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21cfc82 | 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 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 | import Mathlib.Algebra.MvPolynomial.Nilpotent
import Mathlib.Tactic
namespace OAI
/-!
# Uniqueness of normalized polynomial lifts of determinant frames
The bundle in the source is described by `a*c - b*d = 1`, with base
coordinates `a*b`, `d*c`, and `d*b`. This file proves directly in rings
that two frames over the same base differ by a unit. Consequently two
polynomial families over a reduced ring, having the same base and the
same specialization at the zero parameter, agree. In the action-lift
argument this is the algebraic group-law normalization step; the
existence of a lifted polynomial family is a separate obligation.
-/
namespace StableCoordinate
namespace FrameLift
/-- A frame for the rank-one bundle underlying the determinant presentation. -/
structure Frame (R : Type*) [CommRing R] where
a : R
d : R
b : R
c : R
determinant : a * c - b * d = 1
variable {R S : Type*} [CommRing R] [CommRing S]
@[ext]
theorem Frame.ext {F G : Frame R}
(ha : F.a = G.a) (hd : F.d = G.d) (hb : F.b = G.b) (hc : F.c = G.c) :
F = G := by cases F; cases G; simp_all
/-- Extend each coordinate of a frame by a ring homomorphism. -/
def Frame.map (f : R →+* S) (F : Frame R) : Frame S where
a := f F.a
d := f F.d
b := f F.b
c := f F.c
determinant := by simpa only [← map_mul, ← map_sub, map_one] using congrArg f F.determinant
/-- Equality of the three quadric base coordinates. -/
def SameBase (F G : Frame R) : Prop :=
F.a * F.b = G.a * G.b ∧ F.d * F.c = G.d * G.c ∧ F.d * F.b = G.d * G.b
/-- Concrete scalar comparing two frames over the same base. -/
def comparison (F G : Frame R) : R := F.c * G.a - F.b * G.d
theorem sameBase_symm {F G : Frame R} (h : SameBase F G) : SameBase G F :=
⟨h.1.symm, h.2.1.symm, h.2.2.symm⟩
theorem comparison_a {F G : Frame R} (h : SameBase F G) :
comparison F G * F.a = G.a := by
dsimp [comparison]
linear_combination G.a * F.determinant + G.a * h.2.2 - G.d * h.1
theorem comparison_d {F G : Frame R} (h : SameBase F G) :
comparison F G * F.d = G.d := by
dsimp [comparison]
linear_combination G.a * h.2.1 + G.d * G.determinant - G.d * h.2.2
theorem comparison_b {F G : Frame R} (h : SameBase F G) :
comparison G F * F.b = G.b := by
dsimp [comparison]
linear_combination G.c * h.1 + G.b * G.determinant - G.b * h.2.2
theorem comparison_c {F G : Frame R} (h : SameBase F G) :
comparison G F * F.c = G.c := by
dsimp [comparison]
linear_combination G.c * F.determinant + G.c * h.2.2 - G.b * h.2.1
/-- The two comparison scalars are inverse units, without any domain hypothesis. -/
theorem comparison_mul_reverse {F G : Frame R} (h : SameBase F G) :
comparison F G * comparison G F = 1 := by
calc
comparison F G * comparison G F =
comparison F G * comparison G F * (F.a * F.c - F.b * F.d) := by
rw [F.determinant, mul_one]
_ = (comparison F G * F.a) * (comparison G F * F.c) -
(comparison G F * F.b) * (comparison F G * F.d) := by ring
_ = 1 := by
rw [comparison_a h, comparison_c h, comparison_b h, comparison_d h, G.determinant]
/-- The unique unit changing one frame into the other. -/
def comparisonUnit {F G : Frame R} (h : SameBase F G) : Rˣ where
val := comparison F G
inv := comparison G F
val_inv := comparison_mul_reverse h
inv_val := comparison_mul_reverse (sameBase_symm h)
@[simp]
theorem comparison_self (F : Frame R) : comparison F F = 1 := by
simpa only [comparison, mul_comm F.c F.a] using F.determinant
@[simp]
theorem comparison_map (f : R →+* S) (F G : Frame R) :
comparison (F.map f) (G.map f) = f (comparison F G) := by
simp only [comparison, Frame.map, map_sub, map_mul]
/-- The scalar is determined by its action on the two positive-weight coordinates. -/
theorem comparison_unique (F G : Frame R) {q : R}
(ha : q * F.a = G.a) (hd : q * F.d = G.d) : q = comparison F G := by
dsimp [comparison]
linear_combination -q * F.determinant + F.c * ha - F.b * hd
/-- Comparison scalar one forces equality of the entire frame. -/
theorem eq_of_comparison_eq_one {F G : Frame R} (h : SameBase F G)
(hq : comparison F G = 1) : F = G := by
have hrev : comparison G F = 1 := by
simpa only [hq, one_mul] using comparison_mul_reverse h
apply Frame.ext
· simpa only [hq, one_mul] using comparison_a h
· simpa only [hq, one_mul] using comparison_d h
· simpa only [hrev, one_mul] using comparison_b h
· simpa only [hrev, one_mul] using comparison_c h
/-- Two polynomial families of frames over a reduced coefficient ring are equal
if they have the same base coordinates and agree at the zero parameter.
In particular this proves the group law for a normalized polynomial-family lift:
compare the iterated lift over two parameters with the lift at their sum. -/
theorem normalized_polynomial_unique {σ : Type*} [IsReduced R]
{F G : Frame (MvPolynomial σ R)} (h : SameBase F G)
(hzero : F.map MvPolynomial.constantCoeff = G.map MvPolynomial.constantCoeff) :
F = G := by
have hu : IsUnit (comparison F G) := (comparisonUnit h).isUnit
obtain ⟨r, _, hr⟩ := MvPolynomial.isUnit_iff_eq_C_of_isReduced.mp hu
have heval : MvPolynomial.constantCoeff (comparison F G) = 1 := by
rw [← comparison_map, hzero, comparison_self]
have hr1 : r = 1 := by simpa only [hr, MvPolynomial.constantCoeff_C] using heval
apply eq_of_comparison_eq_one h
simpa only [hr1, map_one] using hr
end FrameLift
end StableCoordinate
end OAI
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