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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 | import OAI.Algebra.AuslanderReiten.HomComplex.FiniteFullHomExt
import OAI.Algebra.AuslanderReiten.HomComplex.TensorFullHomZero
import OAI.Algebra.AuslanderReiten.Ext.NativeExtZero
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Abelian
open CochainComplex CochainComplex.HomComplex
open scoped ModuleCat.Algebra
namespace ArExplicit.A
universe u
variable {k : Type u} [Field k] [CharP k 2] (q : k)
open CompleteComparison
variable [HasDerivedCategory (ModuleCat.{u} (A k q))]
def simpleExtZeroEquiv (hq1 : 1+q ≠ 0) : k ≃ₗ[k] Ext (Simple q) (Simple q) 0 :=
(simpleEndEquiv q hq1).symm.trans (nativeExtZeroEquiv (Simple q) (Simple q))
omit [CharP k 2] [HasDerivedCategory (ModuleCat.{u} (A k q))] in
@[simp]
theorem simpleExtZeroEquiv_one (hq1 : 1+q ≠ 0) :
simpleExtZeroEquiv q hq1 1 = Ext.mk₀ (𝟙 (Simple q)) := by
unfold simpleExtZeroEquiv
rw [LinearEquiv.trans_apply, nativeExtZeroEquiv_apply]
congr 1
apply ModuleCat.hom_ext
change (1:k) • (LinearMap.id : Simple q →ₗ[A k q] Simple q) = LinearMap.id
exact one_smul k _
omit [HasDerivedCategory (ModuleCat.{u} (A k q))] in
theorem fullHomZero_eq_scalar (hq1 : 1+q ≠ 0)
(x : CohomologyClass (simpleComplete q) (simpleComplete q) 0) :
x = fullHomZeroScalarEquiv q hq1 x • fullHomIdentity (simpleComplete q) := by
apply (fullHomZeroScalarEquiv q hq1).injective
rw [map_smul]
change _ = _ • fullHomZeroScalarEquiv q hq1 (CohomologyClass.mk (Cocycle.ofHom (𝟙 _)))
rw [fullHomZeroScalarEquiv_identity]
exact (mul_one _).symm
def fullNonnegativeExtEquiv (hq1 : 1+q ≠ 0) : (n : ℕ) →
CohomologyClass (simpleComplete q) (simpleComplete q) (n:ℤ) ≃ₗ[k] Ext (Simple q) (Simple q) n
| 0 => (fullHomZeroScalarEquiv q hq1).trans (simpleExtZeroEquiv q hq1)
| n+1 => FiniteLeftModule.fullPositiveModuleExtEquiv (AFrobeniusTrace k q) (simpleFiniteModule q) n
omit [HasDerivedCategory (ModuleCat.{u} (A k q))] in
@[simp]
theorem fullNonnegativeExtEquiv_identity (hq1 : 1+q ≠ 0) :
fullNonnegativeExtEquiv q hq1 0 (fullHomIdentity (simpleComplete q)) = Ext.mk₀ (𝟙 (Simple q)) := by
change simpleExtZeroEquiv q hq1 (fullHomZeroScalarEquiv q hq1
(CohomologyClass.mk (Cocycle.ofHom (𝟙 _)))) = _
rw [fullHomZeroScalarEquiv_identity, simpleExtZeroEquiv_one]
theorem fullNonnegativeExtEquiv_product_of_eq (hq1 : 1+q ≠ 0) (n m r : ℕ) (hr : n+m=r)
(x : CohomologyClass (simpleComplete q) (simpleComplete q) (n:ℤ))
(y : CohomologyClass (simpleComplete q) (simpleComplete q) (m:ℤ)) :
fullNonnegativeExtEquiv q hq1 r
(TateKoszul.degreeEquiv (k := k) (CohomologyClass (simpleComplete q) (simpleComplete q))
(show (n:ℤ)+(m:ℤ)=(r:ℤ) by omega) (fullHomProduct x y)) =
(fullNonnegativeExtEquiv q hq1 n x).comp (fullNonnegativeExtEquiv q hq1 m y) hr := by
cases n with
| zero =>
obtain rfl : r=m := by omega
rw [fullHomZero_eq_scalar q hq1 x]
simp only [fullHomProduct_smul_left, map_smul,
fullNonnegativeExtEquiv_identity, Ext.smul_comp, Ext.mk₀_id_comp]
erw [fullHomProduct_identity_left]
| succ n =>
cases m with
| zero =>
obtain rfl : r=n+1 := by omega
rw [fullHomZero_eq_scalar q hq1 y]
simp only [fullHomProduct_smul_right, map_smul,
fullNonnegativeExtEquiv_identity, Ext.comp_smul, Ext.comp_mk₀_id]
erw [fullHomProduct_identity_right]
| succ m =>
obtain rfl : r=n+1+m+1 := by omega
exact FiniteLeftModule.fullPositiveModuleExtEquiv_product (AFrobeniusTrace k q)
(simpleFiniteModule q) n m x y
theorem fullNonnegativeExtEquiv_product (hq1 : 1+q ≠ 0) (n m : ℕ)
(x : CohomologyClass (simpleComplete q) (simpleComplete q) (n:ℤ))
(y : CohomologyClass (simpleComplete q) (simpleComplete q) (m:ℤ)) :
fullNonnegativeExtEquiv q hq1 (n+m)
(TateKoszul.degreeEquiv (k := k) (CohomologyClass (simpleComplete q) (simpleComplete q))
(show (n:ℤ)+(m:ℤ)=((n+m:ℕ):ℤ) by omega) (fullHomProduct x y)) =
(fullNonnegativeExtEquiv q hq1 n x).comp (fullNonnegativeExtEquiv q hq1 m y) rfl :=
fullNonnegativeExtEquiv_product_of_eq q hq1 n m (n+m) rfl x y
end ArExplicit.A
end
end OAI
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