Prime ideals of a localization are exactly the primes disjoint from the denominator set

Depends on

• Primes of a localization avoid the denominator set
• A ring map induces a contraction map on prime spectra

Used by

• In a finite-type algebra over a field, radical ideals are intersections of maximal ideals Corollary
• Localisation does not increase Krull dimension Corollary
• The degree of a divisor descends to the Picard group of a normal proper curve Corollary
• The quasi-finite locus of a finite-type algebra is open Corollary
• Under going down and incomparability, lying-over primes have the same finite height Corollary
• A Weil divisor that is not Cartier at the vertex of the quadric cone Counterexample
• The intersection product needs Cartier or complementary-dimension hypotheses Counterexample
• Locally factorial scheme Definition
• Weil divisor normal noetherian scheme Definition
• Pulling a divisor back along the cusp normalization Example
• The punctured affine line as an open finite factorization Example
• A principal localization identifies its spectrum with a distinguished open Lemma
• A regular point lies on one irreducible component Lemma
• Associated primes localize forward Lemma
• Associated primes of a localized finite module come from upstairs Lemma
• Coprime polynomial factorisations lift after an etale localisation Lemma
• Coprime positive-degree plane forms form a regular sequence Lemma
• Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes Lemma
• Fibres of standard smooth algebras are regular of relative dimension Lemma
• Finite local length exactly when no common local branch Lemma
• Finite prime chains lift through module-finite domain extensions without Choice Lemma
• Finite-type field extensions with zero Ω Lemma
• Krull dimension of the holomorphic germ ring Lemma
• Local dimension for a reducible classical algebraic set Lemma
• Local DVRs at the nonzero primes force dimension one Lemma
• Only one saturated step can lie over a fixed contracted prime in R[x] Lemma
• Prime and local-ring correspondence on standard projective charts Lemma
• Reduce the principal ideal theorem to a Noetherian local domain Lemma
• The spectrum of a localisation is the subspace of primes disjoint from the denominator set Lemma
• The symbolic-power step inside the principal ideal theorem Lemma
• A localization of a Dedekind domain is Dedekind or a field Theorem
• A quasi-finite algebra factors openly through a finite algebra Theorem
• Cartier divisors on a normal Noetherian scheme give Weil divisors Theorem
• Comparable primes with the same contraction are equal under an integral map Theorem
• Finite and finite type etale schemes over an algebraically closed field Theorem
• Going down holds for integral extensions over integrally closed domains Theorem
• Height-one localizations of normal Noetherian domains are DVRs Theorem
• Jacobian criterion and openness of the regular locus over a perfect field Theorem
• Lying over for integral ring maps Theorem
• Smooth proper curves, dominant morphisms and function fields Theorem

…and 1 more result.

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 10.17: The spectrum of a ring (standard reference, not scraped)
• A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., §13 and §17 (standard reference, not scraped)