Proper morphisms

Depends on

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

Used by

• Degree is additive on invertible sheaves over a proper curve Corollary
• Euler characteristic in a proper flat family is locally constant Corollary
• Every smooth proper curve admits a projective embedding 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
• The degree of a divisor descends to the Picard group of a normal proper curve Corollary
• Upper semicontinuity of fibre cohomology dimensions Corollary
• A genus-zero curve need not be the projective line Counterexample
• 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
• The fixed point theorem fails without completeness: the additive group acts on the affine line by translations Counterexample
• Under AC, proper integral finite-type schemes over fields with multiple points are not affine Counterexample
• Abelian schemes over a base Definition
• Abelian varieties over a field Definition
• Bivariant Chow operations and bivariant classes Definition
• Cohomology and base-change map Definition
• Complete varieties Definition
• Curves over a field Definition
• Degree divisor proper curve Definition
• Degree of an invertible sheaf on a proper one-dimensional scheme Definition
• Euler characteristic of a coherent sheaf Definition
• Exceptional curves of the first kind and their contractions Definition
• Genus and arithmetic genus of a curve Definition
• Geometric genus of a singular curve Definition
• Normal scheme modifications and normalized point blowups Definition
• Parabolic subgroups of an affine algebraic group Definition
• Principal parts of an invertible sheaf on a curve Definition
• Pushforward of coherent sheaves in algebraic K-theory Definition
• Relative very ampleness in the finite projective-space convention Definition
• Split augmented simplicial schemes and matching objects Definition
• All twists on the projective line Example
• Principal divisors on the projective line have degree zero Example
• Resolving the quadric cone by one blowup Example
• The empty morphism is finite, proper and projective Example
• The quotient of GL2 by the diagonal torus is the complement of the diagonal in P1 x P1 Example
• The twists on the projective line have degree n Example

…and 69 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)