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
• A nonradical ideal need not enlarge every tangent space Counterexample
• A Weil divisor that is not Cartier at the vertex of the quadric cone Counterexample
• Finite type need not be locally free Counterexample
• Quasi-separatedness in pushforward cannot be omitted Counterexample
• The equation must define the intended scheme Counterexample
• The intersection product needs Cartier or complementary-dimension hypotheses Counterexample
• Multiplicity of a hypersurface equation at a rational point Definition
• Multiplicity of a plane curve at a point Definition
• Regular points of locally Noetherian schemes Definition
• Sheaf total quotient rings Definition
• Support of a module Definition
• The height of a prime ideal Definition
• The intrinsic Zariski tangent space Definition
• The local ring A₀ of rational functions regular at q=0 Definition
• A coherent closed-point skyscraper Example
• A cusp local ring is not a DVR Example
• Divisor of a rational function on the projective line 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
• Principal divisors on the projective line have degree zero Example
• Pulling a divisor back along the cusp normalization Example
• S-units of Q 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
• The twists on the projective line have degree n Example
• ℤ₍ₚ₎ consists of rationals with denominator not divisible by p, has maximal ideal pℤ₍ₚ₎, and residue field Fₚ Example
• A meromorphic unit has locally finite nonzero order support Lemma
• A prime lies in the support exactly when some element has annihilator inside it Lemma
• A quasi-compact image stable under specialization is closed Lemma
• A regular point lies on one irreducible component Lemma
• A surjective ring map induces a closed immersion of affine spectra Lemma
• Associated primes lie in the support Lemma
• Choice-free prime factorisation for a monogenic number ring Lemma
• Closed immersions are affine quotients and survive base change Lemma
• Cotangent spaces commute with localization at a rational point Lemma
• Exact principal-open Cech resolution Lemma
• Finite algebras over a strongly transcendental variable are nowhere quasi-finite Lemma

…and 28 more results.

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)