A proof of the sharpness conjectures for all saturated fusion systems

Abstract

We prove the Díaz$\unicode[Arial]{x2013}$Park conjecture for sharpness of saturated fusion systems. More generally, for every saturated fusion system $\mathcal F$ over a finite $p$-group and every Mackey functor for $\mathcal F$ over the localisation $\mathbb{Z}_{(p)}$ of the integers at $p$, the higher limits of its contravariant part over the centric orbit category vanish in all positive degrees. In particular, the subgroup and normaliser decompositions of every $p$-local finite group $(S,\mathcal F,\mathcal L)$ over the $\mathcal F$-centric subgroups are sharp. This answers in the affirmative a question in the list of important open problems in [M. Aschbacher, R. Kessar and B. Oliver, Fusion Systems in Algebra and Topology, Cambridge Univ. Press (2011)], and proves the sharpness conjecture stated in [M. Aschbacher and B. Oliver, Fusion systems, Bull. Amer. Math. Soc. (2016)] and Conjecture 1.2 of [E. Yalç$\unicode[Arial]{x131}$n, Higher limits over the fusion orbit category, Adv. Math. (2022)]. The proof is algebraic and does not rely on the classification of finite simple groups.