Proper morphisms

Depends on

• Separated morphism of schemes
• Locally finite type and finite type morphisms
• Universally closed morphisms

Used by

• Euler characteristic in a proper flat family is locally constant Corollary
• Finite morphisms are proper Corollary
• Finite-dimensional coherent cohomology over a field Corollary
• Global functions on geometrically connected and geometrically reduced proper schemes Corollary
• Proper birational normal curves agree off finitely many points Corollary
• Upper semicontinuity of fibre cohomology dimensions Corollary
• A nonclosed open immersion is not proper Counterexample
• A proper nonprojective scheme from glued projective spaces Counterexample
• Proper cohomology need not be finite for noncoherent sheaves Counterexample
• The affine line is not proper Counterexample
• Under AC, proper integral finite-type schemes over fields with multiple points are not affine Counterexample
• Cohomology and base-change map Definition
• Complete varieties Definition
• Euler characteristic of a coherent sheaf Definition
• Relative very ampleness in the finite projective-space convention Definition
• All twists on the projective line Example
• The empty morphism is finite, proper and projective Example
• Chow lemma for proper Noetherian schemes Lemma
• Closed gluing of two projective three-spaces is proper Lemma
• Euler characteristic is additive in short exact sequences Lemma
• Eventual generation of coherent projective twists Lemma
• Finite projective complex for proper flat coherent cohomology Lemma
• Finite-stage descent of properness for finitely presented schemes Lemma
• Flat field extension commutes with coherent cohomology Lemma
• Morphisms from a proper scheme to a separated one are proper Lemma
• Noetherian approximation of proper flat finitely presented sheaf data Lemma
• Properness is local on the target Lemma
• Properness survives arbitrary base change Lemma
• Properness survives composition Lemma
• Universal finite projective cohomology complex over any base Lemma
• Upper semicontinuity of proper fibre dimension Lemma
• Coherence is essential for proper finiteness Remark
• Projective and proper are distinct notions Remark
• Properness is not a compactness claim on rational points Remark
• A proper quasi-finite morphism is finite Theorem
• Coherent higher direct images under proper morphisms Theorem
• Cohomology and base change for proper flat coherent families Theorem
• Finite coherent cohomology for proper schemes Theorem
• Finite-dimensional projective space is proper over every base Theorem
• Global functions on proper integral schemes form a finite extension of the base field Theorem

…and 4 more results.

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

• Stacks Project, Morphisms of Schemes, §29.42 Definition 29.42.1 (standard reference, not scraped)