Finite group schemes of periodpover a discrete valuation ring

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).

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Finite group schemes of p-rank ≤ 1

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004117000834 ◽

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.

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Extending Endo-monomial Modules

Algebra Colloquium ◽

10.1142/s1005386713000151 ◽

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.

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