Localisation at a prime ideal: $R_{\mathfrak{p}}=(R\setminus \mathfrak{p}{)}^{-1}R$

Depends on

• Multiplicative subsets and the localisation $S^{-1}R$ as equivalence classes of fractions
• Prime ideals and maximal ideals in a commutative ring

Used by

• Assuming the Axiom of Choice, a local ring R is canonically isomorphic to Rₘ at its maximal ideal Corollary
• Primes of a localization at a prime Corollary
• Support of a module Definition
• The height of a prime ideal Definition
• A cusp local ring is not a DVR Example
• F[x]₍ₓ₎ is the ring of rational functions defined at 0, with maximal ideal generated by x and residue field F Example
• Localising cyclic abelian groups and Q/Z at a prime Example
• Localizing a PID at a nonzero prime Example
• The ideal (x,y) in k[x,y]_(x,y) has two minimal generators Example
• The p-primary quotient Q/Z_(p) over Z_(p) shows finite generation is essential in Nakayama Example
• ℤ₍ₚ₎ consists of rationals with denominator not divisible by p, has maximal ideal pℤ₍ₚ₎, and residue field Fₚ Example
• A prime lies in the support exactly when some element has annihilator inside it Lemma
• Associated primes lie in the support Lemma
• Finite algebras over a strongly transcendental variable are nowhere quasi-finite Lemma
• Finite-type field extensions with zero Ω Lemma
• Finite-type maps from Jacobson rings induce finite residue-field extensions at maximal ideals Lemma
• Invertible Jacobian minor gives regular parameters in a polynomial fibre Lemma
• Strong transcendence descends to reduced minimal-prime quotients Lemma
• Unramified residue extensions are finite separable Lemma
• A domain is integrally closed if and only if its prime localisations are, equivalently if and only if its maximal localisations are Theorem
• Algebraic Zariski Main localization at a quasi-finite prime Theorem
• Every localization is flat, and localizing a flat module preserves flatness Theorem
• Height-one localizations of normal Noetherian domains are DVRs Theorem
• Rₚ is local with unique maximal ideal pRₚ Theorem
• The local ring at a point of an affine variety is the localization at its maximal ideal Theorem
• The stalk of the affine structure sheaf at a prime is Aₚ Theorem

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• The Stacks Project, Section 10.18: Local rings (standard reference, not scraped)