Closed immersions of schemes

Depends on

• Morphisms of schemes
• Ideal sheaves

Used by

• Agreement on a schematically dense open Corollary
• Degree is additive on invertible sheaves over a proper curve Corollary
• General hypersurfaces give smooth complete intersections Corollary
• Twisting the exact sequence of an effective Cartier divisor Corollary
• A proper nonprojective scheme from glued projective spaces Counterexample
• Global generation does not imply very ampleness Counterexample
• The canonical map of a hyperelliptic curve is not an embedding Counterexample
• The intersection product needs Cartier or complementary-dimension hypotheses Counterexample
• Unramified of finite presentation does not imply flat or etale Counterexample
• Algebraic cycles and the cycle group of a scheme of finite type over a field Definition
• Embedded deformations of a closed subscheme Definition
• Exceptional curves of the first kind and their contractions Definition
• Exceptional subscheme of a blowup Definition
• Formally smooth morphism Definition
• Formally unramified morphism Definition
• Hilbert function and Euler characteristic on a projective scheme Definition
• Immersion of schemes Definition
• Morphisms and closed subgroup schemes of group schemes Definition
• Morphisms, products and fibre products of algebraic spaces Definition
• Multiple test blow-ups, controlled transforms and resolutions of marked ideals Definition
• Presentations of algebraic spaces Definition
• Projective morphisms before Proj Definition
• Representable morphisms of presheaves and fibrewise properties Definition
• Scheme-theoretic image Definition
• Separated morphism of schemes Definition
• Square-zero extensions, small extensions and first-order thickenings Definition
• Strict transform of a closed subscheme Definition
• The derived subgroup, the derived series and solvable algebraic groups Definition
• Weil divisor normal noetherian scheme Definition
• A closed point immersion is unramified Example
• Additive and infinitesimal group schemes Example
• An irreducible curve can have arbitrarily large tangent dimension Example
• Incidence projection has closed determinantal image Example
• The empty morphism is finite, proper and projective Example
• Two infinitesimal structures at the origin Example
• A section of an invertible sheaf has a canonical zero subscheme Lemma
• A surjective ring map induces a closed immersion of affine spectra Lemma
• An immersion with closed image is a closed immersion Lemma
• Blowups of integral projective complex schemes are projective Lemma
• Canonical resolutions commute with embeddings of ambient smooth schemes Lemma

…and 65 more results.

Dependency tree · two levels

4 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, Morphisms of Schemes, Definition 2.1 (standard reference, not scraped)