Localization (commutative algebra): Difference between revisions

Revision as of 18:21, 26 September 2021

Revision as of 04:10, 28 October 2021

Line 103:

Two classes of localizations are more commonly considered:

* The multiplicative set is the [[complement (set theory)|complement]] of a [[prime ideal]] <math>\mathfrak p</math> of a ring {{mvar|R}}. In this case, one speaks of the "localization at <math>\mathfrak p</math>", or "localization at a point". The resulting ring, denoted <math>R_\mathfrak p</math> is a [[local ring]], and is the algebraic analog of a [[germ (mathematics)#Ring of germs|ring of germs]].

* The set 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.