Generic points of irreducible closed subsets

Depends on

• Affine schemes and their coordinate rings
• Irreducible topological spaces and irreducible subsets in the subspace topology
• A Zariski-closed subset is irreducible exactly when its radical defining ideal is prime, and then it has a unique generic point
• The closure of a prime is its vanishing set

Used by

• Blowing up a nonzero ideal on an integral scheme is birational Corollary
• Proper birational normal curves agree off finitely many points Corollary
• The generic point of the affine line has no relative k-valued coordinate Counterexample
• Under AC, proper integral finite-type schemes over fields with multiple points are not affine Counterexample
• Birational morphisms of integral finite-type schemes Definition
• Order codimension one rational function Definition
• Principal parts of an invertible sheaf on a curve Definition
• Rational section line bundle Definition
• Sheaf total quotient rings Definition
• Weil divisor normal noetherian scheme Definition
• Spec Z has one generic point and closed prime-number points Example
• A meromorphic unit has locally finite nonzero order support Lemma
• Affine neighbourhood containing component generic points Lemma
• Chow lemma for proper Noetherian schemes Lemma
• Cycles of coherent sheaves and of closed subschemes, with flat pullback Lemma
• Dense relative-dimension strata in flat finitely presented fibres Lemma
• Function field of an integral finite-type scheme Lemma
• Proper closed subsets of a curve are finite Lemma
• Regular hyperplane step for coherent support induction Lemma
• Contravariance reverses coordinates and scheme points are not only classical points Remark
• Cartier divisors on a normal Noetherian scheme give Weil divisors Theorem
• Coherent higher direct images under proper morphisms Theorem
• Every irreducible closed subset of an affine spectrum has a unique generic point Theorem
• Global functions on proper integral schemes form a finite extension of the base field Theorem
• Under AC, Cartier and Weil divisors agree on a locally factorial Noetherian integral scheme Theorem

Dependency tree · two levels

15 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

• James S. Milne, Algebraic Geometry, 10.24 (standard reference, not scraped)