normal noetherian ring

Depends on

• Integral closure in an extension ring and integrally closed domains

Used by

• Every smooth proper curve admits a projective embedding Corollary
• Proper birational normal curves agree off finitely many points Corollary
• A normal singular surface: the quadric cone Counterexample
• A Weil divisor that is not Cartier at the vertex of the quadric cone Counterexample
• Finite normalization alone does not make a curve regular Counterexample
• The intersection product needs Cartier or complementary-dimension hypotheses Counterexample
• Canonical bundle and canonical divisors Definition
• Divisors on a smooth proper curve Definition
• Locally factorial scheme Definition
• Normal points and normal varieties Definition
• Normal scheme modifications and normalized point blowups Definition
• Order codimension one rational function Definition
• Weil divisor normal noetherian scheme Definition
• A node is resolved by one point blowup Example
• Affine space is normal Example
• Ramification of the double cover y²=f(x) Example
• Resolving the quadric cone by one blowup Example
• The twists on the projective line have degree n Example
• A meromorphic unit has locally finite nonzero order support Lemma
• A normal Noetherian domain is the intersection of its height-one localizations Lemma
• Blowup charts of the quadric cone at its vertex Lemma
• Indeterminacy of a rational map into an affine scheme is of pure codimension one Lemma
• Normality is checked on affine open charts Lemma
• reduced noetherian total fractions and normal components Lemma
• The finite normalization of a curve factors through the blowup of a closed point Lemma
• The normalization defect is an Euler characteristic and a weighted sum of local lengths Lemma
• A normal curve over a perfect field is nonsingular Theorem
• A normal variety is regular in codimension one Theorem
• Normalization of a reduced curve is finite Theorem
• Normalization of an integral finite-type curve by gluing affine integral closures Theorem
• Resolution of normal surface singularities Theorem
• serre normality criterion Theorem
• Smooth proper curves, dominant morphisms and function fields Theorem
• Under AC, Cartier and Weil divisors agree on a locally factorial Noetherian integral scheme Theorem

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

• Definition 8.34 and Remarks 8.35–8.36, p.56 (standard reference, not scraped)