@[implicit_reducible]

noncomputable instance Lec6.functionField_field (X : AlgebraicGeometry.Scheme) [AlgebraicGeometry.IsIntegral X] : Field ↑X.functionField

The function field K(X) of an integral scheme X is a field (Lec 6, Def 14 setting).

theorem Lec6.functionField_germ_injective (X : AlgebraicGeometry.Scheme) [AlgebraicGeometry.IsIntegral X] (U : X.Opens) [Nonempty ↥↑U] : Function.Injective ⇑(CategoryTheory.ConcreteCategory.hom (X.germToFunctionField U))

The germ map from sections over a nonempty open of an integral scheme into the function field K(X) is injective (Lec 6, Def 14).

instance Lec6.functionField_isFractionRing_affine (R : CommRingCat) [IsDomain ↑R] : IsFractionRing ↑R ↑(AlgebraicGeometry.Spec R).functionField

For affine Spec R with R a domain, the function field is the fraction field of R (Lec 6, Def 14 in the affine case).

theorem Lec6.functionField_isFractionRing_of_affineOpen (X : AlgebraicGeometry.Scheme) [AlgebraicGeometry.IsIntegral X] (U : X.Opens) (hU : AlgebraicGeometry.IsAffineOpen U) [Nonempty ↥↑U] : IsFractionRing ↑(X.presheaf.obj (Opposite.op U)) ↑X.functionField

On an affine open U of an integral scheme X, the function field K(X) is the fraction field of Γ(X, U) (Lec 6, Def 14).

@[reducible, inline]

abbrev Lec6.functionField (R : Type u_1) [CommRing R] [IsDomain R] : Type u_1

The function field of a domain R, defined as its fraction field (Lec 6, Def 14, affine viewpoint).

@[implicit_reducible]

noncomputable instance Lec6.functionField.field (R : Type u_1) [CommRing R] [IsDomain R] : Field (functionField R)

The function field of a domain is a field.

@[implicit_reducible]

instance Lec6.functionField.algebra (R : Type u_1) [CommRing R] [IsDomain R] : Algebra R (functionField R)

The domain R is an algebra over its function field.

instance Lec6.functionField.isFractionRing (R : Type u_1) [CommRing R] [IsDomain R] : IsFractionRing R (functionField R)

The function field of a domain is its fraction field.

noncomputable def Lec6.degreeDominantMap (A : Type u_1) (B : Type u_2) [CommRing A] [CommRing B] [IsDomain A] [IsDomain B] [Algebra B A] [Algebra (FractionRing B) (FractionRing A)] : ℕ

Degree of a dominant map Spec A → Spec B of integral affine schemes, defined as the degree of the corresponding extension of function fields (Lec 6, Def 15).

theorem Lec6.lec6_fiber_card_le_degree {B : Type u_1} [CommRing B] [IsDomain B] [IsIntegrallyClosed B] {A : Type u_2} [CommRing A] [IsDomain A] [Algebra B A] [Module.Finite B A] [NoZeroSMulDivisors B A] (𝔭 : Ideal B) [𝔭.IsPrime] : Nat.card ↑{q : Ideal A | q.IsPrime ∧ Ideal.comap (algebraMap B A) q = 𝔭} ≤ Module.finrank (FractionRing B) (FractionRing A)

Lec 6, Lem 13: for a finite extension B → A with B normal, the number of primes of A lying over a prime 𝔭 of B is at most the degree of the corresponding extension of function fields.

def Lec6.Lec6IsUnramifiedOver (A : Type u_1) (B : Type u_2) [CommRing A] [CommRing B] [IsDomain A] [IsDomain B] [Algebra B A] [Algebra (FractionRing B) (FractionRing A)] (𝔭 : Ideal B) : Prop

A prime 𝔭 of B is unramified in A if the number of primes of A over 𝔭 equals the degree of the function field extension (Lec 6, Def 16).

def Lec6.Lec6IsRamifiedOver (A : Type u_1) (B : Type u_2) [CommRing A] [CommRing B] [IsDomain A] [IsDomain B] [Algebra B A] [Algebra (FractionRing B) (FractionRing A)] (𝔭 : Ideal B) : Prop

A prime 𝔭 of B is ramified in A if it is not unramified (Lec 6, Def 16).

