File size: 3,508 Bytes
82da87f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | import Mathlib.RingTheory.Ideal.Span
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
namespace OAI
/-!
# Explicit global principalization in the quadric bundle
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
|