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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 93 94 95 96 97 98 99 100 101 102 103 104 105 106 | import OAI.AlgebraicGeometry.StableCoordinate.Definitions
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Tactic
namespace OAI
/-!
# Constant affine changes of polynomial coordinates
The automorphisms in this file normalize a nonzero scalar multiple of a
coordinate and a constant translate. They are actual algebra automorphisms
of the whole polynomial ring, with explicit polynomial inverses.
-/
namespace StableCoordinate
open MvPolynomial
section Generic
variable {k σ : Type*} [Field k]
/-- Simultaneously replace every variable `X i` by `a * X i + b`. -/
noncomputable def affineAll (a b : k) (ha : a ≠ 0) :
MvPolynomial σ k ≃ₐ[k] MvPolynomial σ k :=
AlgEquiv.ofAlgHom
(aeval fun i => C a * X i + C b)
(aeval fun i => C a⁻¹ * (X i - C b))
(by
apply algHom_ext
intro i
simp only [AlgHom.comp_apply, map_mul, map_sub, aeval_X, aeval_C,
AlgHom.id_apply, MvPolynomial.algebraMap_eq]
rw [add_sub_cancel_right, ← mul_assoc, ← C_mul, inv_mul_cancel₀ ha, C_1, one_mul])
(by
apply algHom_ext
intro i
simp only [AlgHom.comp_apply, map_mul, map_add, aeval_X, aeval_C,
AlgHom.id_apply, MvPolynomial.algebraMap_eq]
rw [← mul_assoc, ← C_mul, mul_inv_cancel₀ ha, C_1, one_mul, sub_add_cancel])
@[simp] theorem affineAll_X (a b : k) (ha : a ≠ 0) (i : σ) :
affineAll (σ := σ) a b ha (X i) = C a * X i + C b := by
change (aeval fun j => C a * X j + C b) (X i) = _
exact aeval_X _ _
@[simp] theorem affineAll_C (a b : k) (ha : a ≠ 0) (c : k) :
affineAll (σ := σ) a b ha (C c) = C c := by
exact (affineAll (σ := σ) a b ha).commutes c
/-- Its inverse removes the scalar and translate from every variable. -/
@[simp] theorem affineAll_symm_X (a b : k) (ha : a ≠ 0) (i : σ) :
(affineAll (σ := σ) a b ha).symm (X i) = C a⁻¹ * (X i - C b) := by
change (aeval fun j => C a⁻¹ * (X j - C b)) (X i) = _
exact aeval_X _ _
/-- An ambient algebra automorphism taking a polynomial to an affine-linear
coordinate can be normalized to take it to that variable exactly. -/
theorem exists_equiv_eq_X_of_affine
(p : MvPolynomial σ k) (i : σ) (a b : k) (ha : a ≠ 0)
(φ : MvPolynomial σ k ≃ₐ[k] MvPolynomial σ k)
(hφ : φ p = C a * X i + C b) :
∃ ψ : MvPolynomial σ k ≃ₐ[k] MvPolynomial σ k, ψ p = X i := by
refine ⟨φ.trans (affineAll (σ := σ) a b ha).symm, ?_⟩
change (affineAll (σ := σ) a b ha).symm (φ p) = X i
rw [hφ, ← affineAll_X a b ha i, AlgEquiv.symm_apply_apply]
/-- The normalization needed after identifying the generators of two
principal hypersurface ideals. -/
theorem exists_equiv_eq_X_of_sub_eq_scalar_mul
(p : MvPolynomial σ k) (i : σ) (t a : k) (ha : a ≠ 0)
(φ : MvPolynomial σ k ≃ₐ[k] MvPolynomial σ k)
(hφ : φ (p - C t) = C a * X i) :
∃ ψ : MvPolynomial σ k ≃ₐ[k] MvPolynomial σ k, ψ p = X i := by
apply exists_equiv_eq_X_of_affine p i a t ha φ
have hc : φ (C t) = C t := φ.commutes t
rw [map_sub, hc] at hφ
exact sub_eq_iff_eq_add.mp hφ
end Generic
/-- Specialized normalization for the exact `IsCoordinate` definition. -/
theorem isCoordinate_of_equiv_sub_eq_scalar_mul_X
{n : ℕ} (hn : 0 < n) (p : R n) (t a : ℂ) (ha : a ≠ 0)
(φ : R n ≃ₐ[ℂ] R n)
(hφ : φ (p - C t) = C a * X ⟨0, hn⟩) :
IsCoordinate hn p :=
exists_equiv_eq_X_of_sub_eq_scalar_mul p ⟨0, hn⟩ t a ha φ hφ
/-- A constant translate does not alter whether a polynomial is a coordinate. -/
theorem isCoordinate_sub_C_iff {n : ℕ} (hn : 0 < n) (p : R n) (t : ℂ) :
IsCoordinate hn (p - C t) ↔ IsCoordinate hn p := by
constructor
· rintro ⟨φ, hφ⟩
apply isCoordinate_of_equiv_sub_eq_scalar_mul_X hn p t 1 one_ne_zero φ
simpa using hφ
· rintro ⟨φ, hφ⟩
apply exists_equiv_eq_X_of_affine (p - C t) ⟨0, hn⟩ 1 (-t) one_ne_zero φ
rw [map_sub, hφ]
have hc : φ (C t) = C t := φ.commutes t
rw [hc, C_1, one_mul, C_neg, sub_eq_add_neg]
end StableCoordinate
end OAI
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