A Gersten-Witt complex for hermitian Witt groups of coherent algebras over schemes II: Involution of the second kind

2008 ◽  
Vol 4 (2) ◽  
pp. 347-377 ◽  
Author(s):  
Stefan Gille
Keyword(s):  
Krull Dimension ◽  
Semilocal Ring ◽  
Azumaya Algebra ◽  
Noetherian Scheme ◽  
Fixed Ring ◽  
Geometric Type ◽  
Witt Groups

AbstractLet X be a regular noetherian scheme of finite Krull dimension with involution σ and an Azumaya algebra over X with involution τ of the second kind with respect to σ. We construct a hermitian and a skew-hermitian Gersten-Witt complex for (, τ) and show that these complexes are exact if X = Spec R is the spectrum of a regular semilocal ring R of geometric type, such that R is a quadratic étale extension of the fixed ring of σ.

scholarly journals VANISHING OF THE NEGATIVE HOMOTOPY -THEORY OF QUOTIENT SINGULARITIES

2017 ◽  
Vol 18 (3) ◽  
pp. 619-627
Author(s):  
Gonçalo Tabuada
Keyword(s):  
Algebraic Geometry ◽  
Upper Bound ◽  
Homotopy Theory ◽  
Krull Dimension ◽  
Noncommutative Algebraic Geometry ◽  
Noetherian Scheme ◽  
Quotient Singularities ◽  
Cyclic Quotient Singularities

Making use of Gruson–Raynaud’s technique of ‘platification par éclatement’, Kerz and Strunk proved that the negative homotopy$K$-theory groups of a Noetherian scheme$X$of Krull dimension$d$vanish below$-d$. In this article, making use of noncommutative algebraic geometry, we improve this result in the case of quotient singularities by proving that the negative homotopy$K$-theory groups vanish below$-1$. Furthermore, in the case of cyclic quotient singularities, we provide an explicit ‘upper bound’ for the first negative homotopy$K$-theory group.

Commutative rings with infinitely many maximal subrings

2014 ◽  
Vol 13 (07) ◽  
pp. 1450037 ◽  
Author(s):  
Alborz Azarang ◽  
Greg Oman
Keyword(s):  
Direct Product ◽  
Noetherian Ring ◽  
Artinian Ring ◽  
Commutative Rings ◽  
Semilocal Ring ◽  
Finitely Generated ◽  
Fixed Ring ◽  
Irreducible Elements ◽  
Semisimple Ring ◽  
Nonzero Characteristic

It is shown that RgMax (R) is infinite for certain commutative rings, where RgMax (R) denotes the set of all maximal subrings of a ring R. It is observed that whenever R is a ring and D is a UFD subring of R, then | RgMax (R)| ≥ | Irr (D) ∩ U(R)|, where Irr (D) is the set of all non-associate irreducible elements of D and U(R) is the set of all units of R. It is shown that every ring R is either Hilbert or | RgMax (R)| ≥ ℵ0. It is proved that if R is a zero-dimensional (or semilocal) ring with | RgMax (R)| < ℵ0, then R has nonzero characteristic, say n, and R is integral over ℤn. In particular, it is shown that if R is an uncountable artinian ring, then | RgMax (R)| ≥ |R|. It is observed that if R is a noetherian ring with |R| > 2ℵ0, then | RgMax (R)| ≥ 2ℵ0. We determine exactly when a direct product of rings has only finitely many maximal subrings. In particular, it is proved that if a semisimple ring R has only finitely many maximal subrings, then every descending chain ⋯ ⊂ R2 ⊂ R1 ⊂ R0 = R where each Ri is a maximal subring of Ri-1, i ≥ 1, is finite and the last terms of all these chains (possibly with different lengths) are isomorphic to a fixed ring, say S, which is unique (up to isomorphism) with respect to the property that R is finitely generated as an S-module.

Transfer of Krull Dimension, Lying-Over, and Going-Down to the Fixed Ring

2007 ◽  
Vol 35 (4) ◽  
pp. 1227-1247 ◽  
Author(s):  
David E. Dobbs ◽  
Jay Shapiro
Keyword(s):  
Krull Dimension ◽  
Fixed Ring

scholarly journals Impact-Rubbing Dynamic Behavior of Magnetic-Liquid Double Suspension Bearing under Different Protective Bearing Forms

Processes ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 1105
Author(s):  
Jianhua Zhao ◽  
Lanchun Xing ◽  
Sheng Li ◽  
Weidong Yan ◽  
Dianrong Gao ◽  
...  
Keyword(s):  
Dynamic Behavior ◽  
Ball Bearing ◽  
Magnetic Liquid ◽  
Electromagnetic System ◽  
Fixed Ring ◽  
Deep Groove ◽  
Deep Groove Ball Bearing ◽  
Protection Devices ◽  
Hydrostatic System ◽  
The Impact

The magnetic-liquid double suspension bearing (MLDSB) is a new type of suspension bearing, with electromagnetic suspension as the main part and hydrostatic supports as the auxiliary part. It can greatly improve the bearing capacity and stiffness of rotor-bearing systems and is suitable for a medium speed, heavy load, and frequent starting occasions. Compared with the active electromagnetic bearing system, the traditional protective bearing device is replaced by the hydrostatic system in MLDSB, and the impact-rubbing phenomenon can be restrained and buffered. Thus, the probability and degree of friction and wear between the rotor and the magnetic pole are reduced drastically when the electromagnetic system fails. In order to explore the difference in the dynamic behavior law of the impact-rubbing phenomenon between the traditional protection device and hydrostatic system, the dynamic equations of the rotor impact-rubbing in three kinds of protection devices (fixed ring/deep groove ball bearing/hydrostatic system) under electromagnetic failure mode are established, and the axial trajectory and motion law of the rotor are numerically simulated. Finally, the dynamic behavior characteristics of the rotor are compared and analyzed. The results show that: Among the three kinds of protection devices (fixed ring/deep groove ball bearing/hydrostatic system), the hydrostatic system has the least influence on bouncing time, impact-rubbing force, and impact-rubbing degree, and the maximum impact-rubbing force of MLDSB is greatly reduced. Therefore, the protective bear is not required to be installed in the MLDSB. This study provides the basis for the theory of the “gap impact-rubbing” of MLDSB under electromagnetic failure, and helps to identify electromagnetic faults.

scholarly journals BOUNDED AND FULLY BOUNDED MODULES

2011 ◽  
Vol 84 (3) ◽  
pp. 433-440
Author(s):  
A. HAGHANY ◽  
M. MAZROOEI ◽  
M. R. VEDADI
Keyword(s):  
Krull Dimension ◽  
Finitely Generated ◽  
Morita Invariant ◽  
Prime Submodule ◽  
Invariant Properties ◽  
Noetherian Modules

AbstractGeneralizing the concept of right bounded rings, a module MR is called bounded if annR(M/N)≤eRR for all N≤eMR. The module MR is called fully bounded if (M/P) is bounded as a module over R/annR(M/P) for any ℒ2-prime submodule P◃MR. Boundedness and right boundedness are Morita invariant properties. Rings with all modules (fully) bounded are characterized, and it is proved that a ring R is right Artinian if and only if RR has Krull dimension, all R-modules are fully bounded and ideals of R are finitely generated as right ideals. For certain fully bounded ℒ2-Noetherian modules MR, it is shown that the Krull dimension of MR is at most equal to the classical Krull dimension of R when both dimensions exist.

Sign in / Sign up

Export Citation Format

Share Document