Locally finite presentation morphisms

Depends on

• Morphisms of schemes
• Affine open subschemes

Used by

• Euler characteristic in a proper flat family is locally constant Corollary
• Upper semicontinuity of fibre cohomology dimensions Corollary
• Vanishing of the deformation tangent space forces rigidity of deformation classes Corollary
• A flat family with a nodal special fibre is not smooth at the node Counterexample
• Finite type need not mean finite presentation Counterexample
• Flat and finite type is not open without finite presentation Counterexample
• Frobenius on the affine line is finite flat but not smooth Counterexample
• The Lie algebra does not detect nonsmooth group schemes Counterexample
• Abelian schemes over a base Definition
• Deformations of schemes and the infinitesimal deformation functor Definition
• Embedded deformations of a closed subscheme Definition
• Étale morphism of schemes Definition
• Fppf coverings and the fppf site Definition
• Quotient sheaves and representable quotients for pre-relations and group actions Definition
• Smooth morphism of schemes Definition
• The étale locus of a morphism Definition
• The rigidified relative Picard functor and the dual abelian variety Definition
• Compensating h0 and h1 jumps with constant Euler characteristic Example
• Finite field extensions and etaleness Example
• Open immersions are etale Example
• Polynomial rings are flat and smooth Example
• Riemann-Hurwitz for a tame double cover with 2r branch points Example
• The family xy=t Example
• The square-root standard etale chart Example
• An S-rational map defined after a faithfully flat smooth base change is defined Lemma
• Constructible images for finite-presentation affine maps Lemma
• Coprime polynomial factorisations lift after an etale localisation Lemma
• Criterion for a scheme to represent an fppf quotient sheaf Lemma
• Dense relative-dimension strata in flat finitely presented fibres Lemma
• Effective fppf descent for separated locally quasi-finite morphisms Lemma
• Finite-stage descent of finitely presented schemes and their morphisms Lemma
• Flat deformations form a Zariski sheaf of groupoids Lemma
• Flat locally finitely presented restrictions give open subquotients Lemma
• Hilbert divisor charts and the Picard diagonal Lemma
• Local finiteness conditions under base change Lemma
• Noetherian approximation of proper flat finitely presented sheaf data Lemma
• Restriction of an etale equivalence relation Lemma
• Smooth orbits are locally closed and their orbit maps are faithfully flat over every field Lemma
• The product of two projective lines is an integral smooth projective surface Lemma
• Torsors under the additive group over an affine scheme are trivial Lemma

…and 16 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, Definition 22.1 (standard reference, not scraped)