Morphisms of schemes

Depends on

• Schemes
• Morphisms of locally ringed spaces

Used by

• DVR uniqueness need not detect nonseparatedness Counterexample
• Pulling back the equation of a Weil divisor can give zero Counterexample
• Affine morphisms Definition
• Algebraic group actions, orbit maps, orbit subschemes and scheme-theoretic stabilizers Definition
• Algebraic spaces over a scheme, defined as fppf sheaves Definition
• Birational morphisms of integral finite-type schemes Definition
• Closed immersions of schemes Definition
• Deformations of schemes and the infinitesimal deformation functor Definition
• Descent data for schemes over an fppf covering Definition
• Etale neighbourhoods and elementary etale neighbourhoods of a point Definition
• Fibre product of schemes Definition
• Formally smooth morphism Definition
• Formally unramified morphism Definition
• Fppf coverings and the fppf site Definition
• Geometric vector bundle with the sections convention Definition
• Groupoids in schemes, relations and etale equivalence relations Definition
• Limits of one-parameter orbits and concentrator subschemes Definition
• Locally finite presentation morphisms Definition
• Locally finite type and finite type morphisms Definition
• Open and universally open morphisms of schemes Definition
• Open immersions of schemes Definition
• Quasi-compact and quasi-separated morphisms Definition
• Quotient sheaves and representable quotients for pre-relations and group actions Definition
• Rational maps of integral finite-type schemes Definition
• Representable morphisms of presheaves and fibrewise properties Definition
• Scheme-theoretic image Definition
• Schemes and morphisms over a base Definition
• Sheaf of relative Kähler differentials Definition
• The cotangent complex of a morphism of schemes Definition
• The fppf quotient sheaf of a pre-relation Definition
• An irreducible curve can have arbitrarily large tangent dimension Example
• Three axes are not three coplanar lines Example
• A quasi-compact image stable under specialization is closed Lemma
• Canonical resolution under isomorphisms of the ground field Lemma
• Derivative ideals under semilinear ground-field isomorphisms Lemma
• Differentials, open restriction, and the chain rule Lemma
• Flat deformations form a Zariski sheaf of groupoids Lemma
• Fpqc covers are universally submersive Lemma
• Fpqc descent of properness components Lemma
• Morphisms of schemes are local on compatible open covers Lemma

…and 11 more results.

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, Morphisms of Schemes, Section 1 (standard reference, not scraped)