Discriminant as a product of local discriminants

2017 ◽  
Vol 16 (10) ◽  
pp. 1750198 ◽  
Author(s):  
Anuj Jakhar ◽  
Bablesh Jhorar ◽  
Sudesh K. Khanduja ◽  
Neeraj Sangwan
Keyword(s):  
Maximal Ideal ◽  
Valuation Ring ◽  
Approximation Theorem ◽  
Discrete Valuation Ring ◽  
Integral Closure ◽  
Weak Approximation ◽  
Separable Extension ◽  
Ideal Formula ◽  
Valuation Rings ◽  
Basic Equality

Let [Formula: see text] be a discrete valuation ring with maximal ideal [Formula: see text] and [Formula: see text] be the integral closure of [Formula: see text] in a finite separable extension [Formula: see text] of [Formula: see text]. For a maximal ideal [Formula: see text] of [Formula: see text], let [Formula: see text] denote respectively the valuation rings of the completions of [Formula: see text] with respect to [Formula: see text]. The discriminant satisfies a basic equality which says that [Formula: see text]. In this paper, we extend the above equality on replacing [Formula: see text] by the valuation ring of a Krull valuation of arbitrary rank and completion by henselization. In the course of proof, we prove a generalization of the well-known weak Approximation Theorem which is of independent interest as well.

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.

Witts Theorem for Quadratic Forms Over Non-Dyadic Discrete Valuation Rings

1977 ◽  
Vol 29 (5) ◽  
pp. 928-936
Author(s):  
David Mordecai Cohen
Keyword(s):  
Bilinear Form ◽  
Maximal Ideal ◽  
Quadratic Forms ◽  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Symmetric Bilinear Form ◽  
Finitely Generated ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Group Of Isometries

Let R be a discrete valuation ring, with maximal ideal pR, such that ½ ϵ R. Let L be a finitely generated R-module and B : L × L → R a non-degenerate symmetric bilinear form. The module L is called a quadratic module. For notational convenience we shall write xy = B(x, y). Let O(L) be the group of isometries, i.e. all R-linear isomorphisms φ : L → L such that B((φ(x), (φ(y)) = B(x, y).

Polynomials inducing the zero function on chain rings

2018 ◽  
Vol 17 (08) ◽  
pp. 1850160 ◽  
Author(s):  
Mark W. Rogers ◽  
Cameron Wickham
Keyword(s):  
Maximal Ideal ◽  
Principal Ideal ◽  
Valuation Ring ◽  
Residue Field ◽  
Discrete Valuation Ring ◽  
Minimal Set ◽  
Principal Ideal Ring ◽  
Ideal Formula ◽  
Zero Function ◽  
The Ideal

We provide a minimal set of generators for the ideal of polynomials in [Formula: see text] that map the maximal ideal [Formula: see text] into one of its powers [Formula: see text], where [Formula: see text] is a discrete valuation ring with a finite residue field. We use this to provide a minimal set of generators for the ideal of polynomials in [Formula: see text] that send [Formula: see text] to zero, where [Formula: see text] is a finite commutative local principal ideal ring.

scholarly journals A BETTER COMPARISON OF - AND -COHOMOLOGIES

2019 ◽  
Vol 236 ◽  
pp. 183-213
Author(s):  
SHANE KELLY
Keyword(s):  
Finite Rank ◽  
Valuation Ring ◽  
Direct Proof ◽  
Integral Closure ◽  
Motivic Cohomology ◽  
Valuation Rings

In order to work with non-Nagata rings which are Nagata “up-to-completely-decomposed-universal-homeomorphism,” specifically finite rank Hensel valuation rings, we introduce the notions of pseudo-integral closure, pseudo-normalization, and pseudo-Hensel valuation ring. We use this notion to give a shorter and more direct proof that $H_{\operatorname{cdh}}^{n}(X,F_{\operatorname{cdh}})=H_{l\operatorname{dh}}^{n}(X,F_{l\operatorname{dh}})$ for homotopy sheaves $F$ of modules over the $\mathbb{Z}_{(l)}$-linear motivic Eilenberg–Maclane spectrum. This comparison is an alternative to the first half of the author’s volume Astérisque 391 whose main theorem is a cdh-descent result for Voevodsky motives. The motivating new insight is really accepting that Voevodsky’s motivic cohomology (with $\mathbb{Z}[\frac{1}{p}]$-coefficients) is invariant not just for nilpotent thickenings, but for all universal homeomorphisms.

EXPONENT MATRICES AND TILED ORDERS OVER DISCRETE VALUATION RINGS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 997-1012 ◽  
Author(s):  
V. V. KIRICHENKO ◽  
A. V. ZELENSKY ◽  
V. N. ZHURAVLEV
Keyword(s):  
Markov Chains ◽  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Strongly Connected ◽  
Exponent Matrix ◽  
Finite Posets

Exponent matrices appear in the theory of tiled orders over a discrete valuation ring. Many properties of such an order and its quiver are fully determined by its exponent matrix. We prove that an arbitrary strongly connected simply laced quiver with a loop in every vertex is realized as the quiver of a reduced exponent matrix. The relations between exponent matrices and finite posets, Markov chains, and doubly stochastic matrices are discussed.

Constructing the hereditary crossed product order containing a given weak crossed product order and a criterion for weakness

2016 ◽  
Vol 15 (06) ◽  
pp. 1650113
Author(s):  
Christopher James Wilson
Keyword(s):  
Galois Extension ◽  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Integral Closure ◽  
Crossed Product ◽  
Weak Crossed Product ◽  
Product Order

Consider a weak crossed product order [Formula: see text] in [Formula: see text], where [Formula: see text] is the integral closure of a discrete valuation ring [Formula: see text] in a tamely ramified Galois extension [Formula: see text] of the field of fractions of [Formula: see text]. Assume that [Formula: see text] is local. In this paper, we show that [Formula: see text] is hereditary if and only if it is maximal among the weak crossed product orders in [Formula: see text]. We also give an algorithm that constructs, in terms of the basis elements [Formula: see text] and the cocycle [Formula: see text], the unique hereditary weak crossed product order in [Formula: see text] that contains a given [Formula: see text], and we give a criterion for determining whether that hereditary order will have a cocycle that takes nonunit values in [Formula: see text].

