Scissors Congruence Group and the Third Homology of SL2

• Ano de defesa: 2024
• Autor(a) principal: Perez, Elvis Torres
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Biblioteca Digitais de Teses e Dissertações da USP
• Palavras-chave em Português: Algebraic K-theory; Group homology; Homologia de grupos; K-Teoria algébrica; Refined Bloch group; Refined Scissors-congruence group; Refined scissors-congruence group group
• Link de acesso: https://www.teses.usp.br/teses/disponiveis/55/55135/tde-24042024-134525/

The main goal of this work is to study the third integer homology of the special linear group H3(SL2(A);Z) for a commutative ring A and its relationship with the refined scissors congruence group R P1(A) (BLOCH, 2000), (HUTCHINSON, 2013a), (CORONADO; HUTCHINSON, ). An important tool to study the third homology of SL2 is the existence of a refined Bloch-Wigner exact sequence. In this thesis we show that there exist a refined Bloch-Wigner exact sequence over local domains of characteristic 2. In fact, we show that if char(A) = 2, then there exists an exact sequence 0 → Tor(μ(A);μ(A)) → H3(SL2(A);Z) → R B(A) → 0; where R B(A) ⊆ R P1(A) is the refined Bloch group of A. Moreover, we show that if A is a local domain such that -1 is an square, then there exists an exact sequence H3(SM2(A);Z) → H3(SL2(A);Z) → R B(A) → 0; where SM2(A) is the group of monomial matrices in SL2(A). The results of this thesis can be found in (MIRZAII; PÉREZ, a), (MIRZAII; PÉREZ, b).