File size: 2,541 Bytes
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 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 | 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
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