Every affine scheme is quasi-compact

Depends on

• Affine schemes and their coordinate rings
• The Axiom of Choice
• Every distinguished open of an affine spectrum is quasi-compact

Used by

• Finite morphisms are proper Corollary
• Global generation does not imply very ampleness Counterexample
• Proper cohomology need not be finite for noncoherent sheaves Counterexample
• Quasi-separatedness in pushforward cannot be omitted Counterexample
• The Hopf algebra of a split torus and its root-of-unity subgroups Example
• A quasi-compact image stable under specialization is closed Lemma
• A quasi-compact immersion with nonclosed image has a boundary specialization Lemma
• Affineness is local on the target Lemma
• An affine scheme of finite type over a field has a finitely generated coordinate ring Lemma
• Closed gluing of two projective three-spaces is proper Lemma
• Closed immersions are affine quotients and survive base change Lemma
• Constructible subsets stable under generalisation are open in an affine spectrum Lemma
• Eventual generation of coherent projective twists Lemma
• Finite is affine and local on its target Lemma
• Finite projective complex for proper flat coherent cohomology Lemma
• Finite type is affine-local on source and target Lemma
• Flat field extension commutes with coherent cohomology Lemma
• Fpqc covers are universally submersive Lemma
• Fpqc descent of properness components Lemma
• Line bundles on projective three-space and their restrictions Lemma
• Projective space is of finite type over its base Lemma
• Properness survives composition Lemma
• Quasi-compactness is local on the target and survives base change Lemma
• Quasi-separatedness and the diagonal Lemma
• Structure sheaf of a quasi-compact quasi-separated morphism is affine-local Lemma
• The general linear group scheme and its coordinate ring Lemma
• The quotient by a maximal line subbundle is locally free Lemma
• Valuation lifts detect universal closedness Lemma
• Affine acyclicity of quasi-coherent sheaves Theorem
• Affine group schemes of finite type are antiequivalent to finitely generated commutative Hopf algebras Theorem
• Affine schemes are quasi-separated Theorem
• Closed immersions into affine schemes are quotient spectra Theorem
• Coherent sheaves on a locally Noetherian scheme Theorem
• Finite and finite type etale schemes over an algebraically closed field Theorem
• Finite coherent cohomology for proper schemes Theorem
• Locally standard smooth iff flat with geometrically regular fibres Theorem
• Projective n-space has quasi-coherent cohomological dimension at most n Theorem
• Quasi-coherence of pushforward for qcqs morphisms Theorem
• Serre duality for coherent sheaves on projective space Theorem

Dependency tree · two levels

9 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 26.5.1 (standard reference, not scraped)