Prime ideals of a quotient ring are exactly the prime ideals containing the ideal

Depends on

• Primes of a quotient lie over the kernel
• A ring map induces a contraction map on prime spectra

Used by

• A plane intersection with no common component is nonempty and zero-dimensional Corollary
• Dimension of a quotient via chains above an ideal Corollary
• Passing to a quotient does not increase Krull dimension Corollary
• Passing to the reduced quotient does not change the prime spectrum Corollary
• An affine nilpotent thickening Definition
• Delta invariant of a curve singularity Definition
• Dual numbers and their reduced quotient have the same prime set Example
• Minimal and maximal primes of the node ring Example
• The polynomial-dimension formula at fields, Artinian rings, and the zero-ring boundary Example
• A first parameter lowers local dimension by exactly one Lemma
• A prime chain in R extends to a longer chain in R[x] Lemma
• A prime chain in R[x] has length at most one more than its contraction chain Lemma
• A surjective ring map induces a closed immersion of affine spectra Lemma
• Choose a parameter that misses the top-dimensional minimal components Lemma
• Closed immersions are affine quotients and survive base change Lemma
• Constructible subsets stable under generalisation are open in an affine spectrum Lemma
• Finite algebras over a strongly transcendental variable are nowhere quasi-finite Lemma
• Finite local length exactly when no common local branch Lemma
• Finite prime chains lift through module-finite domain extensions without Choice Lemma
• Flat deformations form a Zariski sheaf of groupoids Lemma
• Fpqc covers are universally submersive Lemma
• Height in a quotient measures chains between two primes Lemma
• Only one saturated step can lie over a fixed contracted prime in R[x] Lemma
• Points and topology of a fibre Lemma
• Prime-ideal valuations of a fractional ideal have finite support and add under products Lemma
• Quasi-finite local fibres transfer through quotients and intermediate rings Lemma
• Quotienting by the first minimal prime reduces the remaining height count Lemma
• Reduce the principal ideal theorem to a Noetherian local domain Lemma
• Regular hyperplane step for coherent support induction Lemma
• Schematic closure and agreement on a dense open Lemma
• Support dimension under field extension Lemma
• The order function of a one-dimensional Noetherian local domain Lemma
• The spectrum of a quotient is a closed subspace Lemma
• Algebraic Zariski Main localization at a quasi-finite prime Theorem
• An Artinian ring is canonically the finite product of its localizations at its maximal ideals Theorem
• Closed immersions into affine schemes are quotient spectra Theorem
• Every finite-dimensional Noetherian local ring has a system of parameters Theorem
• Going up for integral ring maps Theorem
• Lying over for integral ring maps Theorem
• Quasi-coherent ideals and closed subschemes 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)