openAI-math-dataset / lean /OAI /Algebra /AuslanderReiten /HomComplex /FullHomZeroRepresentative.lean
| import OAI.Algebra.AuslanderReiten.HomComplex.FullHomMapLinear | |
| import OAI.Algebra.AuslanderReiten.HomComplex.FullHomShiftedIdentity | |
| import OAI.Algebra.AuslanderReiten.HomComplex.FullHomHomotopyMaps | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CochainComplex.HomComplex | |
| namespace ArExplicit | |
| universe u v w | |
| variable {C : Type w} [Category.{v} C] [Preadditive C] | |
| variable {K L : CochainComplex C ℤ} | |
| def fullHomZeroRepresentative (x : CohomologyClass K L 0) : K ⟶ L := | |
| Cocycle.homOf (Classical.choose (CohomologyClass.mk_surjective x)) | |
| @[simp] theorem fullHomZeroRepresentative_class (x : CohomologyClass K L 0) : | |
| CohomologyClass.mk (Cocycle.ofHom (fullHomZeroRepresentative x)) = x := by | |
| rw [fullHomZeroRepresentative, Cocycle.ofHom_homOf_eq_self] | |
| exact Classical.choose_spec (CohomologyClass.mk_surjective x) | |
| variable (k : Type u) [CommRing k] [Linear k C] | |
| include k in | |
| theorem fullHomZero_class_eq_iff (f g : K ⟶ L) : | |
| CohomologyClass.mk (Cocycle.ofHom f) = CohomologyClass.mk (Cocycle.ofHom g) ↔ | |
| (HomotopyCategory.quotient C _).map f = (HomotopyCategory.quotient C _).map g := by | |
| constructor | |
| · intro h | |
| rw [← fullHomPrecomp_identity f, ← fullHomPrecomp_identity g] at h | |
| have hh := congrArg (fullHomShiftedHomotopyEquiv k K L 0) h | |
| rw [fullHomShiftedHomotopyEquiv_precomp, fullHomShiftedHomotopyEquiv_precomp] at hh | |
| change _ ≫ fullHomShiftedHomotopyEquiv k L L 0 (CompleteComparison.fullHomIdentity L) = | |
| _ ≫ fullHomShiftedHomotopyEquiv k L L 0 (CompleteComparison.fullHomIdentity L) at hh | |
| rw [fullHomShiftedHomotopyEquiv_identity] at hh | |
| exact (cancel_mono _).1 hh | |
| · intro h | |
| rw [← fullHomPrecomp_identity f, ← fullHomPrecomp_identity g] | |
| exact fullHomPrecomp_congr k 0 h _ | |
| end ArExplicit | |
| end | |
| end OAI | |