Detecting nilpotence and projectivity over finite unipotent supergroup schemes

Benson D, Iyengar SB, Krause H, Pevtsova J (2021)
Selecta Mathematica. New Series 27(2): 25.

Zeitschriftenaufsatz | Veröffentlicht | Englisch

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Autor*in
Benson, Dave; Iyengar, Srikanth B.; Krause, HenningUniBi; Pevtsova, Julia
Einrichtung
Abstract / Bemerkung
This work concerns the representation theory and cohomology of a finite unipotent supergroup scheme $G$ over a perfect field $k$ of positive characteristic $p\ge 3$. 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 $E\subseteq G_K$, the restriction of $x_K$ 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 $E\subseteq G_K$, the restriction of $M_K$ 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.
Erscheinungsjahr
2021
Zeitschriftentitel
Selecta Mathematica. New Series
Band
27
Ausgabe
2
Art.-Nr.
25
ISSN
1022-1824
eISSN
1420-9020
Page URI
https://pub.uni-bielefeld.de/record/2954674

Zitieren

Benson D, Iyengar SB, Krause H, Pevtsova J. Detecting nilpotence and projectivity over finite unipotent supergroup schemes. Selecta Mathematica. New Series. 2021;27(2): 25.
Benson, D., Iyengar, S. B., Krause, H., & Pevtsova, J. (2021). Detecting nilpotence and projectivity over finite unipotent supergroup schemes. Selecta Mathematica. New Series, 27(2), 25. https://doi.org/10.1007/s00029-021-00632-7
Benson, D., Iyengar, S. B., Krause, H., and Pevtsova, J. (2021). Detecting nilpotence and projectivity over finite unipotent supergroup schemes. Selecta Mathematica. New Series 27:25.
Benson, D., et al., 2021. Detecting nilpotence and projectivity over finite unipotent supergroup schemes. Selecta Mathematica. New Series, 27(2): 25.
D. Benson, et al., “Detecting nilpotence and projectivity over finite unipotent supergroup schemes”, Selecta Mathematica. New Series, vol. 27, 2021, : 25.
Benson, D., Iyengar, S.B., Krause, H., Pevtsova, J.: Detecting nilpotence and projectivity over finite unipotent supergroup schemes. Selecta Mathematica. New Series. 27, : 25 (2021).
Benson, Dave, Iyengar, Srikanth B., Krause, Henning, and Pevtsova, Julia. “Detecting nilpotence and projectivity over finite unipotent supergroup schemes”. Selecta Mathematica. New Series 27.2 (2021): 25.

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arXiv: 1901.08273