def Lec24.h0_P1 (d : ℤ) : ℕ

The dimension h⁰(ℙ¹, O(d)) = max(d + 1, 0).

def Lec24.h1_P1 (d : ℤ) : ℕ

The dimension h¹(ℙ¹, O(d)) = max(-d - 1, 0), dual to h⁰(ℙ¹, O(-d - 2)).

theorem Lec24.euler_char_O_d_P1 (d : ℤ) : ↑(h0_P1 d) - ↑(h1_P1 d) = d + 1

Euler characteristic of O(d) on ℙ¹: χ(O(d)) = h⁰ - h¹ = d + 1.

structure Lec24.LocallyFreeSheafOnP1 : Type

Axiomatic data for a locally free sheaf on ℙ¹_k: its rank and degree together with the cohomology dimensions of its Serre twists, satisfying the expected Euler-characteristic formula.

• rank : ℕ
• degree : ℤ
• h0_twist : ℤ → ℕ
• h1_twist : ℤ → ℕ
• euler_char_twist (d : ℤ) : ↑(self.h0_twist d) - ↑(self.h1_twist d) = self.degree + ↑self.rank * d + ↑self.rank

def Lec24.splitBundle (n : ℕ) (degrees : Fin n → ℤ) : LocallyFreeSheafOnP1

The split bundle ⨁ᵢ O_{ℙ¹}(dᵢ) on ℙ¹, packaged as a LocallyFreeSheafOnP1.

theorem Lec24.grothendieck_birkhoff (E : LocallyFreeSheafOnP1) : ∃ (degrees : Fin E.rank → ℤ), ∑ i : Fin E.rank, degrees i = E.degree ∧ (∀ (d : ℤ), E.h0_twist d = (splitBundle E.rank degrees).h0_twist d) ∧ ∀ (d : ℤ), E.h1_twist d = (splitBundle E.rank degrees).h1_twist d

Cohomological consequence of Grothendieck-Birkhoff: every locally free sheaf on ℙ¹ matches the cohomology data of a unique split bundle with the right rank and degree.

structure Lec24.LocallyFreeSheafData : Type

Numerical data attached to a locally free sheaf on a curve: rank, degree, and the zeroth/first cohomology dimensions.

• rk : ℕ
• deg : ℤ
• h0 : ℕ
• h1 : ℕ

structure Lec24.SmoothCompleteCurveWithSheaves : Type 1

Axiomatic data of a smooth complete curve together with a category of coherent sheaves, their Serre duals, the structure sheaf, the canonical sheaf and the numerical Riemann-Roch and Serre duality relations they satisfy.

• genus : ℕ
• degK : ℤ
• Sheaf : Type
• data : self.Sheaf → LocallyFreeSheafData
• serreDual : self.Sheaf → self.Sheaf
• structureSheaf : self.Sheaf
• canonicalSheaf : self.Sheaf
• structureSheaf_rk : (self.data self.structureSheaf).rk = 1
• structureSheaf_deg : (self.data self.structureSheaf).deg = 0
• structureSheaf_h0 : (self.data self.structureSheaf).h0 = 1
• structureSheaf_h1 : (self.data self.structureSheaf).h1 = self.genus
• canonicalSheaf_rk : (self.data self.canonicalSheaf).rk = 1
• canonicalSheaf_deg : (self.data self.canonicalSheaf).deg = self.degK
• serreDual_rk (E : self.Sheaf) : (self.data (self.serreDual E)).rk = (self.data E).rk
• serreDual_deg (E : self.Sheaf) : (self.data (self.serreDual E)).deg = ↑(self.data E).rk * self.degK - (self.data E).deg
• serreDual_structure : self.data (self.serreDual self.structureSheaf) = self.data self.canonicalSheaf
• serreDual_canonical : self.data (self.serreDual self.canonicalSheaf) = self.data self.structureSheaf
• serreDual_involutive (E : self.Sheaf) : self.data (self.serreDual (self.serreDual E)) = self.data E
• chi_determined_by_rk_deg (E : self.Sheaf) : ↑(self.data E).h0 - ↑(self.data E).h1 = ↑(self.data E).rk * (↑(self.data self.structureSheaf).h0 - ↑(self.data self.structureSheaf).h1) + (self.data E).deg
• serre_duality (E : self.Sheaf) : (self.data E).h0 = (self.data (self.serreDual E)).h1

def Lec24.SmoothCompleteCurveWithSheaves.chi (X : SmoothCompleteCurveWithSheaves) (E : X.Sheaf) : ℤ

