Adjunções entre categorias de álgebras e extensões de quociente bifinito

• Ano de defesa: 2022
• Autor(a) principal: Fernando dos Reis Naves
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: por
• Instituição de defesa: Universidade Federal de Minas Gerais; Brasil; ICEX - INSTITUTO DE CIÊNCIAS EXATAS; Programa de Pós-Graduação em Matemática; UFMG
• Palavras-chave em Português: Álgebra pseudocompacta; adjunção; levantamentos da projeção; conjectura da dimensão finitística; extensões de quociente bifinito; Matemática – Teses; Álgebra pseudocompacta – Teses; Dimensão de representação – Teses
• Link de acesso: http://hdl.handle.net/1843/47367

The objective of this work is divided in two: the study of the correspondence between quivers and algebras via adjunctions and the study of the finitistic dimension conjecture for finite-dimensional algebras, via extensions satisfying homological properties. Our approach to the first problem is to define a correspondence between the category of basic pseudocompact algebras and its full subcategory formed by algebras A such that Jn(A) = 0. Through an equivalence relation on the morphisms of the first, we will study the left adjuncts to A 7→ A/J2(A) for each positive integer n. For example, when we restrict to n = 2, we will prove that the functor that associates each algebra with the complete tensor algebra is left adjoint to F2. For the second problem, given finite-dimensional algebras B ⊆ A, we will control the finitistic dimension of B in terms of that of A, via a homological condition involving A and B. The main result involving finitistic dimension of this thesis is the following: Let B ⊆ A be an extension such that A/B is B-bimodule of finite projective dimension. Then the finitistic dimension of B is finite whenever the finitistic dimension of A is finite. Furthermore, if the global dimension of A is finite, then the global dimension of B is also finite.