openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /LocalAlgebra /FiniteFreeResolutionExt.lean
| import Mathlib.Algebra.Category.ModuleCat.Ext.HasExt | |
| import Mathlib.Algebra.Homology.ShortComplex.ModuleCat | |
| import Mathlib.CategoryTheory.Abelian.Projective.Ext | |
| import Mathlib.LinearAlgebra.FreeModule.Finite.Basic | |
| namespace OAI | |
| noncomputable section | |
| universe w v u | |
| namespace PiExponentSiegelAux.W31 | |
| open Module | |
| open CategoryTheory CategoryTheory.Limits CategoryTheory.Abelian | |
| open HomologicalComplex | |
| section General | |
| variable {C : Type u} [Category.{v} C] [Abelian C] | |
| def projectiveResolutionOfExact (P : ChainComplex C ℕ) (M : C) | |
| (ε : P.X 0 ⟶ M) (hε : P.d 1 0 ≫ ε = 0) | |
| (hzero : (ShortComplex.mk (P.d 1 0) ε hε).Exact) | |
| (hepi : Epi ε) (hexact : ∀ n : ℕ, P.ExactAt (n + 1)) | |
| (hproj : ∀ n : ℕ, CategoryTheory.Projective (P.X n)) : | |
| CategoryTheory.ProjectiveResolution M where | |
| complex := P | |
| projective := hproj | |
| π := (ChainComplex.toSingle₀Equiv P M).symm ⟨ε, hε⟩ | |
| quasiIso := ⟨fun n => by | |
| cases n with | |
| | zero => | |
| rw [ChainComplex.quasiIsoAt₀_iff, | |
| ShortComplex.quasiIso_iff_of_zeros' _ (by simp; rfl) (by simp; rfl) | |
| (by simp; rfl)] | |
| simpa [ChainComplex.toSingle₀Equiv] using! And.intro hzero hepi | |
| | succ n => | |
| rw [quasiIsoAt_iff_exactAt' (hL := ChainComplex.exactAt_succ_single_obj ..)] | |
| exact hexact n⟩ | |
| theorem projectiveResolutionOfExact_complex (P : ChainComplex C ℕ) (M : C) | |
| (ε : P.X 0 ⟶ M) (hε : P.d 1 0 ≫ ε = 0) | |
| (hzero : (ShortComplex.mk (P.d 1 0) ε hε).Exact) | |
| (hepi : Epi ε) (hexact : ∀ n : ℕ, P.ExactAt (n + 1)) | |
| (hproj : ∀ n : ℕ, CategoryTheory.Projective (P.X n)) : | |
| (projectiveResolutionOfExact P M ε hε hzero hepi hexact hproj).complex = P := rfl | |
| variable [HasExt.{w} C] | |
| def resolutionExtAddEquiv (M Y : C) (P : ProjectiveResolution M) (n : ℕ) : | |
| Ext M Y n ≃+ | |
| CochainComplex.HomComplex.CohomologyClass P.cochainComplex | |
| ((CochainComplex.singleFunctor C 0).obj Y) n := | |
| P.extAddEquivCohomologyClass | |
| theorem ext_subsingleton_iff_cocycles_are_boundaries | |
| (M Y : C) (P : ProjectiveResolution M) (n : ℕ) : | |
| Subsingleton (Ext M Y (n + 1)) ↔ | |
| ∀ (f : P.complex.X (n + 1) ⟶ Y), | |
| P.complex.d (n + 2) (n + 1) ≫ f = 0 → | |
| ∃ g : P.complex.X n ⟶ Y, P.complex.d (n + 1) n ≫ g = f := by | |
| constructor | |
| · intro h f hf | |
| exact (P.extMk_eq_zero_iff f (n + 2) rfl hf n rfl).mp | |
| (h.elim _ 0) | |
| · intro h | |
