Extensions of finite group schemes, and Hopf Galois theory over a complete discrete valuation ring

1992 ◽  
Vol 210 (1) ◽  
pp. 37-67 ◽  
Author(s):  
C. Greither
Keyword(s):  
Finite Group ◽  
Valuation Ring ◽  
Galois Theory ◽  
Discrete Valuation Ring ◽  
Group Schemes ◽  
Complete Discrete Valuation Ring ◽  
Finite Group Schemes

The Reduction of an RG–Lattice Modulo pn

1990 ◽  
Vol 42 (2) ◽  
pp. 342-364 ◽  
Author(s):  
Peter Symonds
Keyword(s):  
Finite Group ◽  
Characteristic Zero ◽  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Prime Element ◽  
Class Field ◽  
Complete Discrete Valuation Ring ◽  
Projective Lattice ◽  
A Finite Group

We define the cover of an RG-module V to consist of an RG lattice Ṽ and a homomorphism π : Ṽ→ V such that π induces an isomorphism on Ext*RG(M, —) for any RG-lattice M. Here G is a finite group and, for simplicity in this introduction, R is a complete discrete valuation ring of characteristic zero with prime element p and perfect valuation class field. Let pn(G) be the highest power of p that divides |G| and, given an RG-lattice M, let pn(M) be the smallest power of p such that pn(M) idM : M→M factors through a projective lattice: n(M)≦n(G).

Extending Endo-monomial Modules

2013 ◽  
Vol 20 (01) ◽  
pp. 169-172
Author(s):  
Ziqun Lu ◽  
Jiping Zhang
Keyword(s):  
Finite Group ◽  
Valuation Ring ◽  
Residue Field ◽  
Discrete Valuation Ring ◽  
Characteristic P ◽  
Complete Discrete Valuation Ring ◽  
A Finite Group

Let G be a finite group with a normal Sylow p-subgroup P. Let [Formula: see text] be a complete discrete valuation ring with residue field F of characteristic p. Let M be an indecomposable endo-monomial [Formula: see text]-module. In this paper we prove that M extends to an [Formula: see text]-module if and only if M is G-stable. A similar and well-known version for endo-permutation modules is due to Dade.

scholarly journals Stratification for module categories of finite group schemes

2017 ◽  
Vol 31 (1) ◽  
pp. 265-302 ◽  
Author(s):  
Dave Benson ◽  
Srikanth B. Iyengar ◽  
Henning Krause ◽  
Julia Pevtsova
Keyword(s):  
Finite Group ◽  
Group Schemes ◽  
Finite Group Schemes

Finite group schemes of p-rank ≤ 1

2017 ◽  
Vol 166 (2) ◽  
pp. 297-323
Author(s):  
HAO CHANG ◽  
ROLF FARNSTEINER
Keyword(s):  
Finite Group ◽  
Lie Algebras ◽  
Group Scheme ◽  
Closed Field ◽  
Difficult Case ◽  
Modular Representation Theory ◽  
Group Schemes ◽  
Restricted Lie Algebras ◽  
A Finite Group ◽  
Finite Group Schemes

AbstractLet be a finite group scheme over an algebraically closed field k of characteristic char(k) = p ≥ 3. In generalisation of the familiar notion from the modular representation theory of finite groups, we define the p-rank rkp() of and determine the structure of those group schemes of p-rank 1, whose linearly reductive radical is trivial. The most difficult case concerns infinitesimal groups of height 1, which correspond to restricted Lie algebras. Our results show that group schemes of p-rank ≤ 1 are closely related to those being of finite or domestic representation type.

scholarly journals On the lifting of the Dade group

2019 ◽  
Vol 22 (3) ◽  
pp. 441-451
Author(s):  
Caroline Lassueur ◽  
Jacques Thévenaz
Keyword(s):  
Valuation Ring ◽  
Residue Field ◽  
Discrete Valuation Ring ◽  
Group Homomorphism ◽  
Complete Discrete Valuation Ring ◽  
Dade Group

Abstract For the group of endo-permutation modules of a finite p-group, there is a surjective reduction homomorphism from a complete discrete valuation ring of characteristic 0 to its residue field of characteristic p. We prove that this reduction map always has a section which is a group homomorphism.

scholarly journals Arithmetic 𝒟-modules on the unit disk. With an appendix by Shigeki Matsuda

2011 ◽  
Vol 148 (1) ◽  
pp. 227-268 ◽  
Author(s):  
Richard Crew
Keyword(s):  
Unit Disk ◽  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Mixed Characteristic ◽  
Complete Discrete Valuation Ring ◽  
Canonical Extensions

AbstractLet 𝒱 be a complete discrete valuation ring of mixed characteristic. We classify arithmetic 𝒟-modules on Spf(𝒱[[t]]) up to certain kind of ‘analytic isomorphism’. This result is used to construct canonical extensions (in the sense of Katz and Gabber) for objects of this category.

𝒟-modules arithmétiques sur la variété de drapeaux

2019 ◽  
Vol 2019 (754) ◽  
pp. 1-15
Author(s):  
Christine Huyghe ◽  
Tobias Schmidt
Keyword(s):  
Algebraic Group ◽  
Valuation Ring ◽  
Differential Operators ◽  
Discrete Valuation Ring ◽  
Flag Variety ◽  
Nous Obtenons ◽  
Localization Theorem ◽  
Reductive Algebraic Group ◽  
Rigid Cohomology ◽  
Complete Discrete Valuation Ring

Abstract Soient p un nombre premier, V un anneau de valuation discrète complet d’inégales caractéristiques (0,p) , et G un groupe réductif et deployé sur \operatorname{Spec}V . Nous obtenons un théorème de localisation, en utilisant les distributions arithmétiques, pour le faisceau des opérateurs différentiels arithmétiques sur la variété de drapeaux formelle de G. Nous donnons une application à la cohomologie rigide pour des ouverts dans la variété de drapeaux en caractéristique p. Let p be a prime number, V a complete discrete valuation ring of unequal characteristics (0,p) , and G a connected split reductive algebraic group over \operatorname{Spec}V . We obtain a localization theorem, involving arithmetic distributions, for the sheaf of arithmetic differential operators on the formal flag variety of G. We give an application to the rigid cohomology of open subsets in the characteristic p flag variety.

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