A place of a commutative unital ring has different meanings in the literature:

  • The most common one is an equivalence class of (archimedean or non-archimedean) absolute values. (An absolute value is a non-trivial multiplicative seminorm.) This convention is often used in number theory, where it applies to number or function fields.

  • It can mean an equivalence class of (possibly higher-rank) valuations. This convention is sometimes used in non-archimedean analytic geometry, after Zariski’s work (Zariski–Riemann space).

  • It can mean an equivalence class of morphism to fields. This convention is used in scheme theory, where a place is exactly the same thing as a prime ideal (easy proof).

Other notions of places can be imagined, combining the three above classical examples.

The notion of place in number theory is interesting because it is at the heart of the geometric study of zeta functions. An improvement on this notion is given by the setting of global analytic geometry, which includes trivial seminorms that allow a natural geometric definition of adeles and ideles.

Revised on December 17, 2015 00:19:17 by Urs Schreiber (