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      Higher limits over the fusion orbit category

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      Author(s)
      Yalçın, Ergün
      Date
      2022-06-09
      Source Title
      Advances in Mathematics
      Print ISSN
      0001-8708
      Electronic ISSN
      1090-2082
      Publisher
      Elsevier
      Volume
      406
      Pages
      1 - 69
      Language
      English
      Type
      Article
      Item Usage Stats
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      Abstract
      The fusion orbit category F‾C(G) of a discrete group G over a collection C is the category whose objects are the subgroups H in C, and whose morphisms H→K are given by the G-maps G/H→G/K modulo the action of the centralizer group CG(H). We show that the higher limits over F‾C(G) can be computed using the hypercohomology spectral sequences coming from the Dwyer G-spaces for centralizer and normalizer decompositions for G. If G is the discrete group realizing a saturated fusion system F, then these hypercohomology spectral sequences give two spectral sequences that converge to the cohomology of the centric orbit category Oc(F). This allows us to apply our results to the sharpness problem for the subgroup decomposition of a p-local finite group. We prove that the subgroup decomposition for every p-local finite group is sharp (over F-centric subgroups) if it is sharp for every p-local finite group with nontrivial center. We also show that for every p-local finite group (S,F,L), the subgroup decomposition is sharp if and only if the normalizer decomposition is sharp.
      Keywords
      Fusion systems
      Higher limits
      Orbit category
      p-local finite group
      Cohomology of small categories
      Permalink
      http://hdl.handle.net/11693/111483
      Published Version (Please cite this version)
      https://dx.doi.org/10.1016/j.aim.2022.108482
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