Category:Localization (mathematics)

In mathematics, specifically algebraic geometry and its applications, localization is a way of studying an algebraic object "at" a prime. One may study an object by studying it at every prime (the "local question"), then piecing these together to understand the original object (the "local-to-global question").

The simplest example is solving a Diophantine equation (a polynomial with integer coefficients) by finding solutions mod every prime (properly, finding a p-adic solution for every prime p), then piecing these solutions together, which is called the Hasse principle.

More abstractly, one studies a ring by localizing at a prime ideal, obtaining a local ring. One then often takes the completion.

The geometric terminology ("local" and "global") come from algebraic geometry, and may be called topological algebra (considering algebraic objects as topological spaces, with a notion of "local" and "global"): from the point of view of the spectrum of a ring, the primes are the points of a ring, and thus localization studies a ring (or similar algebraic object) at every point, then the local-to-global question asks to piece these together to understand the entire space.

The failure of local solutions to piece together to form a global solution is a form of obstruction theory, and often yields cohomological invariants, as in sheaf cohomology.

This approach finds applications in algebraic number theory, algebraic geometry, and algebraic topology.