Elimination of quantifiers for ordered valuation rings

1987 ◽  
Vol 52 (1) ◽  
pp. 116-128 ◽  
Author(s):  
M. A. Dickmann
Keyword(s):  
Model Theory ◽  
Maximal Ideal ◽  
Valuation Ring ◽  
Quantifier Elimination ◽  
Point Of View ◽  
List Type ◽  
Divisibility Property ◽  
Valuation Rings ◽  
Commutative Domain ◽  
Image Position

Cherlin and Dickmann [2] proved that the theory RCVR of real closed (valuation) rings admits quantifier-elimination (q.e.) in the language ℒ = {+, −, ·, 0, 1, <, ∣} for ordered rings augmented by the divisibility relation “∣”. The purpose of this paper is to prove a form of converse of this result:Theorem. Let T be a theory of ordered commutative domains (which are not fields), formulated in the language ℒ. In addition we assume that:(1) The symbol “∣” is interpreted as the honest divisibility relation: (2) The following divisibility property holds in T:If T admits q.e. in ℒ, then T = RCVR.We do not know at present whether the restriction imposed by condition (2) can be weakened.The divisibility property (DP) has been considered in the context of ordered valued fields; see [4] for example. It also appears in [2], and has been further studied in Becker [1] from the point of view of model theory. Ordered domains in which (DP) holds are called in [1] convexly ordered valuation rings, for reasons which the proposition below makes clear. The following summarizes the basic properties of these rings:Proposition I [2, Lemma 4]. (1) Let A be a linearly ordered commutative domain. The following are equivalent:(a) A is a convexly ordered valuation ring.(b) Every ideal (or, equivalently, principal ideal) is convex in A.(c) A is a valuation ring convex in its field of fractions quot(A).(d) A is a valuation ring and its maximal ideal MA is convex (in A or, equivalently, in quot (A)).(e) A is a valuation ring and its maximal ideal is bounded by ± 1.

scholarly journals On Dedekind's criterion and monogenicity over Dedekind rings

2003 ◽  
Vol 2003 (71) ◽  
pp. 4455-4464 ◽  
Author(s):  
M. E. Charkani ◽  
O. Lahlou
Keyword(s):  
Number Field ◽  
Valuation Ring ◽  
Minimal Polynomial ◽  
Discrete Valuation Ring ◽  
Integral Closure ◽  
Prime Ideals ◽  
Dedekind Ring ◽  
Ring Of Integers ◽  
Integral Element

We give a practical criterion characterizing the monogenicity of the integral closure of a Dedekind ringR, based on results on the resultantRes(p,pi)of the minimal polynomialpof a primitive integral element and of its irreducible factorspimodulo prime ideals ofR. We obtain a generalization and an improvement of the Dedekind criterion (Cohen, 1996), and we give some applications in the case whereRis a discrete valuation ring or the ring of integers of a number field, generalizing some well-known classical results.

Integral Closure of Noetherian Domains and Intersections of Rees Valuation Rings, (II)

2017 ◽  
Vol 84 (1-2) ◽  
pp. 55
Author(s):  
Paula Kemp ◽  
Louis J. Ratliff, Jr. ◽  
Kishor Shah
Keyword(s):  
Valuation Ring ◽  
Disjoint Union ◽  
Integral Closure ◽  
Primary Ideal ◽  
Unique Factorization ◽  
Unique Factorization Domain ◽  
Valuation Rings ◽  
Primary Ideals ◽  
Rees Valuation

<p>Let 1 &lt; s<sub>1</sub> &lt; . . . &lt; s<sub>k</sub> be integers, and assume that κ ≥ 2 (so s<sub>k</sub> ≤ 3). Then there exists a local UFD (Unique Factorization Domain) (R,M) such that:</p><p>(1) Height(M) = s<sub>k</sub>.</p><p>(2) R = R' = ∩{VI (V,N) € V<sub>j</sub>}, where V<sub>j</sub> (j = 1, . . . , κ) is the set of all of the Rees valuation rings (V,N) of the M-primary ideals such that trd((V I N) I (R I M)) = s<sub>j</sub> - 1.</p><p>(3) With V<sub>1</sub>, . . . , V<sub>κ</sub> as in (2), V<sub>1</sub> ∪ . . . V<sub>κ</sub>is a disjoint union of all of the Rees valuation rings of allof the M-primary ideals, and each M-primary ideal has at least one Rees valuation ring in each V<sub>j</sub> .</p>

A note on valuations associated with a local domain

1955 ◽  
Vol 51 (2) ◽  
pp. 252-253 ◽  
Author(s):  
D. Rees
Keyword(s):  
Maximal Ideal ◽  
Present Note ◽  
Integral Closure ◽  
Local Domain ◽  
Valuation Rings ◽  
Integrally Closed ◽  
Primary Ideals

The purpose of the present note is to prove the following two theorems:Theorem 1. Let Q be an equicharacteristic local domain with maximal ideal m. Let a be any ideal of Q. Then the intersection of all integrally closed m-primary ideals of Q which contain a is the integral closure ā of a.Theorem 2. If Q is as above, and if S denotes the set of valuations on the field of fractions F of Q which are associated with Q, then the intersection of the valuation rings belonging to valuations in S is the integral closure of Q.

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