The Euler characteristic χ(E) = h⁰(E) - h¹(E) of a coherent sheaf on the curve.

theorem Lec24.SmoothCompleteCurveWithSheaves.riemann_roch (X : SmoothCompleteCurveWithSheaves) (E : X.Sheaf) : ↑(X.data E).h0 - ↑(X.data E).h1 = (X.data E).deg + ↑(X.data E).rk * (1 - ↑X.genus)

Riemann-Roch (Theorem 24.2): For a coherent sheaf E on a smooth complete curve of genus g, χ(E) = deg(E) + rank(E) · (1 - g).

theorem Lec24.SmoothCompleteCurveWithSheaves.chi_eq (X : SmoothCompleteCurveWithSheaves) (E : X.Sheaf) : X.chi E = (X.data E).deg + ↑(X.data E).rk * (1 - ↑X.genus)

Restatement of Riemann-Roch in terms of chi: χ(E) = deg E + rk E · (1 - g).

theorem Lec24.SmoothCompleteCurveWithSheaves.chi_structure (X : SmoothCompleteCurveWithSheaves) : X.chi X.structureSheaf = 1 - ↑X.genus

χ(O_X) = 1 - g for the structure sheaf of a smooth complete curve of genus g.

theorem Lec24.serre_duality_dimension (X : SmoothCompleteCurveWithSheaves) (E : X.Sheaf) : (X.data E).h0 = (X.data (X.serreDual E)).h1

Serre duality (Theorem 24.3): h⁰(E) = h¹(E^∨ ⊗ ω_X), here packaged via the Serre dual.

theorem Lec24.serre_duality_reverse (X : SmoothCompleteCurveWithSheaves) (E : X.Sheaf) : (X.data E).h1 = (X.data (X.serreDual E)).h0

Serre duality in reverse: h¹(E) = h⁰(E^∨ ⊗ ω_X).

theorem Lec24.SmoothCompleteCurveWithSheaves.arithmetic_eq_geometric_genus (X : SmoothCompleteCurveWithSheaves) : (X.data X.canonicalSheaf).h0 = X.genus

Arithmetic genus equals geometric genus: h⁰(ω_X) = g.

theorem Lec24.SmoothCompleteCurveWithSheaves.h1_canonical_eq_one (X : SmoothCompleteCurveWithSheaves) : (X.data X.canonicalSheaf).h1 = 1

h¹(ω_X) = 1, the Serre dual of h⁰(O_X) = 1.

theorem Lec24.SmoothCompleteCurveWithSheaves.deg_canonical_eq (X : SmoothCompleteCurveWithSheaves) : X.degK = 2 * ↑X.genus - 2

The degree of the canonical sheaf: deg(ω_X) = 2g - 2.

theorem Lec24.SmoothCompleteCurveWithSheaves.riemann_form (X : SmoothCompleteCurveWithSheaves) (E : X.Sheaf) : ↑(X.data E).h0 - ↑(X.data (X.serreDual E)).h0 = (X.data E).deg + ↑(X.data E).rk * (1 - ↑X.genus)

Riemann form of Riemann-Roch: h⁰(E) - h⁰(E^∨ ⊗ ω_X) = deg E + rk E · (1 - g), combining Riemann-Roch with Serre duality.

theorem Lec24.SmoothCompleteCurveWithSheaves.h0_high_degree (X : SmoothCompleteCurveWithSheaves) (E : X.Sheaf) (hrk : (X.data E).rk = 1) (hvanish : (X.data (X.serreDual E)).h0 = 0) : ↑(X.data E).h0 = (X.data E).deg + 1 - ↑X.genus

High-degree case: for a line bundle L with h⁰(L^∨ ⊗ ω_X) = 0, Riemann-Roch reduces to h⁰(L) = deg L + 1 - g.

structure Lec24.SmoothCurveK0Data : Type 1

Axiomatic data for computing K_0(Coh X) on a smooth curve: a class of sheaves with rank and degree functions, short exact sequences, plus designated structure sheaf and skyscraper-at-a-point with the expected rank/degree values.

