Algebraic K-theory of local rings

• Grant number: 17/26310-7
• Support Opportunities: Scholarships abroad - Research
• Start date: December 01, 2018
• End date: November 30, 2019
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Behrooz Mirzaii
• Grantee: Behrooz Mirzaii
• Host Investigator: Kevin Hutchinson
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Institution abroad: University College Dublin, Ireland

Abstract

In this project we will study algebraic K-theory of local rings. Algebraic K-theory is a subject in mathematics with deep connections with other subjects such as Algebraic Geometry, Algebraic Topology, Homological Algebra, Ring Theory, Number Theory, etc. One can think of algebraic K-theory as higher linear algebra, or as Michel Atiyah calls it "Stable Linear Algebra". The main strategy in this project is not only to study the K-theory of local rings directly, but also to look at its connection and interaction with other subjects. It happens often that a new idea in one subject might lead to a big breakthrough in another subject. In this project we will mainly concentrate on deep connection of algebraic K-theory of local rings with homology of classical groups (in Homological Algebra) and with higher Chow groupsm (in Algebraic Geometry) over local rings. Local rings are an important class of rings and many times problems in Algebraic K-theory can be reduced to the study of the structure of K-groups of such rings. The problems that we will investigate in this project are very fundamental to this subject; some are very old and some are very new, some which the applicant has done some progresses towards their solutions in his previous works. It is fair to say that this project and its objectives are based on the most important questions and conjectures which have not been answered and have been the source of inspiration for some of the most outstanding works in this field. (AU)