Topics in Commutative Algebra: From Classical Foundations to Serre's Local Algebra

• Grant number: 26/17204-8
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: August 01, 2026
• End date: July 31, 2027
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Saeed Tafazolian
• Grantee: Ruan Lucas de Oliveira Gildo
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated research grant: 24/00923-6 - Brazilian Center for Geometry (CBG), AP.CEPID

Abstract

Commutative Algebra constitutes the fundamental algebraic language for core areas of contemporary mathematics, including Algebraic Geometry and Number Theory. This Undergraduate Research project proposes a guided study pathway aimed at mastering the foundations of commutative rings and modules, progressively advancing to sophisticated topics in Jean-Pierre Serre's classic monograph Local Algebra. The schedule of activities is structured into stages covering basic concepts, Nakayama's Lemma, localization, and primary decomposition, followed by dimension theory in Noetherian local rings, the Cohen-Seidenberg theorems, regular local rings, and the Koszul complex. The methodology will consist of prior reading, analytical problem-solving focused on the detailed reconstruction of proofs, periodic advisory meetings, and seminar presentations. The expected outcomes include establishing a solid mathematical foundation and developing autonomy in reading graduate-level texts, thereby preparing the student for further studies at the Master's level. (AU)