@[reducible, inline]

abbrev Lec14DivisorsPicard.cartierDivisorGroupScheme (X : AlgebraicGeometry.Scheme) [AlgebraicGeometry.IsIntegral X] : Type (u + 1)

Cartier divisor group DC(X) of an integral scheme X (Def 30, Lec 14): abbreviation for CartierDivisorScheme.CartierDivisorGroupScheme X.

@[reducible, inline]

abbrev Lec14DivisorsPicard.cartierDivisorGroupAlg (A : Type u_1) [CommRing A] [IsDedekindDomain A] (K : Type u_2) [Field K] [Algebra A K] [IsFractionRing A K] : Type u_2

Algebraic form of the Cartier divisor group attached to a Dedekind domain A with fraction field K.

@[reducible]

noncomputable def Lec14DivisorsPicard.corollary_19_invertible_sheaves_group (R : Type u_1) [CommRing R] : CommGroup (CommRing.Pic R)

Corollary 19 (Lec 14): the Picard group Pic(R) of invertible sheaves on Spec R is naturally a commutative group under tensor product.

def Lec14DivisorsPicard.weilDivisorDegree {Y : Type u_1} [DecidableEq Y] (D : WeilDivisor.Group Y) : ℤ

Degree of a Weil divisor D = Σ nᵢ [Pᵢ]: the integer sum of the coefficients Σ nᵢ.

theorem Lec14DivisorsPicard.weilDivisorDegree_zero {Y : Type u_1} [DecidableEq Y] : weilDivisorDegree 0 = 0

The degree of the zero divisor is 0.

theorem Lec14DivisorsPicard.weilDivisorDegree_add {Y : Type u_1} [DecidableEq Y] (D₁ D₂ : WeilDivisor.Group Y) : weilDivisorDegree (D₁ + D₂) = weilDivisorDegree D₁ + weilDivisorDegree D₂

The degree map is additive: deg(D₁ + D₂) = deg(D₁) + deg(D₂).

def Lec14DivisorsPicard.degreeHom {Y : Type u_1} [DecidableEq Y] : WeilDivisor.Group Y →+ ℤ

Packaged AddMonoidHom version of weilDivisorDegree.

theorem Lec14DivisorsPicard.degree_eq_of_linearlyEquiv {Y : Type u_1} [DecidableEq Y] (D₁ D₂ : WeilDivisor.Group Y) (IsPrincipal : WeilDivisor.Group Y → Prop) (h_princ_deg : ∀ (P : WeilDivisor.Group Y), IsPrincipal P → weilDivisorDegree P = 0) (h_equiv : IsPrincipal (D₁ - D₂)) : weilDivisorDegree D₁ = weilDivisorDegree D₂

Degree is invariant under linear equivalence: assuming that principal divisors have degree zero, linearly equivalent divisors D₁ ∼ D₂ satisfy deg D₁ = deg D₂.