List of Local to Global principles

What are some of the local to global principles in different areas of mathematics?

Solution 1:

Differential Geometry

The existence of partitions of unity allows one to transfer local results to global ones.

The Gauss-Bonnet Theorem relates the Gaussian curvature (a local quantity) to the Euler characteristic (a global one).

Solution 2:

Diophantine Equations

Hasse Condition: If a Diophantine equation is solvable modulo every prime power (locally) as well as in the reals then it is solvable in the integers.

Hasse Principle is that the Hasse Condition holds for all quadratic Diophantine equations.

Solution 3:

Graph Theory

A graph has an Eulerian circuit (global) iff every node has even degree (local).

Solution 4:

Number Theory

The "original" (in terms of giving rise to the name) local-global principle for quadratic forms over number fields, due to Hasse, has already been mentioned. Here are two further local-global principles in which Hasse was involved.

1. Two (finite-dimensional) central simple algebras over a number field $K$ are isomorphic if and only if their base extensions to central simple algebras over $K_v$ are isomorphic for every completion $K_v$ of $K$. This is essentially the Albert-Brauer-Hasse-Noether theorem.

2. Class field theory can be formulated both for number fields and for their completions, called respectively global class field theory and local class field theory. (It can formulated also for function fields over finite fields, but let's not worry about that here.) Historically global class field theory came first and the proofs of local class field theory originally depended on global class field theory. Eventually Hasse was able to develop local class field theory in a self-contained way and then use it to prove global class field theory.

Solution 5:

Complex Analysis

Analytic continuation might be viewed as a local-to-global principle.