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Geometric and arithmetic characterization of [Formula: see text]-module flatness with applications to tensor products
Jian-Gang Tang1,2,3, Huang-Rui Lei1, Miao Liu2
1Division of Mathematics, Sichuan University Jinjiang College, Meishan, Sichuan, China.
Plos One
|October 16, 2025
Summary
This study introduces a new framework for understanding flatness properties in modules using algebraic and geometric methods. It reveals deep connections between different mathematical fields and provides new criteria for module flatness.
Area of Science:
- Algebraic Geometry
- Homological Algebra
- Number Theory
Background:
- Flatness properties are central to module theory in various mathematical fields.
- Understanding tensor products of modules requires sophisticated tools from different areas of mathematics.
Purpose of the Study:
- To establish a unified framework for studying flatness properties and tensor products of modules.
- To develop new criteria for characterizing flatness using diverse mathematical perspectives.
- To explore the interplay between local and global properties of differential systems.
Main Methods:
- Lagrangian geometry
- Homological algebra
- Irregular Hodge theory
- Microlocal analysis
- Irregular Riemann-Hilbert correspondence
- p-adic techniques
Main Results:
- New criteria for flatness characterization.
- A geometric obstruction theory for globalizing pointwise flat modules.
- Fundamental results on the monoidal structure of the derived tensor product category.
- Compatibility theorems for Beilinson-Bernstein localization.
- Arithmetic characterizations of flatness in characteristic p.
Conclusions:
- The study reveals deep connections between algebraic, geometric, and arithmetic contexts for module flatness.
- The developed methods provide new insights into the structure and properties of differential systems.
- This work bridges local and global perspectives in the study of modules and differential systems.
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