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b797a5b | 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 | 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
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