| import Mathlib |
| import OAI.Algebra.FinitisticAsymmetry.Triangular.Socle |
|
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| namespace OAI |
|
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| noncomputable section |
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| open CategoryTheory |
| namespace LittleFinitistic.CummingsConstruction |
|
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| section |
| universe u |
| variable {k C ι : Type u} [Field k] [Ring C] [Algebra k C] |
| variable (χ : C →ₐ[k] (ι → k)) (i : ι) |
|
|
| def cval (r : T χ) : k := χ r.fst.1 i |
| def ehead (r : T χ) : k := (r.fst.2 i).fst |
| def etail (r : T χ) : k := (r.fst.2 i).snd |
| def oval (r : T χ) : k := (SimpleSum.value (TriangularConstruction.offValue r.snd)) i |
|
|
| @[simp] lemma cval_mul (r s : T χ) : cval χ i (r*s) = cval χ i r * cval χ i s := by |
| change χ (r.fst.1 * s.fst.1) i = _ |
| simp [cval] |
| @[simp] lemma ehead_mul (r s : T χ) : ehead χ i (r*s) = ehead χ i r * ehead χ i s := rfl |
| @[simp] lemma etail_mul (r s : T χ) : etail χ i (r*s) = ehead χ i r * etail χ i s + ehead χ i s * etail χ i r := by |
| change _ * _ + _ * _ = _ |
| change (r.fst.2 i).fst * (s.fst.2 i).snd + (r.fst.2 i).snd * (s.fst.2 i).fst = _ |
| dsimp only [ehead, etail] |
| ring |
| @[simp] lemma oval_mul (r s : T χ) : oval χ i (r*s) = ehead χ i r * oval χ i s + cval χ i s * oval χ i r := rfl |
|
|
| def NewInjective (_χ : C →ₐ[k] (ι → k)) (_i : ι) := k × (k × k) |
| namespace NewInjective |
|
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| def value (x : NewInjective χ i) : k × (k × k) := x |
|
|
| @[ext] lemma ext {x y : NewInjective χ i} |
| (h₁ : (value χ i x).1 = (value χ i y).1) |
| (h₂ : (value χ i x).2.1 = (value χ i y).2.1) |
| (h₃ : (value χ i x).2.2 = (value χ i y).2.2) : x = y := |
| Prod.ext h₁ (Prod.ext h₂ h₃) |
| instance : AddCommGroup (NewInjective χ i) := inferInstanceAs (AddCommGroup (k × (k × k))) |
|
|
| def act (r : T χ) (x : NewInjective χ i) : NewInjective χ i := |
| (cval χ i r * (value χ i x).1, |
| ehead χ i r * (value χ i x).2.1, |
| oval χ i r * (value χ i x).1 + etail χ i r * (value χ i x).2.1 + ehead χ i r * (value χ i x).2.2) |
|
|
| instance : SMul (T χ) (NewInjective χ i) := ⟨act χ i⟩ |
|
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| instance : Module (T χ) (NewInjective χ i) where |
| one_smul x := by |
| change (χ 1 i * _, (1 * _, 0 * _ + 0 * _ + 1 * _)) = value χ i x |
| simp |
| mul_smul r s x := by |
| change act χ i (r*s) x = act χ i r (act χ i s x) |
| simp only [act, value, cval_mul, ehead_mul, etail_mul, oval_mul] |
| apply ext <;> dsimp [value, NewInjective] <;> ring |
| smul_zero r := by |
| change act χ i r (0,0,0) = (0,0,0) |
| simp only [act, value, mul_zero, add_zero] |
| rfl |
| smul_add r x y := by |
| apply ext |
| · change cval χ i r * (x.1 + y.1) = cval χ i r*x.1 + cval χ i r*y.1 |
| ring |
| · change ehead χ i r * (x.2.1 + y.2.1) = ehead χ i r*x.2.1 + ehead χ i r*y.2.1 |
| ring |
| · change oval χ i r * (x.1+y.1) + etail χ i r * (x.2.1+y.2.1) + |
| ehead χ i r * (x.2.2+y.2.2) = |
| (oval χ i r*x.1 + etail χ i r*x.2.1 + ehead χ i r*x.2.2) + |
| (oval χ i r*y.1 + etail χ i r*y.2.1 + ehead χ i r*y.2.2) |
| ring |
| add_smul r s x := by |
| apply ext |
| · change χ (r.fst.1+s.fst.1) i * x.1 = χ r.fst.1 i*x.1 + χ s.fst.1 i*x.1 |
| simp [map_add, add_mul] |
| · change ((r.fst.2 i).fst + (s.fst.2 i).fst)*x.2.1 = |
| (r.fst.2 i).fst*x.2.1 + (s.fst.2 i).fst*x.2.1 |
| ring |
| · change (oval χ i r + oval χ i s)*x.1 + |
| (etail χ i r + etail χ i s)*x.2.1 + |
