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ae00908 | 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 | import OAI.Algebra.Finitistic.Selected.SelectedOrdinaryAlgebra
import OAI.Algebra.Finitistic.Ordinary.OrdinaryXDerivedTensor
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits
namespace LittleFinitistic.SelectedOrdinaryResolution
open SelectedOrdinaryAlgebra
/-- The one fixed genuine D-bimodule resolution Q_X used by the source. -/
abbrev complex := OrdinarySimulation.XResolution.complex SelectedRectification.bimodule a length
abbrev target := OrdinarySimulation.XResolution.ordinaryXObject SelectedRectification.bimodule a length
abbrev augmentation := OrdinarySimulation.XResolution.augmentationToX
SelectedRectification.bimodule a length
theorem augmentation_quasiIso : QuasiIso augmentation :=
OrdinarySimulation.XResolution.augmentationToX_quasiIso SelectedRectification.bimodule a length
theorem right_projective (n : ℤ) : Module.Projective Dᵐᵒᵖ
(BimoduleCoefficients ℂ D D (complex.X n)) :=
OrdinarySimulation.XResolution.right_projective SelectedRectification.bimodule a length n
theorem bounded (n : ℤ) (hn : n < -(length : ℤ) ∨ 0 < n) : IsZero (complex.X n) :=
OrdinarySimulation.XResolution.vanishes SelectedRectification.bimodule a length n hn
local instance : HasDerivedCategory.{1} (ModuleCat.{0} D) :=
HasDerivedCategory.standard (ModuleCat.{0} D)
/-- Tensor via the actual bounded right-projective Q_X, with the left D-action retained. -/
abbrev derived := OrdinarySimulation.XResolution.derivedTensor SelectedRectification.bimodule a length
abbrev comparison := OrdinarySimulation.XResolution.derivedTensorComparison
SelectedRectification.bimodule a length
end LittleFinitistic.SelectedOrdinaryResolution
end
end OAI
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