| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Cohomology.LaurentPlane |
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| namespace OAI |
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| section |
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| namespace MaximalSeshadri.LaurentPlane |
| noncomputable section |
| variable {K A R : Type*} [CommRing K] [CommRing A] [CommRing R] |
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| lemma map_unit_zpow (f : A →+* R) (a : Aˣ) (n : ℤ) : |
| f (a ^ n : Aˣ) = ((Units.map f a) ^ n : Rˣ) := by |
| change (((Units.map f.toMonoidHom) (a ^ n) : Rˣ) : R) = _ |
| exact congrArg Units.val (map_zpow (Units.map f.toMonoidHom) a n) |
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| lemma lift_cone_left (k : K →+* R) (u v : Rˣ) (f : A →+* R) |
| (x : Aˣ) (y : A) (hx : Units.map f x = u) (hy : f y = v) |
| (z : ℤ × ℤ) (hz : 0 ≤ z.2) : |
| f (↑(x ^ z.1) * y ^ z.2.toNat) = eval k u v (T z) := by |
| rw [map_mul, map_pow, map_unit_zpow, hx, hy, eval_T, Units.val_mul] |
| have he : z.2 = (z.2.toNat : ℤ) := (Int.toNat_of_nonneg hz).symm |
| rw [he, zpow_natCast, Units.val_pow_eq_pow_val] |
| simp [max_eq_left hz] |
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| lemma lift_cone_right (k : K →+* R) (u v : Rˣ) (f : A →+* R) |
| (x : A) (y : Aˣ) (hx : f x = u) (hy : Units.map f y = v) |
| (z : ℤ × ℤ) (hz : 0 ≤ z.1) : |
| f (x ^ z.1.toNat * ↑(y ^ z.2)) = eval k u v (T z) := by |
| rw [map_mul, map_pow, map_unit_zpow, hx, hy, eval_T, Units.val_mul] |
| have he : z.1 = (z.1.toNat : ℤ) := (Int.toNat_of_nonneg hz).symm |
| rw [he, zpow_natCast, Units.val_pow_eq_pow_val] |
| simp [max_eq_left hz] |
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| lemma lift_cone_top (k : K →+* R) (u v : Rˣ) (f : A →+* R) |
| (x : Aˣ) (y : A) (hx : Units.map f x = u * v⁻¹) (hy : f y = ↑(v⁻¹)) |
| (z : ℤ × ℤ) (hz : z.1 + z.2 ≤ 0) : |
| f (↑(x ^ z.1) * y ^ (-(z.1 + z.2)).toNat) = eval k u v (T z) := by |
| rw [map_mul, map_pow, map_unit_zpow, hx, hy, eval_T] |
| have hn : ((-(z.1 + z.2)).toNat : ℤ) = -(z.1 + z.2) := |
| Int.toNat_of_nonneg (by omega) |
| rw [← Units.val_pow_eq_pow_val, ← zpow_natCast, hn, ← Units.val_mul] |
| congr 1 |
| rw [mul_zpow, mul_assoc, ← zpow_add] |
| have he : z.1 + -(z.1 + z.2) = -z.2 := by omega |
| rw [he] |
| simp |
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| theorem finite_laurent_of_chart {P N M : Type*} [CommRing P] |
| [AddCommGroup N] [AddCommGroup M] [Module P N] [Module R M] |
| [Module.Finite P N] (k : K →+* R) (u v : Rˣ) |
| (e : P →+* Ring K) (F : N →+ M) |
| (hF : ∀ p n, F (p • n) = eval k u v (e p) • F n) |
| (hclear : ∀ m : M, ∃ d : ℕ, ∃ n : N, (↑u * ↑v : R)^d • m = F n) : |
| letI := Module.compHom M (eval k u v) |
| Module.Finite (Ring K) M := by |
| let := Module.compHom M (eval k u v) |
| apply finite_of_clearing e F hF |
| intro m |
| obtain ⟨d,n,hn⟩ := hclear m |
| refine ⟨T (-(d : ℤ),-(d : ℤ)),n,?_⟩ |
| change m = eval k u v (T (-(d : ℤ),-(d : ℤ))) • F n |
| rw [eval_T, ← hn, ← mul_smul] |
| have he : ↑(u ^ (-(d : ℤ)) * v ^ (-(d : ℤ))) * (↑u * ↑v : R)^d = 1 := by |
| rw [← Units.val_mul, ← Units.val_pow_eq_pow_val, ← Units.val_mul] |
| suffices u ^ (-(d : ℤ)) * v ^ (-(d : ℤ)) * (u * v)^d = 1 by |
| exact congrArg Units.val this |
| rw [← mul_zpow, zpow_neg, zpow_natCast] |
| exact inv_mul_cancel _ |
| rw [he, one_smul] |
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| end |
| end MaximalSeshadri.LaurentPlane |
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| end |
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| end OAI |
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