| import Mathlib.RingTheory.Ideal.Span |
| import Mathlib.Tactic.LinearCombination |
| import Mathlib.Tactic.Ring |
|
|
| namespace OAI |
|
|
| /-! |
| |
|
|
| This module gives polynomial identities, valid in every commutative ring, |
| for the principalization in source Section 4, Lemma `bun:principalization`. |
| Unlike the source proof, the argument uses an explicit global Bezout identity |
| and requires neither localization nor prime-spectrum gluing. |
| -/ |
|
|
| namespace StableCoordinate.Bundle |
|
|
| variable {R : Type*} [CommRing R] |
|
|
| /-- Pullback of the source element `s_B` under `x = ab`, `y = dc`. -/ |
| def s (a d b c u : R) : R := d * c - (a * b) * (a * b + u ^ 2) |
|
|
| /-- First generator of the pulled-back distinguished ideal. -/ |
| def f₀ (a d b c u : R) : R := a * b - s a d b c u ^ 2 + u ^ 2 |
|
|
| /-- Second generator of the pulled-back distinguished ideal. -/ |
| def g₀ (a d b c u : R) : R := d * b - s a d b c u * (a * b) |
|
|
| /-- The global generator of the pulled-back ideal. -/ |
| def v (a d b u : R) : R := a ^ 3 * b + a ^ 2 * u ^ 2 - d ^ 2 |
|
|
| /-- First coefficient in the global Bezout relation. -/ |
| def coeffA (a d b : R) : R := a ^ 2 * (1 - 2 * b * d) |
|
|
| /-- Second coefficient in the global Bezout relation. -/ |
| def coeffB (a d b u : R) : R := |
| coeffA a d b * (a * b + u ^ 2) + d ^ 2 * (3 + 2 * b * d) |
|
|
| theorem g₀_factor (a d b c u : R) (h : a * c - b * d = 1) : |
| g₀ a d b c u = b ^ 2 * v a d b u := by |
| unfold g₀ s v |
| linear_combination -(b * d) * h |
|
|
| theorem f₀_factor (a d b c u : R) (h : a * c - b * d = 1) : |
| f₀ a d b c u = (c ^ 2 - b ^ 2 * (a * b + u ^ 2)) * v a d b u := by |
| unfold f₀ s v |
| linear_combination -((a * b + u ^ 2) * (a * c - b * d + 1)) * h |
|
|
| theorem coefficients_bezout (a d b c u : R) (h : a * c - b * d = 1) : |
| coeffA a d b * (c ^ 2 - b ^ 2 * (a * b + u ^ 2)) + |
| coeffB a d b u * b ^ 2 = 1 := by |
| unfold coeffB coeffA |
| linear_combination (1 - 2 * b * d) * (a * c + b * d + 1) * h |
|
|
| theorem generator_bezout (a d b c u : R) (h : a * c - b * d = 1) : |
| coeffA a d b * f₀ a d b c u + coeffB a d b u * g₀ a d b c u = |
| v a d b u := by |
| rw [f₀_factor a d b c u h, g₀_factor a d b c u h] |
| rw [← mul_assoc, ← mul_assoc, ← add_mul, coefficients_bezout a d b c u h, one_mul] |
|
|
| /-- The exact global principalization claimed in Section 4. -/ |
| theorem principalization (a d b c u : R) (h : a * c - b * d = 1) : |
| Ideal.span ({f₀ a d b c u, g₀ a d b c u} : Set R) = |
| Ideal.span ({v a d b u} : Set R) := by |
| apply le_antisymm |
| · apply Ideal.span_le.2 |
| intro z hz |
| rcases Set.mem_insert_iff.mp hz with rfl | hz |
| · apply Ideal.mem_span_singleton.2 |
| exact ⟨c ^ 2 - b ^ 2 * (a * b + u ^ 2), by |
| rw [f₀_factor a d b c u h, mul_comm]⟩ |
| · obtain rfl := Set.mem_singleton_iff.mp hz |
| apply Ideal.mem_span_singleton.2 |
| exact ⟨b ^ 2, by rw [g₀_factor a d b c u h, mul_comm]⟩ |
| · apply Ideal.span_le.2 |
| intro z hz |
| obtain rfl := Set.mem_singleton_iff.mp hz |
| exact Ideal.mem_span_pair.2 |
| ⟨coeffA a d b, coeffB a d b u, generator_bezout a d b c u h⟩ |
| |
| /-- Membership in the pulled-back ideal gives an actual factorization by `v`. -/ |
| theorem exists_factor_of_mem (a d b c u : R) (h : a * c - b * d = 1) |
| {r : R} (hr : r ∈ Ideal.span ({f₀ a d b c u, g₀ a d b c u} : Set R)) : |
| ∃ q : R, r = v a d b u * q := by |
| rw [principalization a d b c u h, Ideal.mem_span_singleton] at hr |
| rcases hr with ⟨q, hq⟩ |
| exact ⟨q, hq⟩ |
| |
| end StableCoordinate.Bundle |
| |
| end OAI |
| |
