}\,.$

5.7. We claim that $\bar{k}$ is actually a $2\hbox{-}$cocycle; explicitly, considering the usual differential map [3, A13.11]

[TABLE]

we claim that $\bar{d}^{{}^{\Omega,{X}}}_{2}(\bar{k})=0\,;$ indeed, with the notation above, for a third ${\cal F}^{{}^{{X}}}\hbox{-}$morphism $\eta\,\colon W\to T$ we get

[TABLE]

and the divisibility of $\bar{\cal M}^{{}^{{X}}}$ forces

[TABLE]

since ${\rm Ker}(\bar{\rho}^{{}^{\Omega,{X}}})$ is Abelian, in the additive notation we obtain

[TABLE]

proving our claim.

5.8. Then, in order to prove the existence of a section $\bar{\sigma}^{{}_{\Omega,{X}}}\,,$ it suffices to show that $\bar{k}$ is a $2\hbox{-}$coboundary and therefore, according to isomorphism  5.4.4 above, it suffices to prove that the image via $\nu_{\bar{h}^{{}^{{X},U}}_{{}_{\Omega}}}\!$ (cf.  5.4.2) of the restriction of $\bar{k}$ to $\tilde{\cal F}^{{}^{{X},U}}$ is a $2\hbox{-}$coboundary. But, for any pair of ${\cal F}^{{}^{{X},U}}\!\hbox{-}$morphisms $\varphi\,\colon R\to Q$ and $\psi\,\colon T\to R\,,$ we have chosen $x_{\varphi}$ in \big{(}N_{\!{\cal M}^{{}^{\Omega,{X}}}\!}(U)\big{)}(Q,R)\,, $x_{\psi}$ in \big{(}N_{\!{\cal M}^{{}^{\Omega,{X}}}\!}(U)\big{)}(R,T) and $x_{\varphi\circ\psi}$ in \big{(}N_{\!{\cal M}^{{}^{\Omega,{X}}}\!}(U)\big{)}(Q,T)\,, so that in equality  5.6.1 the element $k_{\varphi,\psi}$ belongs to \big{(}N_{\!{\cal M}^{{}^{\Omega,{X}}}\!}(U)\big{)}(T) and therefore we get (cf.  5.3.4)

[TABLE]

and therefore we still get

[TABLE]

so that equalities  5.5.3 force $\bar{h}^{{}^{{X},U}}_{{}_{\Omega}}(\bar{k}_{\varphi,\psi})=1\,;$ that is to say, the image via $\nu_{\bar{h}^{{}^{{X},U}}_{{}_{\Omega}}}$ of the restriction of $\bar{k}$ to $\tilde{\cal F}^{{}^{{X},U}}$ is just trivial, proving that $\bar{k}$ is a $2\hbox{-}$coboundary.

5.9. Thus, we have obtained a functorial section $\bar{\sigma}^{{}_{\Omega,{X}}}\,\colon{\cal F}^{{}^{{X}}}\to\bar{\cal M}^{{}^{\Omega,{X}}}$ of $\bar{\rho}^{{}^{\Omega,{X}}}\,;$ actually, $\bar{\sigma}^{{}_{\Omega,{X}}}$ can be modified in order to get an ${\cal F}^{{}^{X}}\!\hbox{-}$locality functorial section [5, 2.9]. Indeed, for any ${\cal F}^{{}^{X}}_{\!P}\hbox{-}$morphism $\zeta\,\colon R\to Q\,,$ choosing $u_{\zeta}$ in $T_{P}(R,Q)$ lifting $\zeta\,,$ both $\bar{\cal M}^{{}^{\Omega,{X}}}\hbox{-}$morphisms $\bar{\sigma}^{{}_{\Omega,{X}}}_{{}_{Q,R}}(\zeta)$ and $\bar{\upsilon}^{{}^{\Omega,{X}}}_{{}_{Q,R}}(u_{\zeta})$ (cf.  4.3) lift $\zeta\,;$ once again, the divisibility of $\bar{\cal M}^{{}^{\Omega,{X}}}$ guarantees the existence and the uniqueness of $\bar{m}_{\zeta}\in{\rm Ker}(\bar{\rho}^{{}^{\Omega,{X}}}_{{}_{R}})$ fulfilling

[TABLE]

actually, it follows easily from  5.5.1 that $\bar{m}_{\zeta}$ only depends on $\tilde{\zeta}\in\tilde{\cal F}_{\!P}(Q,R)$ and, as above, we write $\bar{m}_{\tilde{\xi}}$ instead of $m_{\xi}\,;$ moreover, for a second ${\cal F}^{{}^{X}}_{\!P}\hbox{-}$mor-phism $\xi\,\colon T\to R\,,$ we get

[TABLE]

5.10. Then, always the divisibility of $\bar{\cal M}^{{}^{\Omega,{X}}}$ forces

[TABLE]

and, since ${\rm Ker}(\bar{\rho}^{{}^{\Omega,{X}}}_{T})$ is Abelian (cf. Proposition  3.4,  3.5 and  3.6), in the additive notation we obtain

[TABLE]

that is to say, denoting by $\tilde{\iota}^{{}_{X}}_{P}\,\colon\tilde{\cal F}^{{}^{X}}_{\!P}\i\tilde{\cal F}^{{}^{X}}$ the obvious inclusion functor, the correspondence $\bar{m}$ sending any $\tilde{\cal F}^{{}^{{X}}}_{\!P}\hbox{-}$morphism $\tilde{\zeta}\,\colon R\to Q$ to $\bar{m}_{\tilde{\zeta}}$ defines a $1\hbox{-}$cocycle in {C}^{1}\big{(}\tilde{\cal F}^{{}^{{X}}}_{\!P},{K}{e}{r}(\bar{\rho}^{{}^{\Omega,{X}}})\circ\tilde{\iota}^{{}_{X}}_{P}\big{)}\,; but, since the category $\tilde{\cal F}^{{}^{X}}_{\!P}$ obviously has a final object, we actually have [3, Corollary A4.8]

[TABLE]

consequently, we obtain $\bar{m}=d^{{}^{\Omega,{X}}}_{0}(\bar{w})$ for some element $\bar{w}=(\bar{w}_{Q})_{Q\in{X}}$ in

[TABLE]

In conclusion, equality  5.9.1 becomes

[TABLE]

thus, the new correspondence which, for any pair of subgroups $Q$ and $R$ in ${X}\,,$ sends any $\varphi\in{\cal F}(Q,R)$ to $\bar{w}_{Q}{\cdot}\bar{\sigma}^{{}_{\Omega,{X}}}_{{}_{Q,R}}(\varphi){\cdot}\bar{w}_{R}^{-1}$ defines an ${\cal F}^{{}^{{X}}}\!\hbox{-}$locality functorial section of $\bar{\rho}^{{}^{{X}}}\,.$ We are done.

References

[1] Bob Oliver, A remark on the construction of centric linking systems, arxiv.org/abs/1612.02132 [2]. Lluís Puig, Frobenius categories, Journal of Algebra, 303(2006), 309-357. [3]. Lluís Puig, “Frobenius categories versus Brauer blocks”, Progress in Math. 274(2009), Birkhäuser, Basel. [4]. Lluís Puig, A criterion on trivial homotopy, arxiv.org/abs/1308.3765. [5]. Lluís Puig, Existence, uniqueness and functoriality of the perfect locality over a Frobenius $P\hbox{-}$category, Algebra Colloquium, 23(2016) 541-622.