noncomputable def Lec15PicardDivisorClass.picard_projective_iso_int (k : Type u_1) [Field k] (n : ℕ) : PicardProjective.GradedPicardGroup (Fin (n + 1)) k ≃+ ℤ

The graded Picard group of Pⁿ_k is canonically isomorphic to ℤ, generated by the twisting sheaf O(1).

theorem Lec15PicardDivisorClass.picard_projective_generated_by_O1 (k : Type u_1) [Field k] (n : ℕ) (L : PicardProjective.GradedPicardGroup (Fin (n + 1)) k) : ∃ (d : ℤ), L = PicardProjective.twistingSheafPow n k d

Every element of Pic(Pⁿ_k) is a power O(d) of the twisting sheaf for some integer d.

theorem Lec15PicardDivisorClass.degree_O1_eq_one (k : Type u_1) [Field k] (n : ℕ) : (PicardProjective.gradedPicardGroup_equiv_int k n) (PicardProjective.twistingSheaf n k) = 1

Under the isomorphism Pic(Pⁿ_k) ≃ ℤ, the twisting sheaf O(1) corresponds to 1.

theorem Lec15PicardDivisorClass.prop23a_powerSeries_ufd (k : Type u_1) [Field k] (n : ℕ) : UniqueFactorizationMonoid (MvPowerSeries (Fin n) k)

Proposition 23a (Lec 15): the power-series ring k[[x₁,…,xₙ]] over a field is a UFD.

theorem Lec15PicardDivisorClass.prop23b_noetherian_local_completion_ufd (A : Type u_1) [CommRing A] [IsDomain A] [IsLocalRing A] [IsNoetherianRing A] (hcompl : UniqueFactorizationMonoid (AdicCompletion (IsLocalRing.maximalIdeal A) A)) : UniqueFactorizationMonoid A

Proposition 23b (Lec 15): if the 𝔪-adic completion of a Noetherian local domain A is a UFD, then A itself is a UFD.

@[reducible, inline]

abbrev Lec15PicardDivisorClass.weilLinearlyEquivalent' {Y : Type u_1} (P : LinearEquivDivisors.PrincipalDivisors Y) (D₁ D₂ : DivisorsPicard.WeilDivisorGroup Y) : Prop

Linear equivalence of Weil divisors modulo a chosen subgroup P of principal divisors; abbreviation forwarding to WeilLinearlyEquivalent.

noncomputable def Lec15PicardDivisorClass.lD_dim (k : Type u_1) [Field k] (V : Type u_2) [AddCommGroup V] [Module k V] : ℕ

Dimension ℓ(D) of a linear system: the k-vector space dimension of V.

theorem Lec15PicardDivisorClass.lD_dim_nonneg (k : Type u_1) [Field k] (V : Type u_2) [AddCommGroup V] [Module k V] : 0 ≤ lD_dim k V

The dimension ℓ(D) is a non-negative natural number.

theorem Lec15PicardDivisorClass.lD_dim_eq_zero_iff (k : Type u_1) [Field k] (V : Type u_2) [AddCommGroup V] [Module k V] : lD_dim k V = 0 ↔ Module.finrank k V = 0

ℓ(D) = 0 iff the underlying k-vector space has dimension 0, i.e. is trivial.

@[reducible, inline]

abbrev Lec15PicardDivisorClass.divisorDegree {Y : Type u_1} (D : DivisorsPicard.WeilDivisorGroup Y) : ℤ

Degree of a Weil divisor; abbreviation for WeilDivisor.degree.

theorem Lec15PicardDivisorClass.divisorDegree_zero {Y : Type u_1} : divisorDegree 0 = 0

The zero divisor has degree zero.

theorem Lec15PicardDivisorClass.divisorDegree_add {Y : Type u_1} [DecidableEq Y] (D₁ D₂ : DivisorsPicard.WeilDivisorGroup Y) : divisorDegree (D₁ + D₂) = divisorDegree D₁ + divisorDegree D₂

Degree is additive on Weil divisors.

theorem Lec15PicardDivisorClass.divisorDegree_eq_of_linearlyEquiv {Y : Type u_1} [DecidableEq Y] {P : LinearEquivDivisors.PrincipalDivisors Y} (hP : ∀ D ∈ P, LinearEquivDivisors.WeilDivisor.degree D = 0) {D₁ D₂ : DivisorsPicard.WeilDivisorGroup Y} (h : LinearEquivDivisors.WeilLinearlyEquivalent P D₁ D₂) : divisorDegree D₁ = divisorDegree D₂

