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import OAI.Algebra.FrobeniusObstruction.TensorCohomology
import OAI.Algebra.FrobeniusObstruction.CycleTransfer
namespace OAI
noncomputable section
open scoped BigOperators TensorProduct
namespace BoundaryOnly.FormalObstruction
variable {k : Type*} [Field k]
private theorem generalized_inverse {V : Type*} [AddCommGroup V] [Module k V]
(S : V →ₗ[k] V) : ∃ L : V →ₗ[k] V, ∀ x, S (L (S x)) = S x := by
obtain ⟨g,hg⟩ := S.rangeRestrict.exists_rightInverse_of_surjective S.range_rangeRestrict
obtain ⟨h,hh⟩ := S.range.subtype.exists_leftInverse_of_injective S.range.ker_subtype
refine ⟨g.comp h, fun x => ?_⟩
have hx : h (S x) = S.rangeRestrict x :=
LinearMap.congr_fun hh (S.rangeRestrict x)
change S (g (h (S x))) = S x
rw [hx]
exact congrArg Subtype.val (LinearMap.congr_fun hg (S.rangeRestrict x))
variable {ι : Type*} [Fintype ι] [DecidableEq ι]
variable (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module k (V i)]
def kernelTensorSpace (S : ∀ i, V i →ₗ[k] V i) (J : Finset ι) :
Submodule k (⨂[k] i, V i) :=
Submodule.span k {z | ∃ m : ∀ i, V i,
(∀ i ∈ J, S i (m i) = 0) ∧ PiTensorProduct.tprod k m = z}
def tensorAction (S : ∀ i, V i →ₗ[k] V i) (I : Finset ι) :
(⨂[k] i, V i) →ₗ[k] (⨂[k] i, V i) :=
PiTensorProduct.map fun i => if i ∈ I then S i else LinearMap.id
theorem joint_tensor_kernel (S : ∀ i, V i →ₗ[k] V i) (active : Finset ι) :
(⨅ (I : Finset ι) (_ : I ⊆ active) (_ : I.card = 3),
LinearMap.ker (tensorAction V S I)) =
(⨆ (J : Finset ι) (_ : J ⊆ active) (_ : (active \ J).card ≤ 2),
kernelTensorSpace V S J) := by
classical
let R := ⨆ (J : Finset ι) (_ : J ⊆ active) (_ : (active \ J).card ≤ 2),
kernelTensorSpace V S J
have hR (J : Finset ι) (hJ : J ⊆ active) (hc : (active \ J).card ≤ 2) :
kernelTensorSpace V S J ≤ R := le_iSup_of_le J (le_iSup_of_le hJ (le_iSup_of_le hc le_rfl))
apply le_antisymm
· intro x hx
have hz (I : Finset ι) (hI : I ⊆ active) (hcard : I.card = 3) :
tensorAction V S I x = 0 := by
exact ((Submodule.mem_iInf _).mp ((Submodule.mem_iInf _).mp ((Submodule.mem_iInf _).mp hx I) hI) hcard)
choose L hL using fun i => generalized_inverse (S i)
let Q (i : ι) := (L i).comp (S i)
let P (i : ι) := LinearMap.id - Q i
have hSP (i : ι) (v : V i) : S i (P i v) = 0 := by
change S i (v - L i (S i v)) = 0
rw [map_sub, hL, sub_self]
let F (A : Finset ι) := PiTensorProduct.map fun i => if i ∈ A then Q i else P i
have hsum : ∑ A : Finset ι, F A = LinearMap.id := by
apply PiTensorProduct.ext
ext m
simp only [LinearMap.compMultilinearMap_apply, LinearMap.sum_apply, LinearMap.id_apply]
simp only [F, PiTensorProduct.map_tprod]
