nLab
local-global principle

Contents

Idea

It is a common experience in mathematics that phenomena (“problems”) are usefully decomposed into a “local” and a “global” aspect of rather different nature.

In as far as the situation has an interpretation in geometry then the local aspect is typically given by the formal neighbourhoods of all points, while the global aspect is given by the underlying topology/homotopy type. The following provides some examples.

In number theory and algebraic geometry

In number theory and hence in algebraic geometry over Spec()Spec(\mathbb{Z}), the “local” situation is the formal neighbourhoods of all primes pp (points in the prime spectrum Spec()Spec(\mathbb{Z}) of the integers). These are the formal spectra of the p-adic integers. In terms of number theory this means that a solution to a polynomial equation over the p-adic integers is a “local” approximation to a solution over the integers (the former is a necessary but not a sufficient condition for the latter to exist). In this form the local-global principle is attributed to Helmut Hasse and also called the Hasse principle or similar.

(…)

In conformal field theory

The FRS-theorem shows that every rational 2d CFT is decomposed into local data given by a vertex operator algebra and global data given by a solution of the sewing constraints

More

(…)

References

Revised on April 18, 2016 12:21:12 by Urs Schreiber (131.220.184.222)