Some Remarks on the Tensor Product of Algebras and Applications II

2007 ◽  
Vol 14 (01) ◽  
pp. 143-154 ◽  
Author(s):  
Morton E. Harris
Keyword(s):  
Tensor Product ◽  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Modular Representation ◽  
Group Algebras ◽  
Closed Field ◽  
Modular Representation Theory ◽  
Complete Discrete Valuation Ring ◽  
Algebra Theory ◽  
Green Correspondence

We apply the G-algebra theory to the tensor product of algebras. These considerations are applied to extend the results of Alghamdi and Khammash [1], Khammash [4] and Külshammer [5, Proposition 1.2] on the tensor product of group algebras and modules over an algebraically closed field to lattices over a complete discrete valuation ring. This places these results in the standard integral finite group modular representation theory of G-algebras as pioneered by Puig (cf. [8]). We also study some aspects of covering homomorphisms and the Green correspondence in this context (cf. [8, Sections 20 and 25]).

Analogs of the Green and Puig correspondences for twisted group algebras

2016 ◽  
Vol 19 (1) ◽  
pp. 1-24
Author(s):  
Morton E. Harris
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Finite Groups ◽  
Modular Representation ◽  
Group Algebras ◽  
Modular Representation Theory ◽  
Twisted Group ◽  
Standard Derivation ◽  
Green Correspondence

AbstractIn the modular representation theory of finite groups, we show that the standard derivation of the Green correspondence lifts to a derivation of a Green correspondence for twisted group algebras (Theorem 1.3). Then, from these results we derive a lift of the Puig correspondences for twisted group algebras (Theorem 1.6).Clearly twisted group algebras arise naturally in finite group modular representation theory. We conclude with some suggestions for applications in this mathematical area.

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).

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.

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.

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.

scholarly journals The First Monsky-Washnitzer Cohomology Group

1972 ◽  
Vol 48 ◽  
pp. 99-128
Author(s):  
David Meredith
Keyword(s):  
Finite Type ◽  
Characteristic Zero ◽  
Cohomology Group ◽  
Valuation Ring ◽  
Residue Field ◽  
Discrete Valuation Ring ◽  
Perfect Field ◽  
Quotient Field ◽  
Complete Discrete Valuation Ring

Throughout this paper, k is a perfect field of characteristic p > 0, R is a complete discrete valuation ring with residue field k and quotient field of characteristic zero, and Z is a connected smooth prescheme of finite type over k.

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