@[reducible, inline]

abbrev EulerSequence.GrLinMap (k : Type u) [Field k] (n : ℕ) (M N : QCohProjective.GradedModuleData k n) (d : ℤ) : Type u

A k-linear map between the degree-d components of two graded module data over ℙⁿ.

@[reducible, inline]

abbrev EulerSequence.GrLinEquiv (k : Type u) [Field k] (n : ℕ) (M N : QCohProjective.GradedModuleData k n) (d : ℤ) : Type u

A k-linear equivalence between the degree-d components of two graded module data.

structure EulerSequence.GradedSES (k : Type u) [Field k] (n : ℕ) : Type (u + 1)

A short exact sequence 0 → L → M → R → 0 of graded module data, with degree-wise injectivity, surjectivity, and exactness at the middle.

• left : QCohProjective.GradedModuleData k n
• middle : QCohProjective.GradedModuleData k n
• right : QCohProjective.GradedModuleData k n
• f (d : ℤ) : GrLinMap k n self.left self.middle d
• g (d : ℤ) : GrLinMap k n self.middle self.right d
• f_injective (d : ℤ) : Function.Injective ⇑(self.f d)
• g_surjective (d : ℤ) : Function.Surjective ⇑(self.g d)
• exact_middle (d : ℤ) (x : self.middle.component d) : (self.g d) x = Zero.zero ↔ ∃ (a : self.left.component d), (self.f d) a = x

noncomputable def EulerSequence.directSumTwist (k : Type u) [Field k] (n : ℕ) : QCohProjective.GradedModuleData k n

The graded module O(-1)^{n+1} on ℙⁿ, formed as a (n+1)-fold direct sum of the Serre twist O(-1); this is the middle term of the Euler sequence.

instance EulerSequence.instNonemptyGradedModuleData (k : Type u) [Field k] (n : ℕ) : Nonempty (QCohProjective.GradedModuleData k n)

The type of graded module data on ℙⁿ is inhabited (witnessed by the structure sheaf).

noncomputable def EulerSequence.cotangentSheafGM (k : Type u) [Field k] (n : ℕ) : QCohProjective.GradedModuleData k n

The cotangent sheaf Ω_{ℙⁿ/k} as a graded module datum; sits as the left term in the Euler sequence 0 → Ω_{ℙⁿ} → O(-1)^{n+1} → O → 0.

noncomputable def EulerSequence.canonicalBundleGM (k : Type u) [Field k] (n : ℕ) : QCohProjective.GradedModuleData k n

The canonical bundle K_{ℙⁿ} as a graded module datum, equal to the Serre twist O(-(n+1)).

theorem EulerSequence.eulerSES (k : Type u) [Field k] (n : ℕ) : ∃ (S : GradedSES k n), S.left = cotangentSheafGM k n ∧ S.middle = directSumTwist k n ∧ S.right = TwistSheaf.structureSheafData k n

Proposition 36 (Lecture 20, Euler sequence). There exists a short exact sequence of graded modules 0 → Ω_{ℙⁿ} → O(-1)^{n+1} → O → 0.

theorem EulerSequence.canonical_bundle_eq_twist (k : Type u) [Field k] (n : ℕ) (d : ℤ) : Nonempty (GrLinEquiv k n (canonicalBundleGM k n) (TwistSheaf.serreTwist k n (-(↑n + 1))) d)

The canonical bundle on ℙⁿ agrees degreewise with the Serre twist O(-(n+1)).