Detecting nilpotence and projectivity over finite unipotent supergroup schemes

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

doi:10.1007/s00029-021-00632-7

Cited by 8 publications

(21 citation statements)

References 34 publications

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“…In this paper we develop a theory of π-points for the elementary supergroup schemes E introduced in [13], and classify the localising subcategories of its stable module category. This feeds into the proof of a similar classification for finite unipotent supergroup schemes, presented in [14]. It transpires that rather than flat maps from K[t]/(t p ), we have to consider the K-algebra…”

Section: Introductionmentioning

confidence: 81%

Rank varieties and $π$-points for elementary supergroup schemes

Benson¹,

Iyengar²,

Krause³

et al. 2020

Preprint

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We develop a support theory for elementary supergroup schemes, over a field of positive characteristic p 3, starting with a definition of a πpoint generalising cyclic shifted subgroups of Carlson for elementary abelian groups and π-points of Friedlander and Pevtsova for finite group schemes. These are defined in terms of maps from the graded algebra k[t, τ ]/(t p − τ 2 ), where t has even degree and τ has odd degree. The strength of the theory is demonstrated by classifying the parity change invariant localising subcategories of the stable module category of an elementary supergroup scheme. Contents 16 8. The remaining cases 19 Appendix A. Complete intersections 20 References 24

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Section: Introductionmentioning

confidence: 81%

Rank varieties and $π$-points for elementary supergroup schemes

Benson¹,

Iyengar²,

Krause³

et al. 2020

Preprint

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“…, where E − m,n is the Witt elementary, explicitly described in [13,Definition 3.3]. The main theorem of that paper states that if G is a unipotent finite supergroup scheme over k then a kG-module M is projective if and only if, for all extension fields K, and all elementary sub-supergroup schemes E of G K , the module M K = K ⊗ k M is a projective KE-module.…”

Section: Elementary Supergroup Schemesmentioning

confidence: 99%

“…In [13] we began a program to extend all these results to the world of supergroup schemes. In that work we identified a family of elementary supergroup schemes and proved that the projectivity of modules over a unipotent supergroup scheme can be detected by its restrictions to the elementary ones, possibly defined over extensions fields.…”

Section: Introductionmentioning

confidence: 99%

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Rank varieties and 𝜋-points for elementary supergroup schemes

Benson

Iyengar

Krause

et al. 2021

Trans. Amer. Math. Soc. Ser. B

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We develop a support theory for elementary supergroup schemes, over a field of positive characteristic p ⩾ 3 p\geqslant 3 , starting with a definition of a π \pi -point generalising cyclic shifted subgroups of Carlson for elementary abelian groups and π \pi -points of Friedlander and Pevtsova for finite group schemes. These are defined in terms of maps from the graded algebra k [ t , τ ] / ( t p − τ 2 ) k[t,\tau ]/(t^p-\tau ^2) , where t t has even degree and τ \tau has odd degree. The strength of the theory is demonstrated by classifying the parity change invariant localising subcategories of the stable module category of an elementary supergroup scheme.

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“…In joint work with Iyengar and Krause [6], we study the question of detecting nilpotents in the cohomology of a finite supergroup scheme, or equivalently, a finite dimensional Z/2graded cocommutative Hopf superalgebra. We establish a detecting family in the case of a unipotent finite supergroup scheme which turns out to have a surprisingly more complicated structure than what one sees in the ungraded case in the detection theorems of Quillen and Suslin.…”

Section: Introductionmentioning

confidence: 99%

Representations and cohomology of a family of finite supergroup schemes

Pevtsova

2020

Journal of Algebra

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We examine the cohomology and representation theory of a family of finite supergroup schemes of the form (G − a ×G − a )⋊(G a(r) ×(Z/p) s ). In particular, we show that a certain relation holds in the cohomology ring, and deduce that for finite supergroup schemes having this as a quotient, both cohomology mod nilpotents and projectivity of modules is detected on proper sub-supergroup schemes. This special case feeds into the proof of a more general detection theorem for unipotent finite supergroup schemes, in a separate work of the authors joint with Iyengar and Krause.We also completely determine the cohomology ring in the smallest cases, namelyThe computation uses the local cohomology spectral sequence for group cohomology, which we describe in the context of finite supergroup schemes.

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Detecting nilpotence and projectivity over finite unipotent supergroup schemes

Cited by 8 publications

References 34 publications

Rank varieties and $π$-points for elementary supergroup schemes

Rank varieties and $π$-points for elementary supergroup schemes

Rank varieties and 𝜋-points for elementary supergroup schemes

Representations and cohomology of a family of finite supergroup schemes

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