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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 | import Mathlib.RingTheory.GradedAlgebra.Basic
import Mathlib.RingTheory.Derivation.Basic
import Mathlib.Tactic
namespace OAI
/-!
# A maximal degree shift detected on finitely many algebra generators
For an integer-graded algebra, the values of a derivation on finitely many
generators have only finitely many homogeneous components. Thus a nonzero
derivation has a maximal degree shift which is attained on a generator.
These lemmas supply the finite-support step in the leading-derivation
construction of Sections 3 and 5 of the supplied manuscript.
-/
namespace StableCoordinate
namespace DerivationTopShift
open DirectSum GradedAlgebra
variable {k A ι : Type*} [CommRing k] [CommRing A] [Algebra k A]
variable (𝒜 : ℤ → Submodule k A) [GradedAlgebra 𝒜]
/-- A nonzero element of an internally graded algebra has a nonzero
homogeneous projection. -/
theorem exists_proj_ne_zero {a : A} (ha : a ≠ 0) :
∃ j : ℤ, proj 𝒜 j a ≠ 0 := by
classical
by_contra! h
apply ha
apply (DirectSum.decompose 𝒜).injective
ext j
simpa using h j
/-- A nonzero derivation cannot vanish on all algebra generators. -/
theorem exists_generator_ne_zero (g : ι → A)
(hg : Algebra.adjoin k (Set.range g) = ⊤)
(D : Derivation k A A) (hD : D ≠ 0) :
∃ i, D (g i) ≠ 0 := by
by_contra! h
apply hD
apply Derivation.ext_of_adjoin_eq_top (Set.range g) hg
intro a ha
obtain ⟨i, rfl⟩ := ha
simpa using h i
variable [Fintype ι]
/-- Finitely many generator images admit a maximal degree shift, and for a
nonzero derivation the maximum is attained. No sign restriction is imposed
on generator degrees or on the resulting shift. -/
theorem exists_maximal_generator_shift (g : ι → A) (d : ι → ℤ)
(hg : Algebra.adjoin k (Set.range g) = ⊤)
(D : Derivation k A A) (hD : D ≠ 0) :
∃ m : ℤ,
(∀ i j, d i + m < j → proj 𝒜 j (D (g i)) = 0) ∧
(∃ i, proj 𝒜 (d i + m) (D (g i)) ≠ 0) := by
classical
let S : Finset ℤ := Finset.univ.biUnion fun i =>
((DirectSum.decompose 𝒜 (D (g i))).support).image (fun j => j - d i)
have memS (i : ι) (j : ℤ) (hj : proj 𝒜 j (D (g i)) ≠ 0) : j - d i ∈ S := by
apply Finset.mem_biUnion.mpr
refine ⟨i, Finset.mem_univ _, ?_⟩
apply Finset.mem_image.mpr
exact ⟨j, (GradedAlgebra.mem_support_iff 𝒜 _ _).mpr hj, rfl⟩
have hS : S.Nonempty := by
obtain ⟨i, hi⟩ := exists_generator_ne_zero g hg D hD
obtain ⟨j, hj⟩ := exists_proj_ne_zero 𝒜 hi
exact ⟨j - d i, memS i j hj⟩
refine ⟨S.max' hS, ?_, ?_⟩
· intro i j hj
by_contra hne
have hle := Finset.le_max' S (j - d i) (memS i j hne)
omega
· have hm : S.max' hS ∈ S := Finset.max'_mem S hS
obtain ⟨i, _, hi⟩ := Finset.mem_biUnion.mp hm
obtain ⟨j, hj, heq⟩ := Finset.mem_image.mp hi
refine ⟨i, ?_⟩
have hidx : d i + S.max' hS = j := by omega
rw [hidx]
exact (GradedAlgebra.mem_support_iff 𝒜 _ _).mp hj
/-- For a family of homogeneous component derivations, vanishing outside the
finite shift support can be tested solely on algebra generators. -/
theorem exists_maximal_component (g : ι → A) (d : ι → ℤ)
(hg : Algebra.adjoin k (Set.range g) = ⊤)
(D : Derivation k A A) (hD : D ≠ 0)
(E : ℤ → Derivation k A A)
(hE : ∀ m i, E m (g i) = proj 𝒜 (d i + m) (D (g i))) :
∃ m : ℤ, E m ≠ 0 ∧ ∀ n, m < n → E n = 0 := by
obtain ⟨m, hbound, i, hi⟩ := exists_maximal_generator_shift 𝒜 g d hg D hD
refine ⟨m, ?_, ?_⟩
· intro hzero
apply hi
rw [← hE, hzero]
rfl
· intro n hn
apply Derivation.ext_of_adjoin_eq_top (Set.range g) hg
intro a ha
obtain ⟨i, rfl⟩ := ha
rw [hE]
exact hbound i (d i + n) (by omega)
/-- Once homogeneous components are constructed, finite algebra generation
provides a single upper degree-shift bound on every homogeneous input, with
an attained nonzero top derivation. This is the hypothesis needed for the
leading-component local-nilpotence argument. -/
theorem exists_top_component_and_global_bound
(g : ι → A) (d : ι → ℤ)
(hg : Algebra.adjoin k (Set.range g) = ⊤)
(hhom : ∀ i, g i ∈ 𝒜 (d i))
(D : Derivation k A A) (hD : D ≠ 0)
(E : ℤ → Derivation k A A)
(hE : ∀ m i (x : A), x ∈ 𝒜 i →
E m x = proj 𝒜 (i + m) (D x)) :
∃ m : ℤ, E m ≠ 0 ∧
(∀ n, m < n → E n = 0) ∧
(∀ i (x : A), x ∈ 𝒜 i → ∀ j, i + m < j → proj 𝒜 j (D x) = 0) := by
obtain ⟨m, hm, hmax⟩ := exists_maximal_component 𝒜 g d hg D hD E
(fun m i => hE m (d i) (g i) (hhom i))
refine ⟨m, hm, hmax, ?_⟩
intro i x hx j hj
have heq := hE (j - i) i x hx
have hz := hmax (j - i) (by omega)
rw [hz] at heq
have hidx : i + (j - i) = j := by omega
rw [hidx] at heq
exact heq.symm
end DerivationTopShift
end StableCoordinate
end OAI
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