have hm : (fun i => Q i (m i)) + (fun i => P i (m i)) = m := by
funext i
change Q i (m i) + (m i - Q i (m i)) = m i
abel
conv_rhs => rw [← hm, (PiTensorProduct.tprod k).map_add_univ]
apply Finset.sum_congr rfl
intro A ha
congr 1
funext i
simp only [Finset.piecewise]
split_ifs <;> rfl
have hFx (A : Finset ι) : F A x ∈ R := by
by_cases hc : (active ∩ A).card ≤ 2
· let J := active \ A
have hJ : J ⊆ active := Finset.sdiff_subset
have hjc : (active \ J).card ≤ 2 := by
simpa only [J, Finset.sdiff_sdiff_self_left] using hc
apply hR J hJ hjc
clear hx hz
induction x using PiTensorProduct.induction_on with
| smul_tprod c m =>
rw [map_smul]
apply Submodule.smul_mem
apply Submodule.subset_span
refine ⟨fun i => (if i ∈ A then Q i else P i) (m i), ?_, ?_⟩
· intro i hi
have hiA : i ∉ A := (Finset.mem_sdiff.mp hi).2
simpa only [ite_eq_right hiA] using hSP i (m i)
· exact (PiTensorProduct.map_tprod _ _).symm
| add x y hx hy =>
rw [map_add]
exact Submodule.add_mem _ hx hy
· have hc' : 3 ≤ (active ∩ A).card := by omega
obtain ⟨I,hI,hcard⟩ := Finset.exists_subset_card_eq hc'
have hIa : I ⊆ active := fun i hi => (Finset.mem_inter.mp (hI hi)).1
have hIA : I ⊆ A := fun i hi => (Finset.mem_inter.mp (hI hi)).2
let H := PiTensorProduct.map fun i =>
if i ∈ I then L i else if i ∈ A then Q i else P i
have hcomp : F A = H.comp (tensorAction V S I) := by
dsimp only [F, H, tensorAction]
rw [← PiTensorProduct.map_comp]
congr 1
funext i
by_cases hi : i ∈ I
· simp only [ite_eq_left hi, ite_eq_left (hIA hi), Q]
· simp only [ite_eq_right hi, LinearMap.comp_id]
rw [hcomp, LinearMap.comp_apply, hz I hIa hcard, map_zero]
exact R.zero_mem
change x ∈ R
have hsx := LinearMap.congr_fun hsum x
simp only [LinearMap.sum_apply, LinearMap.id_apply] at hsx
rw [← hsx]
exact Submodule.sum_mem R (fun A _ => hFx A)
· refine iSup_le fun J => iSup_le fun hJ => iSup_le fun hc => ?_
refine le_iInf fun I => le_iInf fun hI => le_iInf fun hcard => ?_
apply Submodule.span_le.mpr
rintro z ⟨m,hm,rfl⟩
have hex : ∃ i ∈ I, i ∈ J := by
by_contra hn
have hsub : I ⊆ active \ J := by
intro i hi
exact Finset.mem_sdiff.mpr ⟨hI hi, fun hij => hn ⟨i,hi,hij⟩⟩
have := Finset.card_le_card hsub
omega
obtain ⟨i,hiI,hiJ⟩ := hex
change tensorAction V S I (PiTensorProduct.tprod k m) = 0
rw [tensorAction, PiTensorProduct.map_tprod]
apply (PiTensorProduct.tprod k).map_coord_zero i
simpa only [ite_eq_left hiI] using hm i hiJ
end BoundaryOnly.FormalObstruction
namespace BoundaryOnly.FormalObstruction
open scoped BigOperators TensorProduct
variable {k ι : Type*} [Field k] [Fintype ι] [LinearOrder ι]
variable {V : ι → Type*} [∀ i, AddCommGroup (V i)] [∀ i, Module k (V i)]
