A proof of the sharpness conjectures for all saturated fusion systems

@misc{zhang2026proof,
  title = {{A proof of the sharpness conjectures for all saturated fusion systems}},
  author = {Baoyu Zhang and Shengmin Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.02043},
  eprint = {2611.02043},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.02043},
}

TY  - UNPB
TI  - A proof of the sharpness conjectures for all saturated fusion systems
AU  - Baoyu Zhang
AU  - Shengmin Zhang
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.02043
UR  - https://arxiv.org/abs/2611.02043
AB  - We prove the Díazunicode[Arial]x2013Park conjecture for sharpness of saturated fusion systems. More generally, for every saturated fusion system ℱ over a finite p-group and every Mackey functor for ℱ over the localisation 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,ℱ,ℒ) over the 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. Yalçunicode[Arial]x131n, 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.
ER  -