The Tor and Ext functors: vanishing properties and related problems

• Grant number: 22/12114-0
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: March 01, 2023
• End date: February 28, 2027
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Victor Hugo Jorge Pérez
• Grantee: Paulo Damião Christo Martins
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated scholarship(s): 24/17809-1 - Structural properties of modules, quasi-projective dimension, and vanishing of Tor and Ext functors, BE.EP.DR

Abstract

This project aims to study the vanishing properties of Tor and Ext derived functors in order to obtain structural, homological or geometric information about modules on commutative rings from homological vanishing properties. There are traditional results of the theory that verify, for example, that if an R-module M satisfies Ext1(M,N)=0, for every R-module N , then M is a projective R-module or if Tor1(M,N)=0, then M is a flat R-module and attempts to improve these conditions. In addition, there are known problems such as the Auslander-Reiten conjecture and under which nulling conditions two R-modules satisfy the depth formula. Traditionally, answers to this type of problems have a strong relationship with other areas of Mathematics, such as Algebraic Geometry and Singularity Theory. Therefore, the project must explore such relationships in a relevant way in order to obtain applications in other areas.