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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 | import OAI.Algebra.AuslanderReiten.Stable.StableExt
import OAI.Algebra.AuslanderReiten.Stable.SourceStableShift
namespace OAI
noncomputable section
namespace ArExplicit
open CategoryTheory CompleteComparison
open CochainComplex CochainComplex.HomComplex
universe u
variable {k R : Type u} [CommRing k] [Ring R] [Algebra k R]
variable {K L : CochainComplex (ModuleCat.{u} R) ℤ}
include k
theorem positiveExtVanish_of_fullHom
(hK : TotallyAcyclicComplex (chainOfCochain K))
(hL : ExactProjectiveComplex (chainOfCochain L))
(h : ∀ n : ℕ, 0 < n → Subsingleton (CohomologyClass K L (n : ℤ))) :
PositiveExtVanish R (Cokernel (chainOfCochain K) 0) (Cokernel (chainOfCochain L) 0) := by
apply positiveExtVanish_of_stableCokernels hK
intro n
let := h (n + 1) (Nat.zero_lt_succ n)
exact (fullHomSourceStableLinearEquiv (k := k) K L ((n + 1 : ℕ) : ℤ) hK hL).symm.injective.subsingleton
theorem not_projective_of_fullHom_nontrivial
(hK : TotallyAcyclicComplex (chainOfCochain K))
(h : Nontrivial (CohomologyClass K K 0)) :
¬ Module.Projective R (Cokernel (chainOfCochain K) 0) := by
intro hp
let := hK.finite 0
let := hp
have : Subsingleton (StableHom R (Cokernel (chainOfCochain K) 0)
(Cokernel (chainOfCochain K) 0)) := ⟨fun a b =>
(StableHom.eq_zero_of_projective_source a).trans (StableHom.eq_zero_of_projective_source b).symm⟩
have : Subsingleton (CohomologyClass K K 0) :=
(fullHomStableZeroLinearEquiv (k := k) K K hK hK.toExactProjectiveComplex).injective.subsingleton
obtain ⟨a, b, hab⟩ := h.exists_pair_ne
exact hab (Subsingleton.elim a b)
end ArExplicit
end
end OAI
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