openAI-math-dataset / lean /OAI /Algebra /AuslanderReiten /Triangular /TriangularComparisonTransport.lean
| import OAI.Algebra.AuslanderReiten.Triangular.TriangularComparison | |
| namespace OAI | |
| noncomputable section | |
| namespace ArExplicit.TriangularInduction | |
| open CategoryTheory CochainComplex.HomComplex | |
| universe u | |
| variable {k R F : Type u} [Field k] [Ring R] [Algebra k R] | |
| variable [AddCommGroup F] [Module k F] [Module R F] [Module Rᵐᵒᵖ F] | |
| variable [SMulCommClass k R F] [SMulCommClass R Rᵐᵒᵖ F] | |
| variable [IsScalarTower k R F] [IsScalarTower k Rᵐᵒᵖ F] [SMulCommClass k Rᵐᵒᵖ F] | |
| attribute [local instance] moduleCatBaseModule moduleCatBaseTower | |
| variable {P : CochainComplex (ModuleCat.{u} R) ℤ} | |
| variable (v : P ⟶ tensorCochain (k := k) (F := F) P) (n : ℤ) | |
| def comparisonSourceEquiv : | |
| (CohomologyClass P P n × CohomologyClass P P n) ≃+ | |
| (CohomologyClass (eCochain (k := k) (F := F) P) | |
| (eCochain (k := k) (F := F) P) n × | |
| CohomologyClass (jCochain (k := k) (F := F) P) | |
| (jCochain (k := k) (F := F) P) n) := | |
| AddEquiv.prodCongr (fullHomCohomologyEquiv (eFunctor k R F) P P n) | |
| (fullHomCohomologyEquiv (jFunctor k R F) P P n) | |
| @[simp] theorem comparisonSourceEquiv_apply (a b : CohomologyClass P P n) : | |
| comparisonSourceEquiv (F := F) (P := P) n (a,b) = | |
| (fullHomCohomologyEquiv (eFunctor k R F) P P n a, | |
| fullHomCohomologyEquiv (jFunctor k R F) P P n b) := rfl | |
| theorem coneComparison_injective_of_tensor | |
| (h : Function.Injective (tensorComparison v n)) : | |
| Function.Injective (coneComparison (crossChainMap v) n) := by | |
| intro x y hxy | |
| obtain ⟨a,rfl⟩ := (comparisonSourceEquiv (F := F) (P := P) n).surjective x | |
| obtain ⟨b,rfl⟩ := (comparisonSourceEquiv (F := F) (P := P) n).surjective y | |
| apply congrArg | |
| apply h | |
| apply (crossCohomologyEquiv P P n).injective | |
| rw [cross_tensorComparison, cross_tensorComparison] | |
| exact hxy | |
| theorem coneComparison_surjective_of_tensor | |
| (h : Function.Surjective (tensorComparison v n)) : | |
| Function.Surjective (coneComparison (crossChainMap v) n) := by | |
| intro y | |
| obtain ⟨x,hx⟩ := h ((crossCohomologyEquiv P P n).symm y) | |
| refine ⟨comparisonSourceEquiv (F := F) (P := P) n x, ?_⟩ | |
| rcases x with ⟨a,b⟩ | |
| rw [comparisonSourceEquiv_apply, ← cross_tensorComparison, hx, AddEquiv.apply_symm_apply] | |
| theorem coneComparison_nonzero_kernel_of_tensor | |
| (h : ∃ x : CohomologyClass P P n × CohomologyClass P P n, | |
| x ≠ 0 ∧ tensorComparison v n x = 0) : | |
| ∃ x : CohomologyClass (eCochain (k := k) (F := F) P) | |
| (eCochain (k := k) (F := F) P) n × | |
| CohomologyClass (jCochain (k := k) (F := F) P) | |
| (jCochain (k := k) (F := F) P) n, | |
| x ≠ 0 ∧ coneComparison (crossChainMap v) n x = 0 := by | |
| obtain ⟨x,hx,h0⟩ := h | |
| refine ⟨comparisonSourceEquiv (F := F) (P := P) n x, ?_, ?_⟩ | |
| · exact fun he => hx ((comparisonSourceEquiv (F := F) (P := P) n).map_eq_zero_iff.mp he) | |
| · rcases x with ⟨a,b⟩ | |
| rw [comparisonSourceEquiv_apply, ← cross_tensorComparison, h0, map_zero] | |
| end ArExplicit.TriangularInduction | |
| end | |
| end OAI | |