Detecting nilpotence and projectivity over finite unipotent supergroup schemes

• Dave Benson ^[1] ; Srikanth B. Iyengar ^[2] ; Henning Krause ^[3] ; Julia Pevtsova ^[4]
  1. [1] University of Aberdeen

     University of Aberdeen

     Reino Unido

     
  2. [2] University of Utah

     University of Utah

     Estados Unidos

     
  3. [3] Bielefeld University

     Bielefeld University

     Kreisfreie Stadt Bielefeld, Alemania

     
  4. [4] University of Washington

     University of Washington

     Estados Unidos

     

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 27, Nº. 2, 2021
• Idioma: inglés
• DOI: 10.1007/s00029-021-00632-7
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• Resumen

  • This work concerns the representation theory and cohomology of a finite unipotent supergroup scheme G over a perfect field k of positive characteristic p≥3. It is proved that an element x in the cohomology of G is nilpotent if and only if for every extension field K of k and every elementary sub-supergroup scheme E⊆GK, the restriction of xK to E is nilpotent. It is also shown that a kG-module M is projective if and only if for every extension field K of k and every elementary sub-supergroup scheme E⊆GK, the restriction of MK to E is projective. The statements are motivated by, and are analogues of, similar results for finite groups and finite group schemes, but the structure of elementary supergroups schemes necessary for detection is more complicated than in either of these cases. One application is a detection theorem for the nilpotence of cohomology, and projectivity of modules, over finite dimensional Hopf subalgebras of the Steenrod algebra.