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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 | import OAI.Algebra.AuslanderReiten.HomComplex.FiniteFullHomZero
import OAI.Algebra.AuslanderReiten.HomComplex.PositiveFullHomProducts
import OAI.Algebra.AuslanderReiten.HomComplex.FullHomDegreeProducts
import OAI.Algebra.AuslanderReiten.Ext.ExtIsoCongr
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Abelian
open CochainComplex CochainComplex.HomComplex
open scoped ModuleCat.Algebra
namespace ArExplicit.FiniteLeftModule
universe u
variable {k R : Type u} [Field k] [Ring R] [Algebra k R] [Module.Finite k R] [CharP k 2]
variable [HasDerivedCategory (ModuleCat.{u} R)]
open CompleteComparison
variable (τ : FrobeniusTrace k R) (M : FiniteLeftModule k R)
def fullPositiveModuleExtEquiv (n : ℕ) :
CohomologyClass (M.completeCochain τ) (M.completeCochain τ) ((n+1:ℕ):ℤ) ≃ₗ[k]
Ext M.carrier M.carrier (n+1) :=
(fullPositiveExtEquiv (k := k) _ _ (M.completeCochain_totallyAcyclic τ).to_chain
(M.completeCochain_totallyAcyclic τ).to_chain.toExactProjectiveComplex n).trans
(selfExtIso (k := k) (M.fullCokernelEquiv τ).toModuleIso (n+1))
theorem fullPositiveModuleExtEquiv_product (n m : ℕ)
(x : CohomologyClass (M.completeCochain τ) (M.completeCochain τ) ((n+1:ℕ):ℤ))
(y : CohomologyClass (M.completeCochain τ) (M.completeCochain τ) ((m+1:ℕ):ℤ)) :
fullPositiveModuleExtEquiv τ M (n+1+m)
(TateKoszul.degreeEquiv (k := k) (CohomologyClass (M.completeCochain τ) (M.completeCochain τ))
(show ((n+1:ℕ):ℤ)+((m+1:ℕ):ℤ)=((n+1+m+1:ℕ):ℤ) by omega) (fullHomProduct x y)) =
(fullPositiveModuleExtEquiv τ M n x).comp (fullPositiveModuleExtEquiv τ M m y) (by omega) := by
obtain ⟨x,rfl⟩ := CohomologyClass.mk_surjective x
obtain ⟨y,rfl⟩ := CohomologyClass.mk_surjective y
rw [fullHomProduct_degree_mk _ x y (ResolutionProducts.natCocycleProduct x y) rfl]
change selfExtIso _ _ (fullPositiveExtEquiv _ _ _ _ _ _) =
(selfExtIso _ _ (fullPositiveExtEquiv _ _ _ _ _ _)).comp
(selfExtIso _ _ (fullPositiveExtEquiv _ _ _ _ _ _)) _
rw [← selfExtIso_comp, fullPositiveExtEquiv_product]
end ArExplicit.FiniteLeftModule
end
end OAI
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