Krull dimension of a nonzero ring

Depends on

• Prime ideals and maximal ideals in a commutative ring

Used by

• A domain finite over a polynomial ring has dimension at least the number of variables Corollary
• A Noetherian local domain has dimension zero exactly when it is a field Corollary
• A plane intersection with no common component is nonempty and zero-dimensional Corollary
• Dimension of a quotient via chains above an ideal Corollary
• Injective integral extensions preserve Krull dimension Corollary
• Localisation does not increase Krull dimension Corollary
• Passing to a quotient does not increase Krull dimension Corollary
• Prime ideals and dimension of a DVR Corollary
• Rings of integers are Dedekind domains Corollary
• The degree of a divisor descends to the Picard group of a normal proper curve Corollary
• Under going down and incomparability, lying-over primes have the same finite height Corollary
• A flat family with a nodal special fibre is not smooth at the node Counterexample
• A genus-zero curve need not be the projective line Counterexample
• Frobenius linear systems have nonreduced general members Counterexample
• Frobenius on the affine line is finite flat but not smooth Counterexample
• Smoothness of the source cannot be dropped in the extension of rational maps Counterexample
• The equation must define the intended scheme Counterexample
• Dedekind domains Definition
• embedding dimension and regular local ring Definition
• Local Krull dimension of a hypersurface germ Definition
• Relative dimension of a smooth morphism at a point Definition
• Systems of parameters and parameter ideals Definition
• The height of a prime ideal Definition
• Total length of a zero-dimensional projective scheme Definition
• Divisor of a rational function on the projective line Example
• Principal divisors on the projective line have degree zero Example
• Pulling a divisor back along the cusp normalization Example
• The polynomial-dimension formula at fields, Artinian rings, and the zero-ring boundary Example
• The ring (ℤ/2)^ℕ is zero-dimensional but not Noetherian Example
• A prime chain in R extends to a longer chain in R[x] Lemma
• A zero-dimensional projective scheme has finitely many closed points with finite-dimensional local rings Lemma
• Choose a parameter that misses the top-dimensional minimal components Lemma
• Coprime positive-degree plane forms form a regular sequence Lemma
• Dense relative-dimension strata in flat finitely presented fibres Lemma
• Divisors on the projective line are classified by degree Lemma
• Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes Lemma
• Field extension preserves the graded pieces and the total length of a zero-dimensional projective quotient Lemma
• Finite local length exactly when no common local branch Lemma
• Graded Nakayama and the Hesselink regularity comparison Lemma
• Height equals local dimension Lemma

…and 20 more results.

Dependency tree · two levels

4 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)