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d7f9a33 | 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 | import OAI.Algebra.AuslanderReiten.Basic
import OAI.Algebra.AuslanderReiten.Complete.CompleteResolutionTail
import Mathlib.CategoryTheory.Abelian.Projective.Ext
namespace OAI
open CategoryTheory
namespace ArExplicit
universe u
variable (R : Type u) [Ring R]
(M N : Type u) [AddCommGroup M] [Module R M]
[AddCommGroup N] [Module R N]
theorem positiveExtVanish_iff_resolution
(P : ProjectiveResolution (ModuleCat.of R M)) :
PositiveExtVanish R M N ↔
∀ (n : ℕ) (f : P.complex.X (n + 1) ⟶ ModuleCat.of R N),
P.complex.d (n + 2) (n + 1) ≫ f = 0 →
∃ g : P.complex.X n ⟶ ModuleCat.of R N,
P.complex.d (n + 1) n ≫ g = f := by
constructor
· intro h n f hf
have := h (n + 1) (Nat.zero_lt_succ n)
exact (P.extMk_eq_zero_iff f (n + 2) rfl hf n rfl).mp
(Subsingleton.elim _ _)
· intro h n hn
obtain ⟨p, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (Nat.ne_of_gt hn)
have hz : ∀ a : CategoryTheory.Abelian.Ext
(ModuleCat.of R M) (ModuleCat.of R N) (p + 1), a = 0 := by
intro a
obtain ⟨f, hf, rfl⟩ := P.extMk_surjective a (p + 2) rfl
exact (P.extMk_eq_zero_iff f (p + 2) rfl hf p rfl).mpr (h p f hf)
exact ⟨fun a b => (hz a).trans (hz b).symm⟩
theorem positiveExtVanish_iff_completeResolution
(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)
(e : M ≃ₗ[R] CompleteCokernel P) :
PositiveExtVanish R M N ↔
∀ (n : ℕ) (f : P.X ((n + 1 : ℕ) : ℤ) →ₗ[R] N),
f.comp (P.d ((n + 2 : ℕ) : ℤ) ((n + 1 : ℕ) : ℤ)).hom = 0 →
∃ g : P.X (n : ℤ) →ₗ[R] N,
f = g.comp (P.d ((n + 1 : ℕ) : ℤ) (n : ℤ)).hom := by
let Q := transportProjectiveResolution (completeCokernelResolution P hproj hex)
e.symm.toModuleIso
refine (positiveExtVanish_iff_resolution R M N Q).trans ⟨?_, ?_⟩
· intro h n f hf
have hfc : Q.complex.d (n + 2) (n + 1) ≫ ModuleCat.ofHom f = 0 := by
change (nonnegativeTail P).d (n + 1 + 1) (n + 1) ≫ ModuleCat.ofHom f = 0
rw [nonnegativeTail_d]
exact ModuleCat.hom_ext hf
obtain ⟨g, hg⟩ := h n (ModuleCat.ofHom f) hfc
refine ⟨g.hom, ?_⟩
change (nonnegativeTail P).d (n + 1) n ≫ g = ModuleCat.ofHom f at hg
rw [nonnegativeTail_d] at hg
exact (congrArg ModuleCat.Hom.hom hg).symm
· intro h n f hf
change (nonnegativeTail P).d (n + 1 + 1) (n + 1) ≫ f = 0 at hf
rw [nonnegativeTail_d] at hf
obtain ⟨g, hg⟩ := h n f.hom (congrArg ModuleCat.Hom.hom hf)
refine ⟨ModuleCat.ofHom g, ?_⟩
change (nonnegativeTail P).d (n + 1) n ≫ ModuleCat.ofHom g = f
rw [nonnegativeTail_d]
exact ModuleCat.hom_ext hg.symm
theorem positiveExtVanish_of_completeResolution
(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)
(hdual : ∀ (n : ℤ) (f : P.X n →ₗ[R] N),
f.comp (P.d (n + 1) n).hom = 0 →
∃ g : P.X (n - 1) →ₗ[R] N, f = g.comp (P.d n (n - 1)).hom)
(e : M ≃ₗ[R] CompleteCokernel P) : PositiveExtVanish R M N := by
let Q := transportProjectiveResolution (completeCokernelResolution P hproj hex)
e.symm.toModuleIso
apply (positiveExtVanish_iff_resolution R M N Q).2
intro n f hf
change (nonnegativeTail P).d (n + 1 + 1) (n + 1) ≫ f = 0 at hf
rw [nonnegativeTail_d] at hf
have aux (i j l : ℤ) (hi : i = j + 1) (hl : l = j - 1)
(f : P.X j →ₗ[R] N) (hf : f.comp (P.d i j).hom = 0) :
∃ g : P.X l →ₗ[R] N, f = g.comp (P.d j l).hom := by
subst i
subst l
exact hdual j f hf
obtain ⟨g, hg⟩ := aux ((n + 1 + 1 : ℕ) : ℤ) ((n + 1 : ℕ) : ℤ) (n : ℤ)
(by omega) (by omega) f.hom (congrArg ModuleCat.Hom.hom hf)
refine ⟨ModuleCat.ofHom g, ?_⟩
change (nonnegativeTail P).d (n + 1) n ≫ ModuleCat.ofHom g = f
rw [nonnegativeTail_d]
exact ModuleCat.hom_ext hg.symm
theorem GorensteinProjective.positiveExt_regular
(h : GorensteinProjective R M) : PositiveExtVanish R M R := by
obtain ⟨P, hp, he, hd, ⟨e⟩⟩ := h
exact positiveExtVanish_of_completeResolution R M R P (fun n => (hp n).2) he hd e
end ArExplicit
end OAI
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