General Theory of Saturation and of Saturated Local Rings I: Saturation of Complete Local Domains of Dimension One Having Arbitrary Coefficient Fields (of Characteristic Zero)

1971 ◽  
Vol 93 (3) ◽  
pp. 573 ◽  
Author(s):  
Oscar Zariski
Keyword(s):  
General Theory ◽  
Characteristic Zero ◽  
Local Rings ◽  
Dimension One

Structures elementarily closed relative to a model for arithmetic

1968 ◽  
Vol 33 (1) ◽  
pp. 101-104
Author(s):  
Eugene W. Madison
Keyword(s):  
General Theory ◽  
Characteristic Zero ◽  
Order Logic ◽  
First Order Logic ◽  
Natural Numbers ◽  
Algebraic Numbers ◽  
First Order

The present paper is a sequel to [1]. It is our purpose to formulate a general theory derived from the methods used to obtain three results for the field of real algebraic numbers in [1]. As there, we shall concern ourselves almost exclusively with fields of characteristic zero; thus we assume a convenient formulation of first order logic with extralogical constants E(x, y), S(x, y, z), F(x, y, z), F(x, y), N(x) and 0, whose intended interpretations are equality, sum, product, y is the successor of x, x ∈ (where is a substructure satisfying all first order truths of the natural numbers) and zero, respectively. In addition, we shall use Q(x, y) for x ≤ y in those cases where our field is ordered, e.g. the field of real algebraic numbers.

The maximal dimension of formal fibers of local rings of an algebraic scheme of finite type

2019 ◽  
Vol 18 (06) ◽  
pp. 1950120
Author(s):  
Đoàn Trung Cu’ò’ng
Keyword(s):  
Finite Type ◽  
Local Ring ◽  
Local Algebra ◽  
Local Rings ◽  
Maximal Dimension ◽  
Algebraic Varieties ◽  
Noetherian Local Ring ◽  
Closed Point ◽  
Dimension One ◽  
Algebraic Scheme

For a scheme [Formula: see text] of finite type over a Noetherian local ring [Formula: see text] with a closed point [Formula: see text] of the special fiber, we show that the maximal dimension of the formal fibers of the local algebra [Formula: see text] equals to [Formula: see text] provided that either [Formula: see text] is complete of dimension one or the dimensions of the formal fibers of [Formula: see text] are less than [Formula: see text]. This extends Matsumura’s theorem for algebraic varieties.

SEMIPRIME QUADRATIC ALGEBRAS OF GELFAND–KIRILLOV DIMENSION ONE

2004 ◽  
Vol 03 (03) ◽  
pp. 283-300 ◽  
Author(s):  
FERRAN CEDÓ ◽  
ERIC JESPERS ◽  
JAN OKNIŃSKI
Keyword(s):  
Positive Integer ◽  
Characteristic Zero ◽  
Prime Ideals ◽  
Quadratic Algebras ◽  
Degeneracy Condition ◽  
Minimal Prime ◽  
Dimension One ◽  
Kirillov Dimension

We consider algebras over a field K with a presentation K<x1,…,xn:R>, where R consists of [Formula: see text] square-free relations of the form xixj=xkxl with every monomial xixj, i≠j, appearing in one of the relations. The description of all four generated algebras of this type that satisfy a certain non-degeneracy condition is given. The structure of one of these algebras is described in detail. In particular, we prove that the Gelfand–Kirillov dimension is one while the algebra is noetherian PI and semiprime in case when the field K has characteristic zero. All minimal prime ideals of the algebra are described. It is also shown that the underlying monoid is a semilattice of cancellative semigroups and its structure is described. For any positive integer m, we construct non-degenerate algebras of the considered type on 4m generators that have Gelfand–Kirillov dimension one and are semiprime noetherian PI algebras.

On 2-dimensional local rings with Artin's approximation property

1983 ◽  
Vol 94 (1) ◽  
pp. 35-52
Author(s):  
M. L. Brown
Keyword(s):  
Characteristic Zero ◽  
Approximation Property ◽  
Local Rings ◽  
Projective Varieties ◽  
Blowing Up

AbstractExtending results of Popescu and Brown, the main result of this paper is that excellent henselian R1 and S1 2-dimensional local rings, at least in characteristic zero, have the approximation property of M. Artin.Most of the paper consists of an extension of Néron's desingularization to rings which are R1 and S1; such a theorem was previously known for factorial domains. The main theorem is then deduced from this desingularization theorem using a theorem of Elkik.Because of cohomological obstructions, the desingularization theorem is proved only for quasi-projective varieties. In the previously known case for factorial domains, these obstructions are always zero and the desingularization can be obtained by blowing up subschemes. The more general desingularization of this paper is obtained by blowing up locally free sheaves instead, the obstructions being zero for this case.

scholarly journals The Structure of Chains of Ulrich Ideals in Cohen-Macaulay Local Rings of Dimension One

2018 ◽  
Vol 44 (1) ◽  
pp. 65-82 ◽  
Author(s):  
Shiro Goto ◽  
Ryotaro Isobe ◽  
Shinya Kumashiro
Keyword(s):  
Local Rings ◽  
Dimension One

The dimension of the category of maximal Cohen-Macaulay modules over Cohen-Macaulay local rings of dimension one

2019 ◽  
Vol 532 ◽  
pp. 8-21 ◽  
Author(s):  
Takesi Kawasaki ◽  
Yukio Nakamura ◽  
Kaori Shimada
Keyword(s):  
Local Rings ◽  
Dimension One

scholarly journals A uniform general Neron desingularization in dimension one

2018 ◽  
Vol 17 (06) ◽  
pp. 1850105
Author(s):  
Asma Khalid ◽  
Gerhard Pfister ◽  
Dorin Popescu
Keyword(s):  
High Power ◽  
Maximal Ideal ◽  
Local Rings ◽  
One Dimensional ◽  
Dimension One

We give a uniform General Neron Desingularization (GND) for one-dimensional local rings with respect to morphisms which coincide modulo a high power of the maximal ideal. The result has interesting applications in the case of Cohen–Macaulay rings.

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