| import Mathlib.Algebra.Module.Submodule.RestrictScalars | |
| import Mathlib.Algebra.Module.Submodule.Union | |
| import Mathlib.LinearAlgebra.Finsupp.LinearCombination | |
| import Mathlib.RingTheory.Ideal.Operations | |
| import Mathlib.RingTheory.Ideal.Span | |
| namespace OAI | |
| namespace PiExponentSiegel.W20 | |
| variable {K M I J : Type*} [Field K] [Infinite K] | |
| [AddCommGroup M] [Module K M] [Finite I] | |
| theorem exists_linearCombination_avoiding_submodules | |
| (g : J → M) (P : I → Submodule K M) | |
| (h : ∀ i, ¬ Submodule.span K (Set.range g) ≤ P i) : | |
| ∃ c : J →₀ K, ∀ i, Finsupp.linearCombination K g c ∉ P i := by | |
| let bad : I → Submodule K (J →₀ K) := | |
| fun i => (P i).comap (Finsupp.linearCombination K g) | |
| have hbad : ∀ i, bad i ≠ ⊤ := by | |
| intro i hi | |
| apply h i | |
| apply Submodule.span_le.mpr | |
| rintro _ ⟨j, rfl⟩ | |
| have hc : Finsupp.single j (1 : K) ∈ bad i := by | |
| rw [hi] | |
| trivial | |
| change Finsupp.linearCombination K g (Finsupp.single j 1) ∈ P i at hc | |
| simpa using hc | |
| obtain ⟨c, hc⟩ := Submodule.exists_forall_notMem_of_forall_ne_top bad hbad | |
| exact ⟨c, hc⟩ | |
| variable {R : Type*} [CommRing R] [Algebra K R] | |
| theorem exists_linearCombination_avoiding_ideals | |
| (g : J → R) (P : I → Ideal R) | |
| (h : ∀ i, ¬ Ideal.span (Set.range g) ≤ P i) : | |
| ∃ c : J →₀ K, ∀ i, Finsupp.linearCombination K g c ∉ P i := by | |
| apply exists_linearCombination_avoiding_submodules g | |
| (fun i => (P i).restrictScalars K) | |
| intro i hi | |
| apply h i | |
| apply Ideal.span_le.mpr | |
| intro x hx | |
| exact hi (Submodule.subset_span hx) | |
| theorem exists_span_element_avoiding_ideals | |
| (g : J → R) (P : I → Ideal R) | |
| (h : ∀ i, ¬ Ideal.span (Set.range g) ≤ P i) : | |
| ∃ x ∈ Submodule.span K (Set.range g), ∀ i, x ∉ P i := by | |
| obtain ⟨c, hc⟩ := exists_linearCombination_avoiding_ideals (K := K) g P h | |
| refine ⟨Finsupp.linearCombination K g c, ?_, hc⟩ | |
| rw [← Finsupp.range_linearCombination] | |
| exact ⟨c, rfl⟩ | |
| theorem exists_linearCombination_avoiding_primes_below_radical | |
| (g : J → R) (m : Ideal R) | |
| (hgen : (Ideal.span (Set.range g)).radical = m) | |
| (P : I → Ideal R) (hprime : ∀ i, (P i).IsPrime) | |
| (hbelow : ∀ i, P i < m) : | |
| ∃ c : J →₀ K, ∀ i, Finsupp.linearCombination K g c ∉ P i := by | |
| apply exists_linearCombination_avoiding_ideals g P | |
| intro i hle | |
| have hmle : m ≤ P i := by | |
| rw [← hgen] | |
| exact (hprime i).radical_le_iff.mpr hle | |
| exact (not_le_of_gt (hbelow i)) hmle | |
| end PiExponentSiegel.W20 | |
| end OAI | |