def Lec6.RamificationLocus (A : Type u_1) (B : Type u_2) [CommRing A] [CommRing B] [Algebra A B] [Module.Finite A B] [Module.Free A B] : Set (PrimeSpectrum A)

The ramification locus in Spec A of a finite free extension A → B, defined as the zero locus of the discriminant (Lec 6, Prop 7 setting).

theorem Lec6.ramificationLocus_isClosed (A : Type u_1) (B : Type u_2) [CommRing A] [CommRing B] [Algebra A B] [Module.Finite A B] [Module.Free A B] : IsClosed (RamificationLocus A B)

Lec 6, Prop 7: the ramification locus is closed in Spec A.

theorem Lec6.ramificationLocus_ne_univ_of_discr_ne_zero (A : Type u_1) (B : Type u_2) [CommRing A] [IsDomain A] [CommRing B] [Algebra A B] [Module.Finite A B] [Module.Free A B] (hd : Algebra.discr A ⇑(Module.Free.chooseBasis A B) ≠ 0) : RamificationLocus A B ≠ Set.univ

If the discriminant of A → B is nonzero, then the ramification locus is a proper subset of Spec A (Lec 6, Prop 7).

theorem Lec6.discr_ne_zero_of_separable_functionField (A : Type u_1) (K : Type u_2) (B : Type u_3) (L : Type u_4) [CommRing A] [IsDomain A] [Field K] [Algebra A K] [IsFractionRing A K] [CommRing B] [IsDomain B] [Field L] [Algebra B L] [IsFractionRing B L] [Algebra A B] [Module.Finite A B] [Module.Free A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsLocalization (Algebra.algebraMapSubmonoid B (nonZeroDivisors A)) L] [Algebra.IsSeparable K L] [Module.Finite K L] : Algebra.discr A ⇑(Module.Free.chooseBasis A B) ≠ 0

Helper for Lec 6, Prop 7: if the function field extension K → L is finite separable, the discriminant of A → B is nonzero.

theorem Lec6.ramificationLocus_ne_univ_of_separable (A : Type u_1) (K : Type u_2) (B : Type u_3) (L : Type u_4) [CommRing A] [IsDomain A] [Field K] [Algebra A K] [IsFractionRing A K] [CommRing B] [IsDomain B] [Field L] [Algebra B L] [IsFractionRing B L] [Algebra A B] [Module.Finite A B] [Module.Free A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsLocalization (Algebra.algebraMapSubmonoid B (nonZeroDivisors A)) L] [Algebra.IsSeparable K L] [Module.Finite K L] : RamificationLocus A B ≠ Set.univ

Lec 6, Prop 7: under separability of the function field extension, the ramification locus is a proper subset.

theorem Lec6.ramificationLocus_properClosed_of_separable (A : Type u_1) (K : Type u_2) (B : Type u_3) (L : Type u_4) [CommRing A] [IsDomain A] [Field K] [Algebra A K] [IsFractionRing A K] [CommRing B] [IsDomain B] [Field L] [Algebra B L] [IsFractionRing B L] [Algebra A B] [Module.Finite A B] [Module.Free A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsLocalization (Algebra.algebraMapSubmonoid B (nonZeroDivisors A)) L] [Algebra.IsSeparable K L] [Module.Finite K L] : IsClosed (RamificationLocus A B) ∧ RamificationLocus A B ≠ Set.univ

Lec 6, Prop 7: the ramification locus is a proper closed subset of Spec A when the function field extension is separable.

theorem Lec6.normalization_is_normal {R : Type u_1} [CommRing R] [IsDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] {L : Type u_3} [Field L] [Algebra K L] [Algebra R L] [IsScalarTower R K L] [FiniteDimensional K L] : IsIntegrallyClosed ↥(integralClosure R L)

The normalization of R in a finite extension of its fraction field is integrally closed (Lec 6, normalization).

theorem Lec6.normalization_map_injective (B : Type u_1) [CommRing B] [IsDomain B] (K : Type u_2) [Field K] [Algebra B K] [IsFractionRing B K] (L : Type u_3) [Field L] [Algebra K L] [Algebra B L] [IsScalarTower B K L] : Function.Injective ⇑(algebraMap B ↥(integralClosure B L))

The structure map from a domain B into its integral closure in an extension L of its fraction field is injective.

