A proof of the sharpness conjectures for all saturated fusion systems

Baoyu Zhang; Shengmin Zhang

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arXiv · 2611.02043

A proof of the sharpness conjectures for all saturated fusion systems

Baoyu Zhang·Shengmin Zhang

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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.

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Baoyu Zhang, Shengmin Zhang. 2026-11-01. A proof of the sharpness conjectures for all saturated fusion systems. https://arxiv.org/abs/2611.02043

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