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b797a5b | 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 | import Mathlib
namespace OAI
namespace PiExponent.GeometrySupport.AssociatedPrimeAvoidance
noncomputable section
open scoped BigOperators
theorem exists_regular_combination
{K ι J : Type*} [Field K] [Infinite K] [Finite ι] [Fintype J]
(R : ι → Type*) [∀ i, CommRing (R i)] [∀ i, Algebra K (R i)]
[∀ i, IsNoetherianRing (R i)] (a : ∀ i, J → R i)
(hspan : ∀ i, Ideal.span (Set.range (a i)) = ⊤) :
∃ c : J → K, ∀ i, (∑ j, c j • a i j) ∈ nonZeroDivisors (R i) := by
classical
let : Fintype ι := Fintype.ofFinite ι
let (i : ι) : Fintype (associatedPrimes (R i) (R i)) :=
(associatedPrimes.finite (R i) (R i)).fintype
let B := (i : ι) × associatedPrimes (R i) (R i)
let L (i : ι) : (J → K) →ₗ[K] R i := Fintype.linearCombination K (a i)
let bad (b : B) : Submodule K (J → K) :=
(b.2.val.restrictScalars K).comap (L b.1)
have hbad : ∀ b : B, bad b ≠ ⊤ := by
intro b hb
apply b.2.property.isPrime.ne_top
apply top_unique
rw [← hspan b.1]
apply Ideal.span_le.mpr
rintro _ ⟨j, rfl⟩
have hm : Pi.single j (1 : K) ∈ bad b := by
rw [hb]
trivial
change Fintype.linearCombination K (a b.1) (Pi.single j 1) ∈ b.2.val at hm
rw [Fintype.linearCombination_apply_single, one_smul] at hm
exact hm
obtain ⟨c, hc⟩ := Submodule.exists_forall_notMem_of_forall_ne_top bad hbad
refine ⟨c, fun i => ?_⟩
by_contra hn
have hz : (∑ j, c j • a i j) ∈ (nonZeroDivisors (R i) : Set (R i))ᶜ := hn
rw [← biUnion_associatedPrimes_eq_compl_nonZeroDivisors (R i)] at hz
obtain ⟨P, hP, hmem⟩ := Set.mem_iUnion₂.mp hz
apply hc ⟨i, ⟨P, hP⟩⟩
change (∑ j, c j • a i j) ∈ P
exact hmem
end
end PiExponent.GeometrySupport.AssociatedPrimeAvoidance
end OAI
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