| (ehead χ i r + ehead χ i s)*x.2.2 = |
| (oval χ i r*x.1 + etail χ i r*x.2.1 + ehead χ i r*x.2.2) + |
| (oval χ i s*x.1 + etail χ i s*x.2.1 + ehead χ i s*x.2.2) |
| ring |
| zero_smul x := by |
| change (χ 0 i * _, (0 * _, 0 * _ + 0 * _ + 0 * _)) = (0,0,0) |
| simp |
|
|
| end NewInjective |
| end |
|
|
| section |
| open CategoryTheory |
| universe u |
| variable {k C ι : Type u} [Field k] [Ring C] [Algebra k C] |
| variable (χ : C →ₐ[k] (ι → k)) (i : ι) [DecidableEq ι] |
|
|
| def newIdem : T χ := ((0, Pi.single i 1), 0) |
|
|
| lemma newIdem_mul (r : T χ) : newIdem χ i * r = |
| oval χ i r • socle χ (.inl i) + etail χ i r • socle χ (.inr i) + |
| ehead χ i r • newIdem χ i := by |
| apply TrivSqZeroExt.ext |
| · apply Prod.ext |
| · change (0 : C) * r.fst.1 = _ |
| change 0 * r.fst.1 = (oval χ i r) • (0 : C) + (etail χ i r) • 0 + (ehead χ i r) • 0 |
| simp |
| · change (Pi.single i 1 : ι → DualNumber k) * r.fst.2 = |
| (oval χ i r) • (0 : ι → DualNumber k) + |
| (etail χ i r) • (Pi.single i (DualNumber.eps : DualNumber k) : ι → DualNumber k) + |
| (ehead χ i r) • (Pi.single i (1 : DualNumber k) : ι → DualNumber k) |
| ext j : 1 |
| by_cases hj : j = i |
| · subst j |
| apply TrivSqZeroExt.ext <;> |
| simp [etail, ehead] |
| · apply TrivSqZeroExt.ext <;> |
| simp [hj] |
| · change (newAugmentation (Pi.single i 1 : ι → DualNumber k) * |
| SimpleSum.value (TriangularConstruction.offValue r.snd) + |
| SimpleSum.rightCharacter χ (MulOpposite.op r.fst.1) * 0) = _ |
| change (newAugmentation (Pi.single i 1 : ι → DualNumber k) * |
| SimpleSum.value (TriangularConstruction.offValue r.snd) + |
| SimpleSum.rightCharacter χ (MulOpposite.op r.fst.1) * 0) = |
| (oval χ i r) • (Pi.single i (1 : k) : ι → k) + (etail χ i r) • 0 + (ehead χ i r) • 0 |
| ext j |
| by_cases hj : j = i |
| · subst j |
| simp [oval, newAugmentation] |
| · simp [oval, newAugmentation, hj] |
|
|
| lemma field_module_injective (V : ModuleCat.{u} k) : Injective V where |
| factors {X Y} g f hf := by |
| obtain ⟨s, hs⟩ := f.hom.exists_leftInverse_of_injective |
| (LinearMap.ker_eq_bot.mpr ((ModuleCat.mono_iff_injective f).mp hf)) |
| refine ⟨ModuleCat.ofHom (g.hom.comp s), ?_⟩ |
| apply ModuleCat.hom_ext |
| change g.hom.comp (s.comp f.hom) = g.hom |
| rw [hs, LinearMap.comp_id] |
|
|
| abbrev coindField := (ModuleCat.coextendScalars (algebraMap k (T χ))).obj (ModuleCat.of k k) |
|
|
| instance coindField_injective : Injective (coindField χ) := |
| (ModuleCat.restrictCoextendScalarsAdj (algebraMap k (T χ))).map_injective _ |
| (field_module_injective (ModuleCat.of k k)) |
|
|
| omit [DecidableEq ι] in |
| lemma coindField_smul_eval |
| (h : (ModuleCat.restrictScalars (algebraMap k (T χ))).obj |
| (ModuleCat.of (T χ) (T χ)) →ₗ[k] k) (a : k) (r : T χ) : |
| h (a • r) = a • h r := by |
| have scalar_eq := h.map_smul a r |
| change h (algebraMap k (T χ) a * r) = a • h r at scalar_eq |
| calc |
| h (a • r) = h (algebraMap k (T χ) a * r) := congrArg h (Algebra.smul_def a r) |
| _ = a • h r := scalar_eq |
|
|
| omit [DecidableEq ι] in |
| lemma coindField_add_eval |
| (h : (ModuleCat.restrictScalars (algebraMap k (T χ))).obj |
| (ModuleCat.of (T χ) (T χ)) →ₗ[k] k) (r s : T χ) : |
| h (r+s) = h r + h s := h.map_add r s |
|
|
| def newInjectiveRetraction : coindField χ ⟶ ModuleCat.of (T χ) (NewInjective χ i) := |
| ModuleCat.ofHom { |
| toFun := fun h => (h (socle χ (.inl i)), h (socle χ (.inr i)), h (newIdem χ i)) |
| map_add' := fun _ _ => rfl |
| map_smul' := fun r h => by |
| apply NewInjective.ext |
