| import OAI.Algebra.Finitistic.Dimension.PDMinimalData |
| import Mathlib.CategoryTheory.Abelian.Projective.Resolution |
| import Mathlib.Algebra.Homology.ShortComplex.ModuleCat |
|
|
| namespace OAI |
| noncomputable section |
| namespace LittleFinitistic.PDMinimal |
| open CategoryTheory CategoryTheory.Limits |
| universe u |
| variable {k A : Type u} [Field k] [Ring A] [Algebra k A] [FiniteDimensional k A] |
| variable (M : FiniteModule k A) |
|
|
| / |
| def complex : ChainComplex (ModuleCat A) ℕ := ChainComplex.of |
| (fun n => (term M n).cat) |
| (fun n => ModuleCat.ofHom (differential M n)) |
| (fun n => ModuleCat.hom_ext (differential_comp M n)) |
|
|
| @[simp] theorem complex_d (n : ℕ) : |
| (complex M).d (n+1) n = ModuleCat.ofHom (differential M n) := by |
| exact ChainComplex.of_d (fun n => (term M n).cat) |
| (fun n => ModuleCat.ofHom (differential M n)) n |
| |
| instance term_projective (n : ℕ) : Projective ((complex M).X n) := by |
| exact (IsProjective.iff_projective _).mp (cover (syzygy M n)).projective |
| |
| lemma complex_exactAt (n : ℕ) : (complex M).ExactAt (n+1) := by |
| rw [HomologicalComplex.exactAt_iff' _ (n+2) (n+1) n (by simp) (by simp)] |
| apply (ShortComplex.moduleCat_exact_iff_range_eq_ker _).mpr |
| change LinearMap.range ((complex M).d (n+2) (n+1)).hom = |
| LinearMap.ker ((complex M).d (n+1) n).hom |
| rw [complex_d, complex_d] |
| exact exact_differential M n |
| |
| lemma complex_d_augment : (complex M).d 1 0 ≫ ModuleCat.ofHom (augment M 0) = 0 := by |
| rw [complex_d] |
| exact ModuleCat.hom_ext (augment_differential M 0) |
| |
| lemma complex_exact_augment : |
| (ShortComplex.mk _ _ (complex_d_augment M)).Exact := by |
| apply (ShortComplex.moduleCat_exact_iff_range_eq_ker _).mpr |
| change LinearMap.range ((complex M).d 1 0).hom = LinearMap.ker (augment M 0) |
| rw [complex_d] |
| exact range_differential M 0 |
| |
| instance augment_epi : Epi (ModuleCat.ofHom (augment M 0)) := |
| (ModuleCat.epi_iff_surjective _).mpr (cover _).surjective |
| |
| / |
| def resolution : ProjectiveResolution M.cat where |
| complex := complex M |
| π := (ChainComplex.toSingle₀Equiv _ _).symm ⟨ModuleCat.ofHom (augment M 0), |
| complex_d_augment M⟩ |
| quasiIso := ⟨fun n => by |
| cases n with |
| | zero => |
| rw [ChainComplex.quasiIsoAt₀_iff, ShortComplex.quasiIso_iff_of_zeros'] |
| · |
| refine (ShortComplex.exact_and_epi_g_iff_of_iso ?_).2 |
| ⟨complex_exact_augment M, augment_epi M⟩ |
| exact ShortComplex.isoMk (Iso.refl _) (Iso.refl _) (Iso.refl _) |
| (by erw [Category.id_comp]) |
| (by erw [Category.id_comp]) |
| all_goals rfl |
| | succ n => |
| rw [quasiIsoAt_iff_exactAt'] |
| · exact complex_exactAt M n |
| · apply ChainComplex.exactAt_succ_single_obj⟩ |
| end LittleFinitistic.PDMinimal |
|
|
| end |
|
|
| end OAI |
|
|
