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

Abstract

Let $p$ be a prime, $P$ a finite p-group and $\mathcal{F}$ 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 ${\mathcal{L}}^{\mathfrak{X}}$ over any up-closed set $\mathfrak{X}$ of $\mathcal{F}$-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 $\mathfrak{X}$ of $\mathcal{F}$-selfcentralizing subgroups of $P$, our arguments are correct, still proving the existence and the uniqueness of the perfect ${\mathcal{F}}^{\mathrm{sc}}$-locality, since it is "extendable". We take advantage to simplify some of our arguments.