| · change h (socle χ (.inl i) * r) = cval χ i r * h (socle χ (.inl i)) |
| rw [socle_mul, coindField_smul_eval] |
| rfl |
| · change h (socle χ (.inr i) * r) = ehead χ i r * h (socle χ (.inr i)) |
| rw [socle_mul, coindField_smul_eval] |
| rfl |
| · change h (newIdem χ i * r) = oval χ i r * h (socle χ (.inl i)) + |
| etail χ i r * h (socle χ (.inr i)) + ehead χ i r * h (newIdem χ i) |
| rw [newIdem_mul, coindField_add_eval, coindField_add_eval, coindField_smul_eval, |
| coindField_smul_eval, coindField_smul_eval] |
| rfl } |
|
|
| end |
|
|
| open CategoryTheory |
| universe u |
| variable {k C ι : Type u} [Field k] [Ring C] [Algebra k C] |
| variable (χ : C →ₐ[k] (ι → k)) (i : ι) [DecidableEq ι] |
|
|
| def lastCoordinate : (ModuleCat.restrictScalars (algebraMap k (T χ))).obj |
| (ModuleCat.of (T χ) (NewInjective χ i)) ⟶ ModuleCat.of k k := ModuleCat.ofHom |
| (X := (ModuleCat.restrictScalars (algebraMap k (T χ))).obj |
| (ModuleCat.of (T χ) (NewInjective χ i))) (Y := ModuleCat.of k k) |
| (show (ModuleCat.restrictScalars (algebraMap k (T χ))).obj |
| (ModuleCat.of (T χ) (NewInjective χ i)) →ₗ[k] k from { |
| toFun := fun x => (NewInjective.value χ i x).2.2 |
| map_add' := fun _ _ => rfl |
| map_smul' := fun a x => by |
| change (NewInjective.act χ i ((algebraMap k (T χ)) a) x).2.2 = a * x.2.2 |
| change (0 * x.1 + 0 * x.2.1 + a * x.2.2) = a * x.2.2 |
| simp }) |
|
|
| def newInjectiveInclusion : ModuleCat.of (T χ) (NewInjective χ i) ⟶ coindField χ := |
| ModuleCat.RestrictionCoextensionAdj.HomEquiv.fromRestriction _ (lastCoordinate χ i) |
|
|
| omit [DecidableEq ι] in |
| lemma newInjectiveInclusion_apply (x : NewInjective χ i) (r : T χ) : |
| (newInjectiveInclusion χ i x) r = (NewInjective.act χ i r x).2.2 := by |
| change (ModuleCat.RestrictionCoextensionAdj.HomEquiv.fromRestriction _ |
| (lastCoordinate χ i)).hom x r = _ |
| rw [ModuleCat.RestrictionCoextensionAdj.HomEquiv.fromRestriction_hom_apply_apply] |
| change (r • x : NewInjective χ i).2.2 = _ |
| rfl |
|
|
| lemma newInjectiveInclusion_retraction : |
| newInjectiveInclusion χ i ≫ newInjectiveRetraction χ i = 𝟙 _ := by |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro x |
| change (newInjectiveRetraction χ i (newInjectiveInclusion χ i x)) = x |
| apply NewInjective.ext (χ := χ) (i := i) |
| · change (newInjectiveInclusion χ i x) (socle χ (.inl i)) = x.1 |
| rw [newInjectiveInclusion_apply] |
| change (NewInjective.act χ i (socle χ (.inl i)) x).2.2 = x.1 |
| change (Pi.single (M := fun _ : ι => k) i 1) i * x.1 + 0 * x.2.1 + 0 * x.2.2 = x.1 |
| simp |
| · change (newInjectiveInclusion χ i x) (socle χ (.inr i)) = x.2.1 |
| rw [newInjectiveInclusion_apply] |
| change (NewInjective.act χ i (socle χ (.inr i)) x).2.2 = x.2.1 |
| simp [NewInjective.act, NewInjective.value, oval, etail, ehead, socle, TriangularConstruction.offValue, SimpleSum.value] |
| try exact Or.inl rfl |
| · change (newInjectiveInclusion χ i x) (newIdem χ i) = x.2.2 |
| rw [newInjectiveInclusion_apply] |
| change (NewInjective.act χ i (newIdem χ i) x).2.2 = x.2.2 |
| simp [NewInjective.act, NewInjective.value, oval, etail, ehead, newIdem, TriangularConstruction.offValue, SimpleSum.value] |
| exact Or.inl rfl |
|
|
| lemma newInjective_injective : Injective (ModuleCat.of (T χ) (NewInjective χ i)) := by |
| exact (show CategoryTheory.Retract (ModuleCat.of (T χ) (NewInjective χ i)) |
| (coindField χ) from ⟨newInjectiveInclusion χ i, newInjectiveRetraction χ i, |
| newInjectiveInclusion_retraction χ i⟩).injective |
|
|
| end LittleFinitistic.CummingsConstruction |
|
|
| end |
|
|
| end OAI |
|
|