theorem Lec6.normalization_finite_module {B : Type u_1} [CommRing B] [IsDomain B] [IsIntegrallyClosed B] [IsNoetherianRing B] (K : Type u_2) [Field K] [Algebra B K] [IsFractionRing B K] (L : Type u_3) [Field L] [Algebra K L] [Algebra B L] [IsScalarTower B K L] [FiniteDimensional K L] [Algebra.IsSeparable K L] : Module.Finite B ↥(integralClosure B L)

The integral closure of a noetherian, integrally closed domain B in a finite separable extension of its fraction field is a finite B-module.

theorem Lec6.integral_closure_noetherian {R : Type u_1} [CommRing R] [IsDomain R] [IsIntegrallyClosed R] [IsNoetherianRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (L : Type u_3) [Field L] [Algebra K L] [Algebra R L] [IsScalarTower R K L] [FiniteDimensional K L] [Algebra.IsSeparable K L] : IsNoetherianRing ↥(integralClosure R L)

The integral closure of a noetherian, integrally closed domain in a finite separable extension of its fraction field is noetherian.

theorem Lec6.normalization_finiteType (k : Type u_1) [Field k] {B : Type u_2} [CommRing B] [IsDomain B] [Algebra k B] [Algebra.FiniteType k B] [IsIntegrallyClosed B] [IsNoetherianRing B] (K : Type u_3) [Field K] [Algebra B K] [IsFractionRing B K] [Algebra k K] [IsScalarTower k B K] (L : Type u_4) [Field L] [Algebra K L] [Algebra B L] [IsScalarTower B K L] [Algebra k L] [IsScalarTower k B L] [FiniteDimensional K L] [Algebra.IsSeparable K L] : Algebra.FiniteType k ↥(integralClosure B L)

The integral closure of a finite-type k-algebra B (assumed nice) in a finite separable extension of its fraction field remains a finite-type k-algebra.

def Lec6.yoneda_fully_faithful (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] : CategoryTheory.coyoneda.FullyFaithful

Lec 6, Lem 14 (Yoneda): the coyoneda embedding is fully faithful.

theorem Lec6.yoneda_full (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] : CategoryTheory.coyoneda.Full

Lec 6, Lem 14 (Yoneda): the coyoneda embedding is full.

theorem Lec6.yoneda_faithful (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] : CategoryTheory.coyoneda.Faithful

Lec 6, Lem 14 (Yoneda): the coyoneda embedding is faithful.

def Lec6.yoneda_injective_on_objects (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] {x y : Cᵒᵖ} (h : CategoryTheory.coyoneda.obj x ≅ CategoryTheory.coyoneda.obj y) : x ≅ y

Lec 6, Lem 14 (Yoneda): the coyoneda embedding reflects isomorphisms, giving an iso x ≅ y from any iso of representable functors.

def Lec6.IsNormalVariety (R : Type u_1) [CommRing R] [IsDomain R] : Prop

An affine variety Spec R is normal if R is an integrally closed domain (Lec 6, normalization).

theorem Lec6.isNormalVariety_of_integrallyClosed (R : Type u_1) [CommRing R] [IsDomain R] [IsIntegrallyClosed R] : IsNormalVariety R

An integrally closed domain defines a normal affine variety.

theorem Lec6.normalization_isNormalVariety {R : Type u_1} [CommRing R] [IsDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] {L : Type u_3} [Field L] [Algebra K L] [Algebra R L] [IsScalarTower R K L] [FiniteDimensional K L] : IsNormalVariety ↥(integralClosure R L)

The normalization of a domain in a finite extension of its fraction field is a normal variety (Lec 6, normalization).

theorem Lec6.dedekindDomain_localRing_isRegularLocalRing (R : Type u_1) [CommRing R] [IsDomain R] [IsDedekindDomain R] [IsLocalRing R] (h : ¬IsField R) : IsRegularLocalRing R

A local Dedekind domain that is not a field is a regular local ring, i.e. a DVR (Lec 6, normalization in dimension one).

theorem Lec6.dedekindDomain_isIntegrallyClosed (R : Type u_1) [CommRing R] [IsDomain R] [IsDedekindDomain R] : IsIntegrallyClosed R

A Dedekind domain is integrally closed.

theorem Lec6.dedekindDomain_isNormalVariety (R : Type u_1) [CommRing R] [IsDomain R] [IsDedekindDomain R] : IsNormalVariety R

A Dedekind domain defines a normal affine variety.