localglobal

Localglobal is a term used in mathematics and related disciplines to denote the interplay between local information and global structure. It is most often encountered as shorthand for local-global principles, which assert that a global property of an object holds precisely when the corresponding property holds in all local or localized settings.

In number theory and algebraic geometry, local-global principles describe how global solvability or global structure can be inferred from local data. A classic example is the Hasse–Minkowski principle for quadratic forms: a quadratic form over a global field (such as the rational numbers) represents zero globally if and only if it represents zero over every completion of that field, including the real numbers and all p-adic fields. This local-to-global viewpoint guides the search for rational solutions to equations by first checking solvability in simpler, localized contexts.

However, local-global principles are not universal. There exist obstructions that prevent a global solution even when all local solutions exist. Such phenomena are studied using tools like the Brauer group and descent theory, which identify and classify the obstructions that block global realization despite local feasibility.

Beyond number theory, localglobal ideas appear in algebraic geometry, topology, and analysis, where local properties and patching techniques inform global structure and behavior. The concept emphasizes that understanding a complex object often reduces to understanding its behavior on simpler, local pieces and how those pieces fit together.

See also: Hasse principle, Brauer–Manin obstruction, descent, patching.