Chain dimension and the empty-space convention

Depends on

• Noetherian topological spaces via ACC on opens or DCC on closed subsets
• Irreducible topological spaces and irreducible subsets in the subspace topology

Used by

• A plane intersection with no common component is nonempty and zero-dimensional Corollary
• Degree is additive on invertible sheaves over a proper curve Corollary
• Proper birational normal curves agree off finitely many points Corollary
• The degree of a divisor descends to the Picard group of a normal proper curve Corollary
• The dimension of a complete linear system Corollary
• A genus-zero curve need not be the projective line Counterexample
• Dimension zero for the empty set loses the empty-fibre distinction 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
• Under AC, proper integral finite-type schemes over fields with multiple points are not affine Counterexample
• Algebraic cycles and the cycle group of a scheme of finite type over a field Definition
• Contact order of two regular components at a point Definition
• Curves over a field Definition
• Degree divisor proper curve Definition
• Degree of an invertible sheaf on a proper one-dimensional scheme Definition
• Delta invariant of a curve singularity Definition
• Genus and arithmetic genus of a curve Definition
• Geometric genus of a singular curve Definition
• Global and local dimension of classical varieties Definition
• Intersection multiplicity of closed subschemes at a point Definition
• Intersection numbers of Cartier divisors on a smooth projective surface Definition
• Order of an ideal sheaf at a point Definition
• Simple normal crossings divisors and simultaneous normal crossings position Definition
• Strict normal crossings divisor on a regular surface Definition
• The canonical divisor of a smooth projective surface Definition
• Principal divisors on the projective line have degree zero Example
• The twists on the projective line have degree n Example
• A nonzero global dual section detects a cohomology class Lemma
• Ample divisors meet nonzero effective divisors positively Lemma
• Closed finite-index subgroups of rational points of smooth connected groups over algebraically closed fields are the whole point group Lemma
• Codimension-one components of a maximal-order support Lemma
• Critical loci have small images in characteristic zero Lemma
• Dimension can be computed on an open cover Lemma
• Dimension of a finite closed union Lemma
• Divisors on the projective line are classified by degree Lemma
• Euler characteristic of a closed point, and invariance under an invertible twist Lemma
• Extension-by-zero generators detect sheaf-cohomology vanishing Lemma
• Fibre dimension and orbit dimension add to the dimension of the group Lemma
• Fibres of a proper birational morphism of regular surfaces Lemma
• Finite-dimensionality of the Riemann-Roch space Lemma

…and 33 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

• Milne §3l Definition 3.39 and §2m Definition 2.48 (standard reference, not scraped)