On the Auslander-Reiten conjecture

• Grant number: 22/03372-5
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: June 01, 2022
• Status: Discontinued
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Victor Hugo Jorge Pérez
• Grantee: Victor Daniel Mendoza Rubio
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated scholarship(s): 23/15733-5 - Vanishing of Ext and Tor, and free modules., BE.EP.DR

Abstract

Let $R$ be a commutative Noetherian ring with unity and $M$ be a finitely generated $R$-module. It is know that if each finitely generated $R$-module $N$ satisfies the condition that the $R$-module $\mbox{Ext}^{i}_R(M,N)$ is the zero module for every integer $i>0$, then $M$ is projective. A question that arises is the following: if such condition holds at least for $N=R$ and $N=M$, is it possible to conclude that $M$ is projective? This problem is known as the Auslander-Reiten conjecture. This conjecture is open for over 44 years, but in this period many cases in which it is valid were found. In this project we are going to identify the main tools used in partial solutions of the Auslander-Reiten conjecture and we intend to generalize some cases in which it is known that it holds. Our main goal is to establish a new non-trivial case in which the Auslander-Reiten conjecture is true and, subsequently, to establish the respective consequences. (AU)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

MENDOZA-RUBIO, VICTOR D.; JORGE-PEREZ, VICTOR H.. On modules whose dual is of finite Gorenstein dimension. COLLECTANEA MATHEMATICA, v. N/A, p. 23-pg., 2025-01-27. (19/21181-0, 22/03372-5)