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2c7d975 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 | 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
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