The height of a prime ideal

Depends on

• Prime ideals and maximal ideals in a commutative ring
• Localisation at a prime ideal: $R_{\mathfrak p}=(R\setminus\mathfrak p)^{-1}R$
• Krull dimension of a nonzero ring

Used by

• A plane intersection with no common component is nonempty and zero-dimensional Corollary
• Height is bounded by the minimal number of local generators Corollary
• Under going down and incomparability, lying-over primes have the same finite height Corollary
• A cusp family defeats the missing source hypothesis Counterexample
• The intersection product needs Cartier or complementary-dimension hypotheses Counterexample
• serre r k and s k conditions Definition
• Weil divisor normal noetherian scheme Definition
• Divisor of a rational function on the projective line Example
• Localisation can strictly lower dimension Example
• Principal divisors on the projective line have degree zero Example
• Pulling a divisor back along the cusp normalization Example
• A prime chain in R[x] has length at most one more than its contraction chain Lemma
• Closed-point local dimension equals ambient irreducible dimension 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
• Flat maps with geometrically regular fibres have standard smooth local presentations Lemma
• Height equals local dimension Lemma
• Krull dimension of the holomorphic germ ring Lemma
• Local fibre-dimension bound from polynomial quasi-finiteness Lemma
• Converse to Krull's height theorem in localised form Theorem
• Height-one localizations of normal Noetherian domains are DVRs Theorem
• Jacobian criterion and openness of the regular locus over a perfect field Theorem
• Under AC, Cartier and Weil divisors agree on a locally factorial Noetherian integral scheme 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

• J. S. Milne, A Primer of Commutative Algebra, v4.03, Definition 3.14 (standard reference, not scraped)
• The Stacks Project, Section 10.60: Dimension of rings (standard reference, not scraped)