openAI-math-dataset / lean /OAI /GroupTheory /PolycyclicRecognition /Finiteness /CofiberResolution.lean
| import OAI.GroupTheory.PolycyclicRecognition.Algebra.RepCofiber | |
| namespace OAI | |
| /-! # Exactness of the cone on an injective map of resolved representations -/ | |
| noncomputable section | |
| namespace PolycyclicRecognition | |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex | |
| attribute [local instance 2000] Rep.hV2 | |
| variable {k G : Type} [CommRing k] [Group G] | |
| {A B : Rep k G} (P : ProjectiveResolution A) (Q : ProjectiveResolution B) | |
| (φ : P.complex ⟶ Q.complex) | |
| theorem repCofiber_boundary_of_lift_fst (i : ℕ) | |
| (x : (repCofiber φ).X (i + 1)) | |
| (hx : ((repCofiber φ).d (i + 1) i).hom x = 0) | |
| (a : P.complex.X (i + 1)) | |
| (ha : (P.complex.d (i + 1) i).hom a = (repCofiberFst φ i).hom x) : | |
| ∃ y : (repCofiber φ).X (i + 2), | |
| ((repCofiber φ).d (i + 2) (i + 1)).hom y = x := by | |
| have hsum := repCofiber_snd_d φ i x | |
| rw [hx, map_zero] at hsum | |
| have hcycle : (Q.complex.d (i + 1) i).hom | |
| ((repCofiberSnd φ (i + 1)).hom x + (φ.f (i + 1)).hom a) = 0 := by | |
| have hc := congrArg (fun f => f.hom a) (φ.comm (i + 1) i) | |
| change (Q.complex.d (i + 1) i).hom ((φ.f (i + 1)).hom a) = | |
| (φ.f i).hom ((P.complex.d (i + 1) i).hom a) at hc | |
| simp only [map_add] | |
| rw [hc, ha] | |
| exact (add_comm _ _).trans hsum.symm | |
| obtain ⟨b, hb⟩ := (rep_shortComplex_exact_iff _).mp (Q.exact_succ i) _ hcycle | |
| let y := (repCofiberInl φ (i + 1)).hom (-a) + | |
| (repCofiberInr φ (i + 2)).hom b | |
| refine ⟨y, repCofiber_ext φ i ?_ ?_⟩ | |
| · rw [repCofiber_fst_d] | |
| simp only [y, map_add, repCofiber_fst_inl, repCofiber_fst_inr, add_zero, | |
| map_neg, ha, neg_neg] | |
| · rw [repCofiber_snd_d] | |
| simp only [y, map_add, repCofiber_fst_inl, repCofiber_fst_inr, | |
| repCofiber_snd_inl, repCofiber_snd_inr, add_zero, map_neg, map_zero] | |
| rw [hb] | |
| abel | |
| theorem repCofiber_boundary_of_mono (f : A ⟶ B) [Mono f] | |
| (hφ : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f) (i : ℕ) | |
| (x : (repCofiber φ).X (i + 1)) | |
| (hx : ((repCofiber φ).d (i + 1) i).hom x = 0) : | |
| ∃ y : (repCofiber φ).X (i + 2), | |
| ((repCofiber φ).d (i + 2) (i + 1)).hom y = x := by | |
| have ha : ∃ a : P.complex.X (i + 1), | |
| (P.complex.d (i + 1) i).hom a = (repCofiberFst φ i).hom x := by | |
| cases i with | |
| | zero => | |
| have hsum := repCofiber_snd_d φ 0 x | |
| rw [hx, map_zero] at hsum | |
| have hzero : (P.π.f 0).hom ((repCofiberFst φ 0).hom x) = 0 := by | |
| apply ((Rep.mono_iff_injective f).mp inferInstance) | |
| erw [map_zero] | |
| have hc := congrArg (fun u => u.hom ((repCofiberFst φ 0).hom x)) hφ | |
| change (Q.π.f 0).hom ((φ.f 0).hom ((repCofiberFst φ 0).hom x)) = | |
| f.hom ((P.π.f 0).hom ((repCofiberFst φ 0).hom x)) at hc | |
| have hd := congrArg (fun u => u.hom ((repCofiberSnd φ 1).hom x)) | |
| Q.complex_d_comp_π_f_zero | |
| change (Q.π.f 0).hom ((Q.complex.d 1 0).hom | |
| ((repCofiberSnd φ 1).hom x)) = 0 at hd | |
| rw [← hc] | |
| have hz := congrArg (Q.π.f 0).hom hsum.symm | |
| simpa only [map_add, hd, add_zero, map_zero] using! hz | |
| exact (rep_shortComplex_exact_iff _).mp P.exact₀ _ hzero | |
| | succ i => | |
| have hz := repCofiber_fst_d φ i x | |
| rw [hx, map_zero] at hz | |
| have hzero := neg_eq_zero.mp hz.symm | |
| exact (rep_shortComplex_exact_iff _).mp (P.exact_succ i) _ hzero | |
| obtain ⟨a, ha⟩ := ha | |
| exact repCofiber_boundary_of_lift_fst P Q φ i x hx a ha | |
| theorem repCofiber_exactAt_of_mono (f : A ⟶ B) [Mono f] | |
| (hφ : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f) (i : ℕ) : | |
| (repCofiber φ).ExactAt (i + 1) := by | |
| rw [HomologicalComplex.exactAt_iff' _ (i + 2) (i + 1) i (by simp) (by simp)] | |
| exact (rep_shortComplex_exact_iff _).mpr | |
| (repCofiber_boundary_of_mono P Q φ f hφ i) | |
| end PolycyclicRecognition | |
| end | |
| end OAI | |