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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 | import OAI.Algebra.AuslanderReiten.Triangular.TriangularProjective
import OAI.Algebra.AuslanderReiten.Complete.CompleteResolution
namespace OAI
namespace ArExplicit.TriangularInduction
variable {k R F : Type*} [CommRing k] [Ring R] [Algebra k R]
variable [AddCommGroup F] [Module k F] [Module R F] [Module Rᵐᵒᵖ F]
variable [SMulCommClass k R F] [SMulCommClass R Rᵐᵒᵖ F]
variable [IsScalarTower k R F] [IsScalarTower k Rᵐᵒᵖ F] [SMulCommClass k Rᵐᵒᵖ F]
variable {K L M U V : Type*}
variable [AddCommGroup K] [Module k K] [Module R K] [IsScalarTower k R K]
variable [AddCommGroup L] [Module k L] [Module R L] [IsScalarTower k R L]
variable [AddCommGroup M] [Module k M] [Module R M] [IsScalarTower k R M]
variable [AddCommGroup U] [Module k U] [Module R U] [IsScalarTower k R U]
variable [AddCommGroup V] [Module k V] [Module R V] [IsScalarTower k R V]
theorem eHomLift (e : K →ₗ[R] L) (d : L →ₗ[R] M) (h : HomExactAt e d)
[Module.Finite R U] [Module.Projective R U]
{ψ : BalancedTensor k R F U →ₗ[R] V}
(g : TriangularE k R F L →ₗ[Triangular R F] TriangularModule k R F U V ψ)
(hg : g.comp (eMap e) = 0) :
∃ G : TriangularE k R F M →ₗ[Triangular R F] TriangularModule k R F U V ψ,
G.comp (eMap d) = g := by
have ha : (TriangularModule.topMap g).comp e = 0 := by
ext x
have hx := congrArg TriangularModule.top (DFunLike.congr_fun hg ⟨x, 0⟩)
simpa only [LinearMap.comp_apply, LinearMap.zero_apply, TriangularModule.top_zero,
eMap, eHom, TriangularModule.hom, LinearMap.coe_mk, AddHom.coe_mk,
map_zero, TriangularModule.topMap_apply] using hx
obtain ⟨b, hb⟩ := h.projective (TriangularModule.topMap g) ha
refine ⟨eHom b, ?_⟩
apply eHomEquiv.injective
apply LinearMap.ext
intro x
exact DFunLike.congr_fun hb x
omit [Module k K] [IsScalarTower k R K] [Module k L] [IsScalarTower k R L]
[Module k M] [IsScalarTower k R M] in
theorem jHomLift (e : K →ₗ[R] L) (d : L →ₗ[R] M) (h : HomExactAt e d)
[Module.Finite R V] [Module.Projective R V]
{ψ : BalancedTensor k R F U →ₗ[R] V}
(g : TriangularJ k R F L →ₗ[Triangular R F] TriangularModule k R F U V ψ)
(hg : g.comp (jMap e) = 0) :
∃ G : TriangularJ k R F M →ₗ[Triangular R F] TriangularModule k R F U V ψ,
G.comp (jMap d) = g := by
have ha : (TriangularModule.bottomMap g).comp e = 0 := by
ext x
exact congrArg TriangularModule.bottom (DFunLike.congr_fun hg ⟨0, x⟩)
obtain ⟨b, hb⟩ := h.projective (TriangularModule.bottomMap g) ha
refine ⟨jHom b, ?_⟩
apply jHomEquiv.injective
apply LinearMap.ext
intro x
exact DFunLike.congr_fun hb x
omit [IsScalarTower k R F] [SMulCommClass k Rᵐᵒᵖ F] in
theorem regular_decomposition (a : Triangular R F) :
eRegularInclusion (eRegularProjection (k := k) a) +
jRegularInclusion (jRegularProjection (k := k) a) = a := by
ext <;> simp [eRegularInclusion, eRegularProjection, jRegularInclusion, jRegularProjection]
theorem eHomExact (e : K →ₗ[R] L) (d : L →ₗ[R] M) (h : HomExactAt e d) :
HomExactAt (eMap (k := k) (F := F) e) (eMap d) := by
intro g hg
let g₁ := (eRegularProjection (k := k)).comp g
have hg₁ : g₁.comp (eMap e) = 0 := by
dsimp [g₁]
rw [LinearMap.comp_assoc, hg, LinearMap.comp_zero]
obtain ⟨G, hG⟩ := eHomLift e d h g₁ hg₁
refine ⟨eRegularInclusion.comp G, ?_⟩
have hg₂ : (jRegularProjection (k := k)).comp g = 0 := hom_E_J_eq_zero _
apply LinearMap.ext
intro x
have hGx := DFunLike.congr_fun hG x
change G (eMap d x) = g₁ x at hGx
change eRegularInclusion (G (eMap d x)) = g x
rw [hGx]
have hsplit := regular_decomposition (k := k) (g x)
have hzero := DFunLike.congr_fun hg₂ x
change jRegularProjection (g x) = 0 at hzero
simpa only [hzero, map_zero, add_zero, g₁, LinearMap.comp_apply] using hsplit
omit [Module k K] [IsScalarTower k R K] [Module k L] [IsScalarTower k R L]
[Module k M] [IsScalarTower k R M] in
theorem jHomExact [Module.Finite R F] [Module.Projective R F]
(e : K →ₗ[R] L) (d : L →ₗ[R] M) (h : HomExactAt e d) :
HomExactAt (jMap (k := k) (F := F) e) (jMap d) := by
intro g hg
let : Module.Finite R (BalancedTensor k R F R) :=
Module.Finite.equiv (BalancedTensor.runitLeft (S := R)).symm
let : Module.Projective R (BalancedTensor k R F R) :=
Module.Projective.of_equiv (BalancedTensor.runitLeft (S := R)).symm
let g₁ := (eRegularProjection (k := k)).comp g
let g₂ := (jRegularProjection (k := k)).comp g
have hg₁ : g₁.comp (jMap e) = 0 := by
dsimp [g₁]
rw [LinearMap.comp_assoc, hg, LinearMap.comp_zero]
have hg₂ : g₂.comp (jMap e) = 0 := by
dsimp [g₂]
rw [LinearMap.comp_assoc, hg, LinearMap.comp_zero]
obtain ⟨G₁, hG₁⟩ := jHomLift e d h g₁ hg₁
obtain ⟨G₂, hG₂⟩ := jHomLift e d h g₂ hg₂
refine ⟨eRegularInclusion.comp G₁ + jRegularInclusion.comp G₂, ?_⟩
apply LinearMap.ext
intro x
change eRegularInclusion (G₁ (jMap d x)) + jRegularInclusion (G₂ (jMap d x)) = g x
have h₁ : G₁ (jMap d x) = g₁ x := DFunLike.congr_fun hG₁ x
have h₂ : G₂ (jMap d x) = g₂ x := DFunLike.congr_fun hG₂ x
rw [h₁, h₂]
exact regular_decomposition (k := k) (g x)
end ArExplicit.TriangularInduction
end OAI
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