theorem select_special_fiber_tensor
(S P D r h : SignedTensor.EndFamily (k := k) V)
(active : Finset ι) (hcard : 5 ≤ active.card)
(hP2 : ∀ i, P i * P i = 1)
(hDP : ∀ i, D i * P i = -(P i * D i))
(hDr : ∀ i, D i * r i = r i * D i)
(hPr : ∀ i, P i * r i = r i * P i)
(hPh : ∀ i, P i * h i = -(h i * P i))
(hDh : ∀ i, D i * h i + h i * D i = 1 - r i)
(hNr : ∀ i ∈ active, ∀ x, S i x = 0 →
S i (r i x) = 0 ∧ S i (D i (r i x)) = 0)
(φ : (⨂[k] i, V i) →ₗ[k] k)
(hφD : ∀ x, φ (SignedTensor.differential V P D x) = 0)
(z : ⨂[k] i, V i)
(hDz : SignedTensor.differential V P D z = 0)
(hφz : φ z ≠ 0)
(hz : ∀ I : Finset ι, I ⊆ active → I.card = 3 → tensorAction V S I z = 0) :
∃ (J : Finset ι) (m : ∀ i, V i),
J ⊆ active ∧ 3 ≤ J.card ∧
(∀ i ∈ J, S i (m i) = 0 ∧ S i (D i (m i)) = 0) ∧
φ (PiTensorProduct.tprod k m) ≠ 0 := by
classical
let Good : Set (⨂[k] i, V i) := {x | ∃ (J : Finset ι) (m : ∀ i, V i),
J ⊆ active ∧ (active \ J).card ≤ 2 ∧
(∀ i ∈ J, S i (m i) = 0 ∧ S i (D i (m i)) = 0) ∧
PiTensorProduct.tprod k m = x}
let T := Submodule.span k Good
have hm : (⨅ (I : Finset ι) (_ : I ⊆ active) (_ : I.card = 3),
LinearMap.ker (tensorAction V S I)) ≤ T.comap (PiTensorProduct.map r) := by
rw [joint_tensor_kernel]
refine iSup_le fun J => iSup_le fun hJ => iSup_le fun hJc => ?_
apply Submodule.span_le.mpr
rintro _ ⟨m,hm,rfl⟩
change PiTensorProduct.map r (PiTensorProduct.tprod k m) ∈ T
rw [PiTensorProduct.map_tprod]
apply Submodule.subset_span
refine ⟨J,fun i => r i (m i),hJ,hJc,?_,rfl⟩
intro i hi
exact hNr i (hJ hi) (m i) (hm i hi)
have hzmem : z ∈ (⨅ (I : Finset ι) (_ : I ⊆ active) (_ : I.card = 3),
LinearMap.ker (tensorAction V S I)) := by
simp only [Submodule.mem_iInf, LinearMap.mem_ker]
exact hz
have hRz : PiTensorProduct.map r z ∈ T := hm hzmem
have hpair : φ (PiTensorProduct.map r z) = φ z := by
have H := LinearMap.congr_fun
(SignedTensor.tensor_homotopy P D r h hP2 hDP hDr hPr hPh hDh) z
change SignedTensor.differential V P D (SignedTensor.homotopy V P r h z) +
SignedTensor.homotopy V P r h (SignedTensor.differential V P D z) =
z - PiTensorProduct.map r z at H
have Hφ := congrArg φ H
rw [hDz, map_zero, add_zero, hφD, map_sub] at Hφ
exact (sub_eq_zero.mp Hφ.symm).symm
have hex : ∃ x ∈ Good, φ x ≠ 0 := by
by_contra hn
have hle : T ≤ LinearMap.ker φ := by
apply Submodule.span_le.mpr
intro x hx
change φ x = 0
by_contra hnx
exact hn ⟨x,hx,hnx⟩
apply hφz
rw [← hpair]
exact hle hRz
obtain ⟨_,⟨J,m,hJ,hJc,hm,rfl⟩,hp⟩ := hex
refine ⟨J,m,hJ,?_,hm,hp⟩
have hc := Finset.card_sdiff_of_subset hJ
have hle := Finset.card_le_card hJ
omega
end BoundaryOnly.FormalObstruction
end
end OAI
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