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7f5fab8 | 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 | import OAI.AlgebraicGeometry.CommutingDerivations.DerivationOperations
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.MvPolynomial.Degrees
import Mathlib.LinearAlgebra.LinearIndependent.Defs
namespace OAI
/-! The actual polynomial coordinate partials: commuting, locally nilpotent,
linearly independent over the polynomial ring, and with exact constant common
kernel in characteristic zero. No localization equivalence is assumed here. -/
noncomputable section
namespace AbhyankarSathaye.CommutingDerivations
open MvPolynomial
variable {B σ : Type*} [CommRing B]
theorem coordinate_partials_commute (i j : σ) (p : MvPolynomial σ B) :
pderiv i (pderiv j p) = pderiv j (pderiv i p) := by
classical
by_cases hij : i = j
· subst j
rfl
ext m
simp [coeff_pderiv, Finsupp.add_apply, hij, Ne.symm hij,
add_comm, add_left_comm, mul_comm, mul_left_comm]
/-- Vanishing of all coefficients at exponent at least n implies annihilation
by the nth partial iterate. This works over every commutative ring. -/
theorem coordinate_partial_iterate_eq_zero (i : σ) (n : ℕ) (p : MvPolynomial σ B)
(hp : ∀ m : σ →₀ ℕ, n ≤ m i → p.coeff m = 0) :
(pderiv i : MvPolynomial σ B → MvPolynomial σ B)^[n] p = 0 := by
classical
induction n generalizing p with
| zero =>
apply MvPolynomial.ext
intro m
simpa using hp m (Nat.zero_le _)
| succ n ih =>
rw [Function.iterate_succ_apply]
apply ih
intro m hm
rw [coeff_pderiv]
have hz : p.coeff (m + Finsupp.single i 1) = 0 := by
apply hp
simpa using Nat.succ_le_succ hm
rw [hz, zero_mul]
/-- An explicit pointwise bound: one more than the largest exponent of i
among the finitely many supported monomials. -/
theorem coordinate_partial_nilpotence_bound (i : σ) (p : MvPolynomial σ B) :
(pderiv i : MvPolynomial σ B → MvPolynomial σ B)^[
p.support.sup (fun m => m i) + 1] p = 0 := by
classical
apply coordinate_partial_iterate_eq_zero
intro m hm
by_contra hc
have hmem : m ∈ p.support := mem_support_iff.mpr hc
have hle : m i ≤ p.support.sup (fun a => a i) :=
Finset.le_sup (f := fun a : σ →₀ ℕ => a i) hmem
exact (Nat.not_succ_le_self _) (hm.trans hle)
theorem coordinate_partial_locallyNilpotent (i : σ) :
LocallyNilpotent (pderiv i : Derivation B (MvPolynomial σ B) (MvPolynomial σ B)) := by
intro p
exact ⟨p.support.sup (fun m => m i) + 1, coordinate_partial_nilpotence_bound i p⟩
/-- Exact coefficient consequence of the common-kernel condition. -/
theorem coefficient_eq_zero_of_all_partials_zero [IsDomain B] [CharZero B]
(p : MvPolynomial σ B) (hp : ∀ i, pderiv i p = 0)
(m : σ →₀ ℕ) (hm : m ≠ 0) : p.coeff m = 0 := by
classical
have hex : ∃ i, m i ≠ 0 := by
by_contra hh
apply hm
ext i
change m i = 0
by_contra hi
exact hh ⟨i, hi⟩
obtain ⟨i, hi⟩ := hex
have he := congrArg (fun polynomial : MvPolynomial σ B =>
polynomial.coeff (m - Finsupp.single i 1)) (hp i)
rw [coeff_pderiv, Finsupp.sub_add_single_one_cancel hi,
AddMonoidAlgebra.coeff_zero] at he
have hfactor : (((m - Finsupp.single i 1 : σ →₀ ℕ) i : ℕ) : B) + 1 ≠ 0 := by
rw [← Nat.cast_succ]
exact Nat.cast_ne_zero.mpr (Nat.succ_ne_zero _)
exact (mul_eq_zero.mp he).resolve_right hfactor
/-- The common kernel is literally the coefficient ring, with witness equal
to the constant coefficient. In particular this applies to Laurent coefficients
over C, once that coefficient algebra's domain and characteristic are installed. -/
theorem coordinate_partials_common_kernel [IsDomain B] [CharZero B]
(p : MvPolynomial σ B) :
(∀ i, pderiv i p = 0) ↔ ∃ b : B, p = C b := by
classical
constructor
· intro hp
refine ⟨p.coeff 0, ?_⟩
ext m
by_cases hm : m = 0
· subst m
simp
· rw [coefficient_eq_zero_of_all_partials_zero p hp m hm, coeff_C_of_ne_zero hm]
· rintro ⟨b, rfl⟩ i
exact pderiv_C
theorem coordinate_partials_linearIndependent [Fintype σ] :
LinearIndependent (MvPolynomial σ B)
(fun i : σ => (pderiv i : Derivation B (MvPolynomial σ B) (MvPolynomial σ B))) := by
classical
apply Fintype.linearIndependent_iff.mpr
intro a ha j
let ev : Derivation B (MvPolynomial σ B) (MvPolynomial σ B) →+ MvPolynomial σ B :=
{ toFun := fun D => D (X j)
map_zero' := rfl
map_add' := fun _ _ => rfl }
have he := congrArg ev ha
rw [map_sum, map_zero] at he
simpa [ev, Derivation.smul_apply, smul_eq_mul, pderiv_X, Pi.single_apply] using he
end AbhyankarSathaye.CommutingDerivations
end
end OAI
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