| import Mathlib.Algebra.Category.ModuleCat.Projective |
| import Mathlib.Algebra.Homology.ShortComplex.ModuleCat |
| import Mathlib.CategoryTheory.Abelian.Projective.Resolution |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory |
| namespace ArExplicit |
| universe u |
| variable {R : Type u} [Ring R] |
|
|
| def nonnegativeTail (P : ChainComplex (ModuleCat.{u} R) ℤ) : |
| ChainComplex (ModuleCat.{u} R) ℕ := |
| ChainComplex.of (fun n => P.X (n : ℤ)) |
| (fun n => P.d ((n + 1 : ℕ) : ℤ) (n : ℤ)) |
| (fun _n => P.d_comp_d _ _ _) |
|
|
| @[simp] |
| theorem nonnegativeTail_d (P : ChainComplex (ModuleCat.{u} R) ℤ) (n : ℕ) : |
| (nonnegativeTail P).d (n + 1) n = P.d ((n + 1 : ℕ) : ℤ) (n : ℤ) := by |
| simp [nonnegativeTail] |
|
|
| abbrev CompleteCokernel (P : ChainComplex (ModuleCat.{u} R) ℤ) := |
| P.X 0 ⧸ LinearMap.range (P.d 1 0).hom |
|
|
| def nonnegativeTailAugmentation (P : ChainComplex (ModuleCat.{u} R) ℤ) : |
| nonnegativeTail P ⟶ (ChainComplex.single₀ (ModuleCat.{u} R)).obj |
| (ModuleCat.of R (CompleteCokernel P)) := |
| (ChainComplex.toSingle₀Equiv _ _).symm |
| ⟨ModuleCat.ofHom (LinearMap.range (P.d 1 0).hom).mkQ, by |
| rw [nonnegativeTail_d] |
| apply ModuleCat.hom_ext |
| exact LinearMap.range_mkQ_comp _⟩ |
|
|
| @[simp] |
| theorem nonnegativeTailAugmentation_f_zero (P : ChainComplex (ModuleCat.{u} R) ℤ) : |
| (nonnegativeTailAugmentation P).f 0 = |
| ModuleCat.ofHom (LinearMap.range (P.d 1 0).hom).mkQ := by |
| apply ChainComplex.toSingle₀Equiv_symm_apply_f_zero |
|
|
| theorem nonnegativeTail_exactAt (P : ChainComplex (ModuleCat.{u} R) ℤ) |
| (hex : ∀ n : ℤ, LinearMap.range (P.d (n + 1) n).hom = |
| LinearMap.ker (P.d n (n - 1)).hom) (n : ℕ) : |
| (nonnegativeTail P).ExactAt (n + 1) := by |
| rw [HomologicalComplex.exactAt_iff' _ (n + 1 + 1) (n + 1) n (by simp) (by simp)] |
| rw [ShortComplex.moduleCat_exact_iff_range_eq_ker] |
| change LinearMap.range ((nonnegativeTail P).d (n + 1 + 1) (n + 1)).hom = |
| LinearMap.ker ((nonnegativeTail P).d (n + 1) n).hom |
| rw [nonnegativeTail_d, nonnegativeTail_d] |
| have aux (i j l : ℤ) (hi : i = j + 1) (hl : l = j - 1) : |
| LinearMap.range (P.d i j).hom = LinearMap.ker (P.d j l).hom := by |
| subst i |
| subst l |
| exact hex j |
| exact aux _ _ _ (by omega) (by omega) |
|
|
| def completeCokernelResolution (P : ChainComplex (ModuleCat.{u} R) ℤ) |
| (hproj : ∀ n : ℤ, Module.Projective R (P.X n)) |
| (hex : ∀ n : ℤ, LinearMap.range (P.d (n + 1) n).hom = |
| LinearMap.ker (P.d n (n - 1)).hom) : |
| ProjectiveResolution (ModuleCat.of R (CompleteCokernel P)) where |
| complex := nonnegativeTail P |
| projective n := by |
| let := hproj (n : ℤ) |
| exact (P.X (n : ℤ)).projective_of_categoryTheory_projective |
| π := nonnegativeTailAugmentation P |
| quasiIso := ⟨fun n => by |
| cases n with |
| | zero => |
| rw [ChainComplex.quasiIsoAt₀_iff, ShortComplex.quasiIso_iff_of_zeros'] |
| · constructor |
| · rw [ShortComplex.moduleCat_exact_iff_range_eq_ker] |
| change LinearMap.range ((nonnegativeTail P).d 1 0).hom = |
| LinearMap.ker ((nonnegativeTailAugmentation P).f 0).hom |
| rw [nonnegativeTail_d, nonnegativeTailAugmentation_f_zero] |
| exact (Submodule.ker_mkQ _).symm |
| · rw [ModuleCat.epi_iff_surjective] |
| change Function.Surjective (LinearMap.range (P.d 1 0).hom).mkQ |
| exact Submodule.mkQ_surjective _ |
| · change (nonnegativeTail P).d 0 0 = 0 |
| exact (nonnegativeTail P).shape _ _ (by simp) |
| · rfl |
| · rfl |
| | succ n => |
| rw [quasiIsoAt_iff_exactAt'] |
| · exact nonnegativeTail_exactAt P hex n |
| · exact ChainComplex.exactAt_succ_single_obj _ _⟩ |
|
|
| def transportProjectiveResolution {X Y : ModuleCat.{u} R} |
| (Q : ProjectiveResolution X) (e : X ≅ Y) : ProjectiveResolution Y where |
| complex := Q.complex |
| projective := Q.projective |
| π := Q.π ≫ (ChainComplex.single₀ (ModuleCat.{u} R)).map e.hom |
|
|
| end ArExplicit |
|
|
| end |
|
|
| end OAI |
|
|
