| import OAI.Algebra.AuslanderReiten.Resolutions.Resolution |
| import OAI.Algebra.AuslanderReiten.Parameters |
| 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.Resolution |
|
|
| variable {k : Type*} [Field k] [CharP k 2] (q : k) |
|
|
| local notation "C" => BaseAlgebra k q |
|
|
| abbrev Simple := (F q) ⧸ LinearMap.range (dU q) |
|
|
| def objects : ℕ → ModuleCat C |
| | 0 => ModuleCat.of C (F q) |
| | _ + 1 => ModuleCat.of C (E q) |
|
|
| def differential : (n : ℕ) → objects q (n + 1) ⟶ objects q n |
| | 0 => ModuleCat.ofHom (dU q) |
| | n + 1 => ModuleCat.ofHom (dEll q (q ^ n)) |
|
|
| theorem differential_comp (n : ℕ) : differential q (n + 1) ≫ differential q n = 0 := by |
| cases n with |
| | zero => |
| apply ModuleCat.hom_ext |
| change (dU q).comp (dEll q (q ^ 0)) = 0 |
| simpa only [pow_zero] using dU_comp_dEll_one q |
| | succ n => |
| apply ModuleCat.hom_ext |
| change (dEll q (q ^ n)).comp (dEll q (q ^ (n + 1))) = 0 |
| simpa only [pow_succ'] using dEll_comp_successor q (q ^ n) |
|
|
| def complex : ChainComplex (ModuleCat C) ℕ := |
| ChainComplex.of (objects q) (differential q) (differential_comp q) |
|
|
| instance complex_projective (n : ℕ) : CategoryTheory.Projective ((complex q).X n) := by |
| cases n <;> dsimp [complex, objects] <;> infer_instance |
|
|
| instance complex_scalar (n : ℕ) : Module k ((complex q).X n) := by |
| cases n with |
| | zero => exact (inferInstance : Module k (F q)) |
| | succ n => exact (inferInstance : Module k (E q)) |
|
|
| instance complex_finite (n : ℕ) : Module.Finite k ((complex q).X n) := by |
| cases n with |
| | zero => exact (inferInstance : Module.Finite k (F q)) |
| | succ n => exact (inferInstance : Module.Finite k (E q)) |
|
|
| def NoPositivePowerOne : Prop := ∀ m : ℕ, 0 < m → 1 + q ^ m ≠ 0 |
|
|
| omit [CharP k 2] in |
| theorem denominator_power_ne (h : NoPositivePowerOne q) (n : ℕ) : |
| 1 + q ^ 2 * q ^ n ≠ 0 := by |
| rw [← pow_add] |
| exact h (2 + n) (by omega) |
|
|
| theorem differential_exact (h : NoPositivePowerOne q) (n : ℕ) : |
| Function.Exact (differential q (n + 1)) (differential q n) := by |
| cases n with |
| | zero => |
| change Function.Exact (dEll q (q ^ 0)) (dU q) |
| simpa only [pow_zero] using exact_dU q (by simpa using h 1 (by decide)) (h 2 (by decide)) |
| | succ n => |
| change Function.Exact (dEll q (q ^ (n + 1))) (dEll q (q ^ n)) |
| simpa only [pow_succ'] using exact_dEll q (q ^ n) |
| (denominator_power_ne q h n) |
| (by simpa [pow_succ'] using denominator_power_ne q h (n + 1)) |
|
|
| theorem complex_exactAt_succ (h : NoPositivePowerOne q) (n : ℕ) : |
| (complex q).ExactAt (n + 1) := by |
| rw [HomologicalComplex.exactAt_iff' _ (n + 1 + 1) (n + 1) n (by simp) (by simp)] |
| apply (ShortComplex.ShortExact.moduleCat_exact_iff_function_exact _).mpr |
| simpa [complex, HomologicalComplex.sc', HomologicalComplex.shortComplexFunctor', |
| ChainComplex.of_d] using! differential_exact q h n |
|
|
| def augmentation : (complex q).X 0 ⟶ ModuleCat.of C (Simple q) := |
| ModuleCat.ofHom (LinearMap.range (dU q)).mkQ |
|
|
| instance augmentation_epi : Epi (augmentation q) := |
| (ModuleCat.epi_iff_surjective _).mpr (Submodule.mkQ_surjective _) |
|
|
| theorem differential_augmentation : (complex q).d 1 0 ≫ augmentation q = 0 := by |
| apply ModuleCat.hom_ext |
| change (LinearMap.range (dU q)).mkQ.comp (dU q) = 0 |
| exact LinearMap.range_mkQ_comp (dU q) |
|
|
| def toSimple : complex q ⟶ (ChainComplex.single₀ (ModuleCat C)).obj (ModuleCat.of C (Simple q)) := |
| (ChainComplex.toSingle₀Equiv _ _).symm ⟨augmentation q, differential_augmentation q⟩ |
|
|
| @[simp] theorem toSimple_zero : (toSimple q).f 0 = augmentation q := |
| ChainComplex.toSingle₀Equiv_symm_apply_f_zero _ _ |
|
|
| theorem augmentation_exact : |
| (ShortComplex.mk ((complex q).d 1 0) (augmentation q) (differential_augmentation q)).Exact := by |
| apply (ShortComplex.ShortExact.moduleCat_exact_iff_function_exact _).mpr |
| change Function.Exact (dU q) (LinearMap.range (dU q)).mkQ |
| exact LinearMap.exact_map_mkQ_range (dU q) |
|
|
| theorem toSimple_quasiIso (h : NoPositivePowerOne q) : QuasiIso (toSimple q) := by |
| constructor |
| intro n |
| cases n with |
| | zero => |
| rw [ChainComplex.quasiIsoAt₀_iff, ShortComplex.quasiIso_iff_of_zeros'] |
| · change (ShortComplex.mk ((complex q).d 1 0) ((toSimple q).f 0) _).Exact ∧ |
| Epi ((toSimple q).f 0) |
| simpa only [toSimple_zero, ChainComplex.single₀_obj_zero] using And.intro (augmentation_exact q) (augmentation_epi q) |
| · rfl |
| · rfl |
| · rfl |
| | succ n => |
| rw [quasiIsoAt_iff_exactAt'] |
| · exact complex_exactAt_succ q h n |
| · exact ChainComplex.exactAt_succ_single_obj _ _ |
|
|
| def projectiveResolution (h : NoPositivePowerOne q) : |
| CategoryTheory.ProjectiveResolution (ModuleCat.of C (Simple q)) where |
| complex := complex q |
| π := toSimple q |
| quasiIso := toSimple_quasiIso q h |
|
|
| theorem baseField_noPositivePowerOne : NoPositivePowerOne ArExplicit.q := by |
| intro m hm |
| simpa [add_comm] using ArExplicit.q_pow_add_one_ne_zero m hm |
|
|
| def baseFieldProjectiveResolution : |
| CategoryTheory.ProjectiveResolution |
| (ModuleCat.of (BaseAlgebra BaseField ArExplicit.q) (Simple ArExplicit.q)) := |
| projectiveResolution ArExplicit.q baseField_noPositivePowerOne |
|
|
| end ArExplicit.Resolution |
|
|
| end |
|
|
| end OAI |
|
|
