Motivic cohomology and characterizations of the Bloch-Kato conjecture

• Grant number: 19/09534-4
• Support Opportunities: Scholarships in Brazil - Master
• Start date: August 01, 2019
• End date: October 31, 2021
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Hugo Luiz Mariano
• Grantee: Daniel de Almeida Souza
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

This master's project in mathematics has the aim of deepening the student's contact with algebraic geometry - already developed at the level of scientific initiation - through the study of motivic cohomology and A1-homotopy theory, which were introduced mainly by V. Voevodsky, F. Morel, S. Bloch, and A. Suslin. These areas formed the basis for the proofs of Milnor conjecture (Voevodsky, 1996), proposed in the 1970s after Milnor's studies on the theory of quadratic forms and algebraic K-theory. Later, the result was generalized to a proof of the celebrated Bloch-Kato conjecture (Voevodsky, 2008), using methods from the recent theory of motivic cohomology, a powerful invariant for algebraic varieties extending the classical Chow ring. The proposed project will introduce motivic cohomology theory for algebraic varieties, reformulate the conjectures of Milnor and Bloch-Kato as a comparison isomorphism between motivic and étale cohomologies - thus transforming the conjectures into particular cases of a deep problem of geometric nature -, and will deal with the triple equivalence between Bloch-Kato, the extended Theorem 90 of Hilbert (an important case reduction), and the Beilinson-Lichtenbaum conjecture (a generalization to a bigger class of varieties). The student will then be solidly prepared to act at research level on recent and relevant topics in algebraic geometry. (AU)

Academic Publications

(References retrieved automatically from State of São Paulo Research Institutions)

SOUZA, Daniel de Almeida. Motivic cohomology, Milnor K-theory, and Galois cohomology. 2022. Master's Dissertation - Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI) São Paulo.