Dave Benson, Srikanth B. Iyengar, Henning Krause, Julia Pevtsova
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.
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