openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /LocalAlgebra /AssociatedPrimeAvoidance.lean
| 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 | |