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Master's Dissertation
DOI
https://doi.org/10.11606/D.55.2021.tde-27092021-110718
Document
Author
Full name
Leonardo Soares Moço
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2021
Supervisor
Committee
Levcovitz, Daniel (President)
Coutinho, Severino Collier
Miranda Neto, Cleto Brasileiro
Novacoski, Josnei Antonio
Title in Portuguese
Derivações simples de álgebras afins
Keywords in Portuguese
Álgebra comutativa
Derivações
Derivações simples
Abstract in Portuguese
As derivações que não deixam ideais invariantes no anel de polinômios em uma variável sobre um corpo K são bem conhecidas, entretanto, quando estamos sobre duas ou mais variáveis ou sobre quocientes dessas álgebras polinomiais ainda não é possível caracterizá-las. Dada uma derivação d sobre uma K-álgebra R, são apresentados nesse trabalho alguns resultados a respeito da d-simplicidade de R, isto é, da ausência de ideais de R não triviais que são invariantes por d. Entre eles citamos o teorema de Shamsuddin e a correspondência entre d-simplicidade e o comportamento de espaços tangentes de conjuntos algébricos afins. A parte mais importante, o capítulo 4, apresenta alguns exemplos geométricos que ilustram bem propriedades de subconjuntos algébricos afins relacionadas à derivações de seu anel de coordenadas. Os exemplos e resultados aqui apresentados fazem parte da tese de doutorado de J. Archer (ARCHER, 1981).
Title in English
Simple derivations of affine algebras
Keywords in English
Commutative algebra
Derivations
Simple derivations
Abstract in English
The derivations that do not leave invariant ideals are well known for the case of polynomial rings in one variable over a field K. However, over two or more variables or over quotients of polynomial algebras, it is still not possible to classify them. Given a derivation d on a Kalgebra R, this work presents some results regarding the d-simplicity of R, that is, the absence of non-trivial R ideals that are invariant by the action of d, like Shamsuddins theorem, and correspondences between such property and the behavior of tangent spaces to affine algebraic sets. The most important part, chapter 4, has some geometric examples that well illustrate properties of affine algebraic subsets related to derivations of their coordinate ring. The examples and results presented here are part of J. Archers doctoral thesis (ARCHER, 1981).
 
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Publishing Date
2021-09-27
 
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