On homological dimensions and related problems

• Grant number: 25/19657-7
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: August 01, 2026
• End date: July 31, 2029
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Eduardo do Nascimento Marcos
• Grantee: Victor Daniel Mendoza Rubio
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 24/14914-9 - Structures and representations of algebraic systems and their applications, AP.TEM

Abstract

In Commutative Algebra, several homological dimensions extending the classical notions of projective and injective dimensions have been introduced. This project considers the study of homological dimensions in different directions. On the one hand, we propose to construct new homological dimensions that generalize the classical ones, following the recently introduced quasi-projective and quasi-injective dimensions, as well as their extensions with respect to semidualizing modules. We also seek to establish characterizations of certain classes of rings via homological dimensions, along with criteria for detecting the finiteness of the homological dimensions of modules. In particular, in this direction, we will seek to obtain results via Frobenius endomorphisms and, more broadly, through large classes of homomorphisms such as contracting endomorphisms and $h_2$-vanishing homomorphisms. In addition, concerning criteria for detecting finiteness of homological dimension of modules, the Auslander-Reiten conjecture and Tor-persistence will also be considered in this study. Another focus will be the study of the consequences of having certain types of modules with finite homological dimension, in particular Tor modules, which will lead to new homological criteria. Additionally, we expect to obtain applications of the results obtained in Singularity Theory and Algebraic Geometry. (AU)