Localization (commutative algebra): Difference between revisions

Revision as of 14:04, 13 February 2024

Revision as of 14:09, 13 February 2024

Line 106:

* The multiplicative set consists of all powers of an element {{mvar|t}} of a ring {{mvar|R}}. The resulting ring is commonly denoted <math>R_t,</math> and its spectrum is the Zariski open set of the prime ideals that do not contain {{mvar|t}}. Thus the localization is the analog of the restriction of a topological space to a neighborhood of a point (every prime ideal has a [[neighborhood basis]] consisting of Zariski open sets of this form).

{{anchor|away from}}In [[number theory]] and [[algebraic topology]], when working over the ring <math>\Z</math> of the [[integer]]s, one refers to a property relative to an integer {{mvar|n}} as a property true ''at'' {{mvar|n}} or ''away'' from {{mvar|n}}, depending on the localization that is considered. "'''Away from''' {{mvar|n}}" means that the property is considered after localization by the powers of {{mvar|n}}, and, if {{mvar|p}} is a [[prime number]], "at {{mvar|p}}" means that the property is considered after localization at the prime ideal <math>p\Z</math>. This terminology can be explained by the fact that, if {{mvar|p}} is prime, the nonzero prime ideals of the localization of <math>\Z</math> are either the [[singleton set]] {{math|{{mset|p}}}} or its complement in the set of prime numbers.

== Localization and saturation of ideals ==