The underlying space of an affine spectrum

Depends on

• The prime spectrum and vanishing sets
• Principal distinguished subsets of the prime spectrum
• A ringed space
• A locally ringed space

Used by

• Euler characteristic in a proper flat family is locally constant Corollary
• Global functions on geometrically connected and geometrically reduced proper schemes 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 non-quasi-coherent module with H1 on an affine scheme Counterexample
• A nonclosed open immersion is not proper Counterexample
• A nonseparated affine cover can have nonaffine intersection Counterexample
• A scheme is not determined by its underlying topological space Counterexample
• A Weil divisor that is not Cartier at the vertex of the quadric cone Counterexample
• Finite type need not be locally free Counterexample
• Flat and finite type is not open without finite presentation Counterexample
• Global generation does not imply very ampleness Counterexample
• Proper cohomology need not be finite for noncoherent sheaves Counterexample
• Pulling back the equation of a Weil divisor can give zero Counterexample
• Quasi-separatedness in pushforward cannot be omitted Counterexample
• Stalk and fibre are different Counterexample
• The affine line is not proper Counterexample
• A one-vertex framed quiver variety and its Hecke correspondence Definition
• Affine schemes and their coordinate rings Definition
• Affine-local quasi-coherent algebras before general sheaf theory Definition
• Constructible subsets of a scheme Definition
• Hilbert function and Euler characteristic on a projective scheme Definition
• Module sheaf on an affine scheme Definition
• Points of Proj of a graded ring Definition
• Sheaf total quotient rings Definition
• The localization presheaf on distinguished opens Definition
• The scheme-theoretic tangent cone at a point Definition
• Total length of a zero-dimensional projective scheme Definition
• Two-affine projective line and its twists Definition
• A coherent closed-point skyscraper Example
• An upper jump of h0 in a flat projective family Example
• Divisor of a rational function on the projective line Example
• Finite power map of the affine line Example
• Open immersions are etale Example
• Principal divisors on the projective line have degree zero Example
• Pulling a divisor back along the cusp normalization Example
• The empty morphism is finite, proper and projective Example
• The spectrum of a field is a one-point affine scheme Example
• The twists on the projective line have degree n Example
• The zero ring has empty spectrum Example

…and 54 more results.

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, Section 26.5 (standard reference, not scraped)