If every principal divisor in P has degree zero, then the degree is an invariant of the linear equivalence class of a divisor.

noncomputable def Lec15PicardDivisorClass.goal119_weilDivisor_sum_of_points (R : Type u_1) [CommRing R] [IsDomain R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] : Additive (FractionalIdeal (nonZeroDivisors R) K)ˣ →+ IsDedekindDomain.HeightOneSpectrum R →₀ ℤ

Smooth-curve case: the additive homomorphism from invertible fractional ideals to the free abelian group on height-one primes (i.e. closed points), realizing a fractional ideal as a finite formal sum of points.

noncomputable def Lec15PicardDivisorClass.goal119_weilDivisor_sum_of_points_equiv (R : Type u_1) [CommRing R] [IsDomain R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] : Additive (FractionalIdeal (nonZeroDivisors R) K)ˣ ≃+ (IsDedekindDomain.HeightOneSpectrum R →₀ ℤ)

Upgraded version of goal119_weilDivisor_sum_of_points: this map is in fact an isomorphism of abelian groups.

noncomputable def Lec15PicardDivisorClass.goal119_weilDivisorClassGroup_iso_pic (R : Type u_1) [CommRing R] [IsDomain R] [IsDedekindDomain R] : ClassGroup R ≃* CommRing.Pic R

For a Dedekind domain R, the divisor class group Cl(R) is isomorphic to the Picard group Pic(R) of invertible sheaves.

theorem Lec15PicardDivisorClass.bezout_thm (k : Type u_1) [Field k] (g : Polynomial (Polynomial k)) (hg_monic : g.Monic) (hg_irred : Irreducible g) (f : Polynomial k) (hf : (algebraMap (Polynomial k) (AdjoinRoot g)) f ≠ 0) : Module.finrank k (AdjoinRoot g ⧸ Ideal.span {(algebraMap (Polynomial k) (AdjoinRoot g)) f}) = g.natDegree * f.natDegree

Bézout's theorem (algebraic form): for g monic irreducible in k[x][y] defining a smooth curve and f ∈ k[x] non-vanishing on its image, the intersection multiplicity is deg(g) · deg(f).

theorem Lec15PicardDivisorClass.line_line_bezout (k : Type u_1) [Field k] : Module.finrank k (Polynomial (Polynomial k) ⧸ Ideal.span {Polynomial.C Polynomial.X, Polynomial.X}) = 1

Simplest case of Bézout: two distinct lines x = 0 and y = 0 in 𝔸²_k meet in a single reduced point.

theorem Lec15PicardDivisorClass.prop24_principal_divisor_degree_zero (k : Type u_1) [Field k] {S : Type u_2} [CommRing S] [IsDomain S] {ι : Type u_3} [Fintype ι] [Algebra (Polynomial k) S] [Algebra k S] [IsScalarTower k (Polynomial k) S] [IsCompleteCurveAlg k S] (b : Module.Basis ι (Polynomial k) S) {p q : Polynomial k} (hp : p ≠ 0) (hq : q ≠ 0) (hpd : IsPrincipalDivisorData k S p q) : PrincipalDivisorDegZero.principalDivisorDeg k S (Ideal.span {(algebraMap (Polynomial k) S) p}) (Ideal.span {(algebraMap (Polynomial k) S) q}) = 0

Proposition 24 (Lec 15): on a complete curve, every principal divisor has degree zero.

theorem Lec15PicardDivisorClass.prop24_degree_formula (k : Type u_1) [Field k] {S : Type u_2} [CommRing S] [IsDomain S] {ι : Type u_3} [Fintype ι] [Algebra (Polynomial k) S] [Algebra k S] [IsScalarTower k (Polynomial k) S] (b : Module.Basis ι (Polynomial k) S) {p q : Polynomial k} (hp : p ≠ 0) (hq : q ≠ 0) : PrincipalDivisorDegZero.principalDivisorDeg k S (Ideal.span {(algebraMap (Polynomial k) S) p}) (Ideal.span {(algebraMap (Polynomial k) S) q}) = ↑(Fintype.card ι) * (↑p.natDegree - ↑q.natDegree)

Degree formula (Prop 24): deg ((p) - (q)) = card(ι) · (deg p - deg q) where card(ι) is the rank of the algebra S over k[x].

