Sections and restrictions on distinguished opens of an affine scheme

Depends on

• The localization construction extends to the structure sheaf on Spec A
• Principal distinguished subsets of the prime spectrum
• Universal property of localisation: maps that invert $S$ factor uniquely through $S^{-1}R$

Used by

• A nonseparated affine cover can have nonaffine intersection Counterexample
• Finite type need not be locally free Counterexample
• Proper cohomology need not be finite for noncoherent sheaves Counterexample
• Under AC, proper integral finite-type schemes over fields with multiple points are not affine Counterexample
• Module sheaf on an affine scheme Definition
• Principal parts of an invertible sheaf on a curve Definition
• Two-affine projective line and its twists Definition
• Two-affine Mayer–Vietoris on the projective line Example
• A principal localization identifies its spectrum with a distinguished open Lemma
• Affine charts recover the algebraic module of differentials Lemma
• Affine pushforward algebra localizes Lemma
• Affineness from a finite principal cover Lemma
• An affine scheme of finite type over a field has a finitely generated coordinate ring Lemma
• An associated sheaf restricts to an associated sheaf on an affine open Lemma
• Closed gluing of two projective three-spaces is proper Lemma
• Closed immersions are affine quotients and survive base change Lemma
• Finite is affine and local on its target Lemma
• Intersections of affine opens admit principal affine covers Lemma
• Nonzero maps from an invertible sheaf to a locally free sheaf are injective Lemma
• Pushouts of closed immersions exist Lemma
• Schematic closure and agreement on a dense open Lemma
• Structure sheaf of a quasi-compact quasi-separated morphism is affine-local Lemma
• The spectrum of a finite product ring is the disjoint union of the factor spectra Lemma
• Valuation lifts detect universal closedness Lemma
• Affine quasi-coherent sheaves are modules Theorem
• Affine schemes are contravariantly equivalent to commutative rings Theorem
• Closed immersions into affine schemes are quotient spectra Theorem
• Extension of scalars of a scheme along a field extension Theorem
• Global functions on Spec A recover A Theorem
• High powers of an ample line bundle embed a proper scheme Theorem
• Projective n-space has quasi-coherent cohomological dimension at most n Theorem
• Scheme-theoretic image of a quasi-compact morphism Theorem
• The associated module sheaf exists Theorem
• The stalk of the affine structure sheaf at a prime is Aₚ Theorem

Dependency tree · two levels

11 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

• The Stacks Project, Lemma 26.5.4 (standard reference, not scraped)