| apply subsingleton_of_forall_eq 0 | |
| intro e | |
| obtain ⟨f, hf, rfl⟩ := P.extMk_surjective e (n + 2) rfl | |
| exact (P.extMk_eq_zero_iff f (n + 2) rfl hf n rfl).mpr (h f hf) | |
| theorem ext_subsingleton_of_isZero_resolution_term | |
| (M Y : C) (P : ProjectiveResolution M) (n : ℕ) | |
| (hterm : IsZero (P.complex.X n)) : Subsingleton (Ext M Y n) := by | |
| apply subsingleton_of_forall_eq 0 | |
| intro e | |
| obtain ⟨f, hf, rfl⟩ := P.extMk_surjective e (n + 1) rfl | |
| have hfzero : f = 0 := hterm.eq_of_src f 0 | |
| subst f | |
| exact P.extMk_zero (n + 1) rfl | |
| end General | |
| variable {R : Type u} [CommRing R] | |
| theorem module_exactAt_succ_of_range_eq_ker (P : ChainComplex (ModuleCat.{u} R) ℕ) | |
| (n : ℕ) | |
| (h : LinearMap.range (P.d (n + 2) (n + 1)).hom = | |
| LinearMap.ker (P.d (n + 1) n).hom) : P.ExactAt (n + 1) := by | |
| rw [HomologicalComplex.exactAt_iff' _ (n + 2) (n + 1) n (by simp) (by simp)] | |
| rw [ShortComplex.moduleCat_exact_iff_range_eq_ker] | |
| exact h | |
| def finiteFreeResolutionOfExact (P : ChainComplex (ModuleCat.{u} R) ℕ) | |
| (M : ModuleCat.{u} R) (ε : P.X 0 ⟶ M) (hε : P.d 1 0 ≫ ε = 0) | |
| (hzero : LinearMap.range (P.d 1 0).hom = LinearMap.ker ε.hom) | |
| (hepi : Function.Surjective ε.hom) | |
| (hexact : ∀ n : ℕ, LinearMap.range (P.d (n + 2) (n + 1)).hom = | |
| LinearMap.ker (P.d (n + 1) n).hom) | |
| (rank : ℕ → ℕ) (basis : ∀ n : ℕ, Basis (Fin (rank n)) R (P.X n)) : | |
| ProjectiveResolution M := | |
| projectiveResolutionOfExact P M ε hε | |
| ((ShortComplex.moduleCat_exact_iff_range_eq_ker _).mpr hzero) | |
| ((ModuleCat.epi_iff_surjective ε).mpr hepi) | |
| (fun n => module_exactAt_succ_of_range_eq_ker P n (hexact n)) | |
| (fun n => ModuleCat.projective_of_free (basis n)) | |
| def moduleFreeResolutionOfExact (P : ChainComplex (ModuleCat.{u} R) ℕ) | |
| (M : ModuleCat.{u} R) (ε : P.X 0 ⟶ M) (hε : P.d 1 0 ≫ ε = 0) | |
| (hzero : LinearMap.range (P.d 1 0).hom = LinearMap.ker ε.hom) | |
| (hepi : Function.Surjective ε.hom) | |
| (hexact : ∀ n : ℕ, LinearMap.range (P.d (n + 2) (n + 1)).hom = | |
| LinearMap.ker (P.d (n + 1) n).hom) | |
| (hfree : ∀ n : ℕ, Module.Free R (P.X n)) : ProjectiveResolution M := | |
| projectiveResolutionOfExact P M ε hε | |
| ((ShortComplex.moduleCat_exact_iff_range_eq_ker _).mpr hzero) | |
| ((ModuleCat.epi_iff_surjective ε).mpr hepi) | |
| (fun n => module_exactAt_succ_of_range_eq_ker P n (hexact n)) | |
| (fun n => by | |
| let := hfree n | |
| exact ModuleCat.projective_of_free (Module.Free.chooseBasis R (P.X n))) | |
| end PiExponentSiegelAux.W31 | |
| end | |
| end OAI | |