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 $\cal F$ 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.