# Category:Localization (mathematics)

Computing portal |

Here is a list of articles in the category **Localization (mathematics)** of the Computing portal that unifies foundations of mathematics and computations using computers. 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.

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.

## Pages in category "Localization (mathematics)"

The following 12 pages are in this category, out of 12 total.

### D

- Discrete valuation ring
*(computing)*

### H

- Hasse principle
*(computing)*

### L

- Local analysis
*(computing)* - Local ring
*(computing)* - Localization (algebra)
*(computing)* - Localization (commutative algebra)
*(computing)* - Localization of a category
*(computing)* - Localization of a module
*(computing)* - Localization of a ring
*(computing)* - Localization of a topological space
*(computing)*

### S

- Semi-local ring
*(computing)*

### V

- Valuation ring
*(computing)*