• CohSheaf : Type
• rank : self.CohSheaf → ℕ
• deg : self.CohSheaf → ℤ
• SES : self.CohSheaf → self.CohSheaf → self.CohSheaf → Prop
• O_X : self.CohSheaf
• O_x : self.CohSheaf
• O_X_rank : self.rank self.O_X = 1
• O_X_deg : self.deg self.O_X = 0
• O_x_rank : self.rank self.O_x = 0
• O_x_deg : self.deg self.O_x = 1
• rank_additive (A B C : self.CohSheaf) : self.SES A B C → self.rank B = self.rank A + self.rank C
• deg_additive (A B C : self.CohSheaf) : self.SES A B C → self.deg B = self.deg A + self.deg C

def Lec24.sesRelations (X : SmoothCurveK0Data) : AddSubgroup (FreeAbelianGroup X.CohSheaf)

The subgroup of ℤ[CohSheaf] generated by the relations [A] + [C] - [B] coming from short exact sequences 0 → A → B → C → 0. Quotienting yields K_0(Coh X).

@[reducible, inline]

abbrev Lec24.K0Coh (X : SmoothCurveK0Data) : Type

The Grothendieck group K_0(Coh X): the free abelian group on coherent sheaves modulo the short-exact-sequence relations.

def Lec24.K0Coh.classOf (X : SmoothCurveK0Data) (F : X.CohSheaf) : K0Coh X

The class [F] of a coherent sheaf F in K_0(Coh X).

def Lec24.rankDegToK0 (X : SmoothCurveK0Data) : ℤ × ℤ →+ K0Coh X

The group homomorphism ℤ ⊕ ℤ → K_0(Coh X) sending (r, d) ↦ r·[O_X] + d·[O_x], which is shown by Lemma 35 to be surjective.

theorem Lec24.classOf_in_range (X : SmoothCurveK0Data) (torsion_decomp : ∀ (F : X.CohSheaf), X.rank F = 0 → ∃ (n : ℤ), K0Coh.classOf X F = n • K0Coh.classOf X X.O_x) (rank_reduction : ∀ (F : X.CohSheaf), 0 < X.rank F → ∃ (d : ℤ) (Q : X.CohSheaf), X.rank Q < X.rank F ∧ K0Coh.classOf X F = K0Coh.classOf X X.O_X + d • K0Coh.classOf X X.O_x + K0Coh.classOf X Q) (F : X.CohSheaf) : K0Coh.classOf X F ∈ (rankDegToK0 X).range

Strong-induction helper: assuming torsion sheaves decompose as multiples of [O_x] and rank can always be reduced via an O_X-direct-summand, every class [F] lies in the image of (r, d) ↦ r[O_X] + d[O_x].

theorem Lec24.K0_generators_lemma35 (X : SmoothCurveK0Data) (torsion_decomp : ∀ (F : X.CohSheaf), X.rank F = 0 → ∃ (n : ℤ), K0Coh.classOf X F = n • K0Coh.classOf X X.O_x) (rank_reduction : ∀ (F : X.CohSheaf), 0 < X.rank F → ∃ (d : ℤ) (Q : X.CohSheaf), X.rank Q < X.rank F ∧ K0Coh.classOf X F = K0Coh.classOf X X.O_X + d • K0Coh.classOf X X.O_x + K0Coh.classOf X Q) : Function.Surjective ⇑(rankDegToK0 X)

Lemma 35 (generators of K_0): On a smooth curve, the classes [O_X] and [O_x] generate K_0(Coh X), i.e. the rank-degree map is surjective.

theorem Lec24.goal_179_lemma35 (X : SmoothCurveK0Data) : Function.Surjective ⇑(rankDegToK0 X)

Lemma 35 (final form): The rank-degree map ℤ × ℤ → K_0(Coh X) is surjective for any smooth curve X.

theorem Lec24.goal_185 (X : SmoothCompleteCurveWithSheaves) : X.degK = 2 * ↑X.genus - 2

Goal 185 (degree of canonical sheaf): deg(ω_X) = 2g - 2, a corollary of Riemann-Roch applied to ω_X and Serre duality.

def Lec24.SmoothCompleteCurveWithSheaves.arithmeticGenus_pa (X : SmoothCompleteCurveWithSheaves) : ℤ

The arithmetic genus p_a := 1 - χ(O_X).

def Lec24.SmoothCompleteCurveWithSheaves.geometricGenus_pg (X : SmoothCompleteCurveWithSheaves) : ℕ

The geometric genus p_g := h⁰(ω_X).

theorem Lec24.SmoothCompleteCurveWithSheaves.pa_eq_genus (X : SmoothCompleteCurveWithSheaves) : X.arithmeticGenus_pa = ↑X.genus

The arithmetic genus agrees with g: p_a = g.

theorem Lec24.SmoothCompleteCurveWithSheaves.pg_eq_genus (X : SmoothCompleteCurveWithSheaves) : X.geometricGenus_pg = X.genus

The geometric genus agrees with g: p_g = g.