theorem Lec15PicardDivisorClass.prop24_affine_formula (k : Type u_1) [Field k] (p q : Polynomial k) : PrincipalDivisorDegZero.principalDivisorDeg k (Polynomial k) (Ideal.span {p}) (Ideal.span {q}) = ↑p.natDegree - ↑q.natDegree

Specialization of the degree formula to the affine line: deg ((p) - (q)) = deg(p) - deg(q) for p, q ∈ k[x].

theorem Lec15PicardDivisorClass.prop25_principal_divisor_degree_zero_curve (k : Type u_1) [Field k] {S : Type u_2} [CommRing S] [IsDomain S] {ι : Type u_3} [Fintype ι] [Algebra (Polynomial k) S] [Algebra k S] [IsScalarTower k (Polynomial k) S] (b : Module.Basis ι (Polynomial k) S) (c₁ c₂ : k) : PrincipalDivisorDegZero.principalDivisorDeg k S (Ideal.span {(algebraMap (Polynomial k) S) (Polynomial.X - Polynomial.C c₁)}) (Ideal.span {(algebraMap (Polynomial k) S) (Polynomial.X - Polynomial.C c₂)}) = 0

Proposition 25 (Lec 15): principal divisor associated to a translation (x - c₁) - (x - c₂) has degree zero on a curve covered by k[x].

theorem Lec15PicardDivisorClass.prop25_fiber_dimension_eq_rank (k : Type u_1) [Field k] {S : Type u_2} [CommRing S] [IsDomain S] {ι : Type u_3} [Fintype ι] [Algebra (Polynomial k) S] [Algebra k S] [IsScalarTower k (Polynomial k) S] (b : Module.Basis ι (Polynomial k) S) (c : k) : Module.finrank k (S ⧸ Ideal.span {(algebraMap (Polynomial k) S) (Polynomial.X - Polynomial.C c)}) = Fintype.card ι

The fiber S / (x - c) over a closed point x = c has k-dimension equal to the rank of S as a k[x]-module.

theorem Lec15PicardDivisorClass.prop25_fiber_dimension_constant (k : Type u_1) [Field k] {S : Type u_2} [CommRing S] [IsDomain S] {ι : Type u_3} [Fintype ι] [Algebra (Polynomial k) S] [Algebra k S] [IsScalarTower k (Polynomial k) S] (b : Module.Basis ι (Polynomial k) S) (c₁ c₂ : k) : Module.finrank k (S ⧸ Ideal.span {(algebraMap (Polynomial k) S) (Polynomial.X - Polynomial.C c₁)}) = Module.finrank k (S ⧸ Ideal.span {(algebraMap (Polynomial k) S) (Polynomial.X - Polynomial.C c₂)})

Constancy of the fiber dimension: dim_k(S / (x - c)) is independent of c.

theorem Lec15PicardDivisorClass.prop25_fiber_dimension_formula (k : Type u_1) [Field k] {S : Type u_2} [CommRing S] [IsDomain S] {ι : Type u_3} [Fintype ι] [Algebra (Polynomial k) S] [Algebra k S] [IsScalarTower k (Polynomial k) S] (b : Module.Basis ι (Polynomial k) S) {p : Polynomial k} (hp : p ≠ 0) : Module.finrank k (S ⧸ Ideal.span {(algebraMap (Polynomial k) S) p}) = Fintype.card ι * p.natDegree

General fiber dimension formula: dim_k(S / (p)) = rank(S over k[x]) · deg(p).

noncomputable def Lec15PicardDivisorClass.completeLinearSystemProjDim (k : Type u_1) [Field k] (V : Type u_2) [AddCommGroup V] [Module k V] : ℤ

Projective dimension dim |D| = ℓ(D) - 1 of the complete linear system associated to a divisor D, as an integer (so that dim ∅ = -1).

theorem Lec15PicardDivisorClass.completeLinearSystemProjDim_eq (k : Type u_1) [Field k] (V : Type u_2) [AddCommGroup V] [Module k V] : completeLinearSystemProjDim k V = ↑(lD_dim k V) - 1

Unfolding: dim |D| = ℓ(D) - 1 in the integer expression.

theorem Lec15PicardDivisorClass.completeLinearSystem_nonempty_of_lD_pos (k : Type u_1) [Field k] (V : Type u_2) [AddCommGroup V] [Module k V] (h : 1 ≤ lD_dim k V) : completeLinearSystemProjDim k V ≥ 0

If ℓ(D) ≥ 1 then the complete linear system |D| has non-negative projective dimension (in particular is non-empty as a projective space).