scholarly journals Boundedness results for finite flat group schemes over discrete valuation rings of mixed characteristic

2012 ◽  
Vol 132 (9) ◽  
pp. 2003-2019
Author(s):  
Adrian Vasiu ◽  
Thomas Zink
Keyword(s):  
Mixed Characteristic ◽  
Group Schemes ◽  
Discrete Valuation Rings ◽  
Valuation Rings

scholarly journals On the Structure of Affine Flat Group Schemes Over Discrete Valuation Rings, II

2020 ◽  
Author(s):  
Phùng Hô Hai ◽  
João Pedro dos Santos
Keyword(s):  
Discrete Valuation Ring ◽  
Free Module ◽  
Galois Groups ◽  
Affine Group ◽  
Group Schemes ◽  
Infinite Type ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
D Modules ◽  
Tannakian Categories

Abstract In the first part of this work [ 12], we studied affine group schemes over a discrete valuation ring (DVR) by means of Neron blowups. We also showed how to apply these findings to throw light on the group schemes coming from Tannakian categories of $\mathcal{D}$-modules. In the present work, we follow up this theme. We show that a certain class of affine group schemes of “infinite type,” Neron blowups of formal subgroups, are quite typical. We also explain how these group schemes appear naturally in Tannakian categories of $\mathcal{D}$-modules. To conclude, we isolate a Tannakian property of affine group schemes, named prudence, which allows one to verify if the underlying ring of functions is a free module over the base ring. This is then successfully applied to obtain a general result on the structure of differential Galois groups over complete DVRs.

scholarly journals On cuspidal representations of general linear groups over discrete valuation rings

2010 ◽  
Vol 175 (1) ◽  
pp. 391-420 ◽  
Author(s):  
Anne-Marie Aubert ◽  
Uri Onn ◽  
Amritanshu Prasad ◽  
Alexander Stasinski
Keyword(s):  
General Linear ◽  
Linear Groups ◽  
General Linear Groups ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Cuspidal Representations

Right Invariant Right Hereditary Rings

1974 ◽  
Vol 26 (5) ◽  
pp. 1186-1191 ◽  
Author(s):  
H. H. Brungs
Keyword(s):  
Principal Ideal ◽  
Prime Ideals ◽  
Maximal Orders ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Dedekind Domains ◽  
Left Order ◽  
Principal Ideal Domains

Let R be a right hereditary domain in which all right ideals are two-sided (i.e., R is right invariant). We show that R is the intersection of generalized discrete valuation rings and that every right ideal is the product of prime ideals. This class of rings seems comparable with (and contains) the class of commutative Dedekind domains, but the rings considered here are in general not maximal orders and not Dedekind rings in the terminology of Robson [9]. The left order of a right ideal of such a ring is a ring of the same kind and the class contains right principal ideal domains in which the maximal right ideals are two-sided [6].

Polynomial index in discrete valuation rings

2019 ◽  
Vol 56 (2) ◽  
pp. 260-266
Author(s):  
Mohamed E. Charkani ◽  
Abdulaziz Deajim
Keyword(s):  
Lower Bound ◽  
Valuation Ring ◽  
Irreducible Polynomial ◽  
Discrete Valuation Ring ◽  
Field Extension ◽  
Integral Closure ◽  
Quotient Field ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Adic Valuation

Abstract Let R be a discrete valuation ring, its nonzero prime ideal, P ∈R[X] a monic irreducible polynomial, and K the quotient field of R. We give in this paper a lower bound for the -adic valuation of the index of P over R in terms of the degrees of the monic irreducible factors of the reduction of P modulo . By localization, the same result holds true over Dedekind rings. As an important immediate application, when the lower bound is greater than zero, we conclude that no root of P generates a power basis for the integral closure of R in the field extension of K defined by P.

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