scholarly journals A Remark on Quadrics in Projective Klingenberg Spaces over a Certain Local Algebra

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2168
Author(s):  
Marek Jukl
Keyword(s):  
Local Ring ◽  
Linear Algebra ◽  
Nilpotent Element ◽  
Local Algebra ◽  
Geometric Description

This article is devoted to some polar properties of quadrics in the projective Klingenberg spaces over a local ring which is a linear algebra generated by one nilpotent element. In this case, polar subspaces are described; the notion “degree of neighborhood” is used for the geometric description of polar subspaces of quadrics. The polarity induced by a quadric is also studied.

scholarly journals A Remark on Structure of Projective Klingenberg Spaces over a Certain Local Algebra

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 702 ◽  
Author(s):  
Marek Jukl
Keyword(s):  
Local Ring ◽  
Linear Algebra ◽  
Nilpotent Element ◽  
Local Algebra ◽  
Geometric Description

This article is devoted to the projective Klingenberg spaces over a local ring, which is a linear algebra generated by one nilpotent element. In this case, subspaces of such Klingenberg spaces are described. The notion of the “degree of neighborhood” is introduced. Using this, we present the geometric description of subsets of points of a projective Klingenberg space whose arithmetical representatives need not belong to a free submodule.

scholarly journals The Centralizer of a Nilpotent Section

2008 ◽  
Vol 190 ◽  
pp. 129-181 ◽  
Author(s):  
George J. McNinch
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Local Ring ◽  
Nilpotent Element ◽  
Group Scheme ◽  
Closed Field ◽  
Good Characteristic ◽  
Component Group ◽  
Levi Factor ◽  
Subgroup Scheme

Let F be an algebraically closed field and let G be a semisimple F-algebraic group for which the characteristic of F is very good. If X ∈ Lie(G) = Lie(G)(F) is a nilpotent element in the Lie algebra of G, and if C is the centralizer in G of X, we show that (i) the root datum of a Levi factor of C, and (ii) the component group C/C° both depend only on the Bala-Carter label of X; i.e. both are independent of very good characteristic. The result in case (ii) depends on the known case when G is (simple and) of adjoint type.The proofs are achieved by studying the centralizer of a nilpotent section X in the Lie algebra of a suitable semisimple group scheme over a Noetherian, normal, local ring . When the centralizer of X is equidimensional on Spec(), a crucial result is that locally in the étale topology there is a smooth -subgroup scheme L of such that Lt is a Levi factor of for each t ∈ Spec ().

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.

scholarly journals On Modules over Local Rings

2016 ◽  
Vol 24 (1) ◽  
pp. 217-230
Author(s):  
Fatma Özen Erdoğan ◽  
Süleyman Çiftçi ◽  
Atilla Akpιnar
Keyword(s):  
Local Ring ◽  
Linear Algebra ◽  
Local Rings ◽  
The Real

Abstract This paper is dealed with a special local ring A and modules over A. Some properties of modules, that are constructed over the real plural algebra, are investigated. Moreover a module is constructed over the linear algebra of matrix Mmm(ℝ) and one of its basis is found.

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