Scheme-theoretic fiber #
Main result #
AlgebraicGeometry.Scheme.Hom.fiber
:f.fiber y
is the scheme theoretic fiber off
aty
.AlgebraicGeometry.Scheme.Hom.fiberHomeo
:f.fiber y
is homeomorphic tof ⁻¹' {y}
.AlgebraicGeometry.Scheme.Hom.finite_preimage
: Finite morphisms have finite fibers.AlgebraicGeometry.Scheme.Hom.discrete_fiber
: Finite morphisms have discrete fibers.
f.fiber y
is the scheme theoretic fiber of f
at y
.
Equations
- f.fiber y = CategoryTheory.Limits.pullback f (Y.fromSpecResidueField y)
Instances For
Equations
- AlgebraicGeometry.instCanonicallyOver_1 f y = { hom := f.fiberι y }
The canonical map from the scheme theoretic fiber to the residue field.
Equations
Instances For
@[reducible]
def
AlgebraicGeometry.Scheme.Hom.fiberOverSpecResidueField
{X Y : Scheme}
(f : X.Hom Y)
(y : ↥Y)
:
(f.fiber y).Over (Spec (Y.residueField y))
The fiber of f
at y
is naturally a κ(y)
-scheme.
Equations
- f.fiberOverSpecResidueField y = { hom := f.fiberToSpecResidueField y }
Instances For
theorem
AlgebraicGeometry.Scheme.Hom.fiberToSpecResidueField_apply
{X Y : Scheme}
(f : X.Hom Y)
(y : ↥Y)
(x : ↥(f.fiber y))
:
The scheme theoretic fiber of f
at y
is homeomorphic to f ⁻¹' {y}
.
Equations
- f.fiberHomeo y = ⋯.toHomeomorph.trans (Homeomorph.setCongr ⋯)
Instances For
@[simp]
theorem
AlgebraicGeometry.Scheme.Hom.fiberHomeo_apply
{X Y : Scheme}
(f : X.Hom Y)
(y : ↥Y)
(x : ↥(f.fiber y))
:
@[simp]
theorem
AlgebraicGeometry.Scheme.Hom.fiberι_fiberHomeo_symm
{X Y : Scheme}
(f : X.Hom Y)
(y : ↥Y)
(x : ↑(⇑(CategoryTheory.ConcreteCategory.hom f.base) ⁻¹' {y}))
:
def
AlgebraicGeometry.Scheme.Hom.asFiber
{X Y : Scheme}
(f : X.Hom Y)
(x : ↥X)
:
↥(f.fiber ((CategoryTheory.ConcreteCategory.hom f.base) x))
A point x
as a point in the fiber of f
at f x
.
Equations
- f.asFiber x = (f.fiberHomeo ((CategoryTheory.ConcreteCategory.hom f.base) x)).symm ⟨x, ⋯⟩
Instances For
@[simp]
theorem
AlgebraicGeometry.Scheme.Hom.fiberι_asFiber
{X Y : Scheme}
(f : X.Hom Y)
(x : ↥X)
:
(CategoryTheory.ConcreteCategory.hom (f.fiberι ((CategoryTheory.ConcreteCategory.hom f.base) x)).base) (f.asFiber x) = x
instance
AlgebraicGeometry.instCompactSpaceCarrierCarrierCommRingCatFiberOfQuasiCompact
{X Y : Scheme}
(f : X ⟶ Y)
[QuasiCompact f]
(y : ↥Y)
:
CompactSpace ↥(Scheme.Hom.fiber f y)
theorem
AlgebraicGeometry.QuasiCompact.isCompact_preimage_singleton
{X Y : Scheme}
(f : X ⟶ Y)
[QuasiCompact f]
(y : ↥Y)
:
instance
AlgebraicGeometry.instIsAffineFiberOfIsAffineHom
{X Y : Scheme}
(f : X ⟶ Y)
[IsAffineHom f]
(y : ↥Y)
:
IsAffine (Scheme.Hom.fiber f y)
instance
AlgebraicGeometry.instJacobsonSpaceCarrierCarrierCommRingCatFiberOfLocallyOfFiniteType
{X Y : Scheme}
(f : X ⟶ Y)
(y : ↥Y)
[LocallyOfFiniteType f]
:
JacobsonSpace ↥(Scheme.Hom.fiber f y)
instance
AlgebraicGeometry.instFiniteCarrierCarrierCommRingCatFiberOfIsFinite
{X Y : Scheme}
(f : X ⟶ Y)
(y : ↥Y)
[IsFinite f]
:
Finite ↥(Scheme.Hom.fiber f y)
instance
AlgebraicGeometry.Scheme.Hom.discrete_fiber
{X Y : Scheme}
(f : X ⟶ Y)
(y : ↥Y)
[IsFinite f]
:
DiscreteTopology ↥(fiber f y)