The depth of an extension of valued fields

• Grant number: 24/08989-6
• Support Opportunities: Scholarships abroad - Research
• Start date: January 01, 2025
• End date: May 31, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Josnei Antonio Novacoski
• Grantee: Josnei Antonio Novacoski
• Host Investigator: Enric Nart
• Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil
• Institution abroad: Universitat Autònoma de Barcelona (UAB), Spain

Abstract

The main goal of this visit is to continue the studies about extensions of valued fields via key polynomials. More precisely, we will study the depth of a finite valued field extension. The main application we have in mind is the study of the module of K\"ahler differentials of an extension of valuation rings, whose corresponding field extension $L/K$ is simple and algebraic. Recently, it has been proved that if $L/K$ has depth one, then this module is the quotient of two fractional ideals of $L$. The next step, which has started already, is to obtain results that determine whether a given extension has depth one. This work is a natural continuation of the work developed during the research project JP-FAPESP (process number 2017/17835-9) of which Novacoski was the main researcher.