nLab local-global principle

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 $Spec(\mathbb{Z})$, the “local” situation is the formal neighbourhoods of all primes $p$ (points in the prime spectrum $Spec(\mathbb{Z})$ 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.

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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…

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Related concepts

• descent

References

• Barry Mazur, On the passage from local to global in number theory, Bulletin of the AMS Vol 29, Number 1, 1993 (pdf)

• Wikipedia, Hasse principle