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216fb31 | 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 | import OAI.Algebra.Finitistic.Localization.BoundedHomotopy
import Mathlib.Algebra.Category.ModuleCat.Projective
import Mathlib.Algebra.Category.ModuleCat.Biproducts
import Mathlib.RingTheory.Finiteness.Prod
import Mathlib.RingTheory.Finiteness.Basic
namespace OAI
noncomputable section
open CategoryTheory Limits
namespace LittleFinitistic.ProjectiveComplexes
universe u
variable (B : Type u) [Ring B]
/-- Right B-modules are left modules over the opposite ring. -/
abbrev RightModule := ModuleCat.{u} Bᵐᵒᵖ
def finiteProjective : ObjectProperty (RightModule B) :=
fun M => Module.Finite Bᵐᵒᵖ M ∧ Module.Projective Bᵐᵒᵖ M
instance : (finiteProjective B).IsClosedUnderIsomorphisms where
of_iso e h := by
let := h.1
let := h.2
exact ⟨Module.Finite.equiv e.toLinearEquiv,
Module.Projective.of_equiv e.toLinearEquiv⟩
instance : (finiteProjective B).ContainsZero where
exists_zero := ⟨ModuleCat.of Bᵐᵒᵖ PUnit, ModuleCat.isZero_of_subsingleton _,
by constructor <;> infer_instance⟩
instance : (finiteProjective B).IsClosedUnderBinaryProducts where
limitsOfShape_le := by
rintro X ⟨p⟩
let M := p.diag.obj (.mk .left)
let N := p.diag.obj (.mk .right)
have hM := p.prop_diag_obj (.mk .left)
have hN := p.prop_diag_obj (.mk .right)
let := hM.1
let := hM.2
let := hN.1
let := hN.2
have hp : finiteProjective B (ModuleCat.of Bᵐᵒᵖ (M × N)) :=
⟨inferInstance, inferInstance⟩
apply (finiteProjective B).prop_of_iso _ hp
exact IsLimit.conePointUniqueUpToIso (ModuleCat.binaryProductLimitCone M N).isLimit
((IsLimit.postcomposeHomEquiv (diagramIsoPair p.diag) _).2 p.isLimit)
instance : (finiteProjective B).IsClosedUnderFiniteProducts := .mk'
/-- The additive category of finitely generated projective right B-modules. -/
abbrev Proj := (finiteProjective B).FullSubcategory
instance : HasBinaryBiproducts (Proj B) := .of_hasBinaryProducts
/-- The actual category of bounded complexes of finitely generated projectives. -/
abbrev Complex := BoundedHomotopy.Complex (Proj B)
/-- Section 3's K = Kᵇ(proj-B), with homotopy classes of chain maps. -/
abbrev K := BoundedHomotopy.K (Proj B)
example : HasBinaryBiproducts (Proj B) := inferInstance
example : IsTriangulated (K B) := inferInstance
/-- The passage from bounded projective complexes to their homotopy category. -/
abbrev quotient : Complex B ⥤ K B := BoundedHomotopy.quotient (Proj B)
end LittleFinitistic.ProjectiveComplexes
end
end OAI
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