Open immersions of schemes

Depends on

• Morphisms of schemes
• Affine open subschemes

Used by

• Proper birational normal curves agree off finitely many points Corollary
• A nonclosed open immersion is not proper Counterexample
• Global generation does not imply very ampleness Counterexample
• Under AC, proper integral finite-type schemes over fields with multiple points are not affine Counterexample
• Fppf coverings and the fppf site Definition
• Immersion of schemes Definition
• Morphisms, products and fibre products of algebraic spaces Definition
• Rational maps of integral finite-type schemes Definition
• Relative projective space from standard charts Definition
• Representable morphisms of presheaves and fibrewise properties Definition
• S-dense open subschemes and S-rational maps Definition
• Open immersions are etale Example
• The affine line with doubled origin Example
• The projective line from two affine charts Example
• The scheme-theoretic image of a dense open immersion Example
• Valuative extension of projective coordinates Example
• An étale universally injective morphism is an open immersion Lemma
• Birational morphisms restrict to isomorphisms between principal affine opens Lemma
• Blowups restrict to open subschemes of the base Lemma
• Chow lemma for proper Noetherian schemes Lemma
• Differentials, open restriction, and the chain rule Lemma
• Gluing algebraic spaces along open subfunctors Lemma
• Modification preserves open cohomology images Lemma
• Open and closed immersions are separated Lemma
• Open immersions are monomorphisms Lemma
• Open restrictions of the canonical desingularization Lemma
• Restricting fibre products to open subschemes Lemma
• Schematic closure and agreement on a dense open Lemma
• SNC projective compactifications of smooth quasiprojective varieties Lemma
• Tangent vectors at rational points are dual-number points Lemma
• Torsors under the additive group over an affine scheme are trivial Lemma
• Valuation lifts detect universal closedness Lemma
• An unramified morphism has an open diagonal Theorem
• Finite-dimensional projective space is proper over every base Theorem
• Immersions are monomorphisms locally of finite type Theorem
• Resolution of singularities in characteristic zero Theorem
• Smoothness survives base change and composition Theorem

Dependency tree · two levels

5 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, Schemes, Section 10 (standard reference, not scraped)