noncomputable def canonicalModule (k : Type u) [Field k] (d : ℕ) : Submodule (MvPolynomial (Fin (d + 1)) k) (ExteriorAlgebra (MvPolynomial (Fin (d + 1)) k) Ω[MvPolynomial (Fin (d + 1)) k⁄k])

Definition 37 (Lecture 19). The canonical module ω = ∧^{d+1} Ω of the polynomial ring k[x_0,…,x_d], the algebraic model of the canonical bundle on affine (d+1)-space.

theorem canonicalModule_rank_one (k : Type u) [Field k] (d : ℕ) : Module.finrank (MvPolynomial (Fin (d + 1)) k) ↥(canonicalModule k d) = 1

The canonical module on affine (d+1)-space is free of rank one (the top exterior power of a rank d+1 module).

theorem canonicalModule_free (k : Type u) [Field k] (d : ℕ) : Module.Free (MvPolynomial (Fin (d + 1)) k) ↥(canonicalModule k d)

The canonical module on affine (d+1)-space is a free module.

noncomputable def canonicalSheafInPicard (n : ℕ) (k : Type u_1) [Field k] : PicardProjective.GradedPicardGroup (Fin (n + 1)) k

The canonical sheaf ω_{ℙⁿ} viewed as an element of the graded Picard group of ℙⁿ.

theorem euler_sequence_canonical_degree {k : Type u_1} [Field k] (n : ℕ) : (PicardProjective.gradedPicardGroup_equiv_int k n) (canonicalSheafInPicard n k) = -(↑n + 1)

The canonical bundle on ℙⁿ has degree -(n+1) in Pic(ℙⁿ) ≃ ℤ; this is the consequence of the Euler sequence taken with top exterior powers.

theorem canonical_projective_space_eq {k : Type u_1} [Field k] (n : ℕ) : canonicalSheafInPicard n k = PicardProjective.twistingSheafPow n k (-(↑n + 1))

The canonical sheaf on ℙⁿ is isomorphic to the twist O(-(n+1)).

theorem canonical_projective_space_degree {k : Type u_1} [Field k] (n : ℕ) : (PicardProjective.gradedPicardGroup_equiv_int k n) (canonicalSheafInPicard n k) = -(↑n + 1)

Restated degree formula: the image of K_{ℙⁿ} in Pic(ℙⁿ) ≃ ℤ is -(n+1).