Motivic cohomology, Milnor K-theory, and Galois cohomology

• Autor(a) principal: Souza, Daniel de Almeida
• Data de Publicação: 2022
• Tipo de documento: Dissertação
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: https://www.teses.usp.br/teses/disponiveis/45/45131/tde-11052022-151747/
• topic: Cohomologia galoisiana; Cohomologia motívica; Galois cohomology; K-teoria de Milnor; Milnor K-theory; Motivic cohomology

This dissertation presents one of the possible foundations, based on motivic complexes, for the motivic cohomology of smooth varieties over a given base field $k$. Its basic properties are discussed, as well as its relation to Milnor K-theory and to certain Galois cohomology groups of $k$. In particular, we discuss the formulation in terms of motivic cohomology of the norm residue homomorphism, which compares the Milnor K-theory groups modulo a prime number $l$ different from the characteristic of $k$ with the Galois cohomology groups with coefficients in tensor powers of the module of $l$-th roots of unity. Finally, we list some preliminary results used for characterizing the Bloch-Kato conjecture in terms of certain statements of motivic nature.