Principal distinguished subsets of the prime spectrum

Depends on

• The prime spectrum and vanishing sets

Used by

• Proper birational normal curves agree off finitely many points Corollary
• The quasi-finite locus of a finite-type algebra is open Corollary
• The spectrum of a principal localisation is the distinguished open D(f) Corollary
• A nonclosed open immersion is not proper Counterexample
• A nonseparated affine cover can have nonaffine intersection Counterexample
• Quasi-finite does not imply finite Counterexample
• Quasi-separatedness in pushforward cannot be omitted Counterexample
• The localization presheaf on distinguished opens Definition
• The underlying space of an affine spectrum Definition
• A coherent closed-point skyscraper Example
• Finite power map of the affine line Example
• The distinguished subset D(x) matches the primes of the localization at x Example
• The spectrum of the integers has one generic point, closed points (p), and basic opens D(n) Example
• A finite unit-ideal expression yields a finite distinguished-open subcover Lemma
• A quasi-compact image stable under specialization is closed Lemma
• A quasi-compact immersion with nonclosed image has a boundary specialization Lemma
• An affine scheme of finite type over a field has a finitely generated coordinate ring Lemma
• An idempotent partitions the spectrum into complementary clopen subsets Lemma
• Birational morphisms restrict to isomorphisms between principal affine opens Lemma
• Closed gluing of two projective three-spaces is proper Lemma
• Closed immersion preserves cohomology and coherent pushforward Lemma
• Closed immersions are affine quotients and survive base change Lemma
• Constructible subsets stable under generalisation are open in an affine spectrum Lemma
• Distinguished-subset covers detect radicals Lemma
• Distinguished-subset identities Lemma
• Every point of a Zariski-open set has a distinguished-open neighbourhood inside it Lemma
• Every Zariski-open subset is a union of distinguished opens Lemma
• Function field of an integral finite-type scheme Lemma
• Pushouts of closed immersions exist Lemma
• Schematic closure and agreement on a dense open Lemma
• The spectrum of a finite product ring is the disjoint union of the factor spectra Lemma
• Valuation lifts detect universal closedness Lemma
• A quasi-finite algebra factors openly through a finite algebra Theorem
• Closed immersions into affine schemes are quotient spectra Theorem
• Global functions on proper integral schemes form a finite extension of the base field Theorem
• Normalization of an integral finite-type curve by gluing affine integral closures Theorem
• Projective n-space has quasi-coherent cohomological dimension at most n Theorem
• Sections and restrictions on distinguished opens of an affine scheme Theorem
• The smooth locus is open Theorem

Dependency tree · two levels

3 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, §14 The spectrum of a ring (standard reference, not scraped)
• The Stacks Project, Section 10.17: The spectrum of a ring (standard reference, not scraped)