Affine-domain dimension equals transcendence degree

Depends on

• Noether normalisation yields module finiteness over a polynomial subring
• Transcendence degree is additive in finite towers
• A finite affine extension of a polynomial ring has dimension at least the number of variables
• A finite affine extension of a polynomial ring has dimension at most the number of variables
• Algebraicity is transitive in towers of field extensions

Used by

• A genus-one curve with a rational point embeds as a plane cubic Corollary
• Every smooth proper curve admits a projective embedding Corollary
• Height plus quotient dimension equals ambient dimension in an affine domain Corollary
• Normalization resolves the singularities of a projective curve Corollary
• The genus of a smooth plane curve in terms of its degree Corollary
• A genus-zero curve need not be the projective line Counterexample
• Blowing up a point on a singular surface need not be smooth Counterexample
• Normalization of a non-normal surface is not a point blowup Counterexample
• Smoothness of the source cannot be dropped in the extension of rational maps Counterexample
• The intersection product needs Cartier or complementary-dimension hypotheses Counterexample
• The Riemann inequality is not an equality for special divisors Counterexample
• Geometric genus of a singular curve Definition
• Gonality Definition
• Relative dimension of a smooth morphism at a point Definition
• A smooth conic is a projective line once it has a rational point Example
• A system of parameters need not minimally generate the maximal ideal Example
• Cuspidal cubic: delta invariant and normalization Example
• Nodal cubic: arithmetic genus one, delta one, geometric genus zero Example
• Riemann-Hurwitz for a tame double cover with 2r branch points Example
• The intersection pairing on the projective plane Example
• A divisorial valuation restricts to a divisorial valuation or the trivial valuation Lemma
• Affine morphisms have one-sided perverse exactness Lemma
• Complex algebraic boundary points admit normalized curve arcs Lemma
• Cycles of coherent sheaves and of closed subschemes, with flat pullback Lemma
• Dense relative-dimension strata in flat finitely presented fibres Lemma
• Divisors on the projective line are classified by degree Lemma
• Fibres of standard smooth algebras are regular of relative dimension Lemma
• Fibrewise regular sequences persist openly in equidimensional Cohen-Macaulay fibres Lemma
• Finite relative integral-closure charts for classical quasi-finite morphisms Lemma
• Flat pullback of cycles and of rational equivalence Lemma
• Local fibre dimension equals local ring dimension plus residue transcendence degree Lemma
• Prime-to-characteristic torsion bound for affine commutative groups Lemma
• Proper normal curve rational function map Lemma
• Proper pushforward of cycles and the norm formula Lemma
• Regular hyperplane step for coherent support induction Lemma
• Dimension equals transcendence degree Theorem
• Fibres have pure expected dimension over a dense open Theorem
• Line bundles of degree at least 2g+1 are very ample Theorem
• Nonzero multiplication on an abelian variety is finite and faithfully flat Theorem
• Point blowups of regular surfaces stay regular, with rational exceptional fibre over the residue field Theorem

…and 2 more results.

Dependency tree · two levels

16 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, §§18, 21 (standard reference, not scraped)
• The Stacks Project, Section 10.116: Dimension of finite type algebras over fields, reprise (standard reference, not scraped)
• Melvin Hochster, Dimension theory and systems of parameters (standard reference, not scraped)