| import Mathlib.Algebra.Homology.HomologicalComplex | |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.RegularSequenceResolution | |
| namespace OAI | |
| namespace PiExponentSiegelAux.W30 | |
| open CategoryTheory | |
| universe u | |
| variable {R : Type u} [CommRing R] | |
| lemma coneBottom_comp {X Y Z : Type*} [AddCommGroup X] [AddCommGroup Y] | |
| [AddCommGroup Z] [Module R X] [Module R Y] [Module R Z] | |
| (f : X →ₗ[R] Y) (g : Y →ₗ[R] Z) (r : R) (hgf : g.comp f = 0) : | |
| (coneBottom g r).comp (coneDifferential f g r) = 0 := by | |
| apply LinearMap.ext | |
| intro xy | |
| have hz : g (f xy.1) = 0 := LinearMap.congr_fun hgf xy.1 | |
| change g (f xy.1 + r • xy.2) + r • (-g xy.2) = 0 | |
| simp [map_add, map_smul, hz] | |
| def scalarConeObject (C : ChainComplex (ModuleCat.{u} R) ℕ) : ℕ → ModuleCat.{u} R | |
| | 0 => C.X 0 | |
| | n + 1 => ModuleCat.of R (C.X (n + 1) × C.X n) | |
| def scalarConeMap (C : ChainComplex (ModuleCat.{u} R) ℕ) (r : R) : | |
| ∀ n, scalarConeObject C (n + 1) ⟶ scalarConeObject C n | |
| | 0 => ModuleCat.ofHom (coneBottom (C.d 1 0).hom r) | |
| | n + 1 => ModuleCat.ofHom | |
| (coneDifferential (C.d (n + 2) (n + 1)).hom (C.d (n + 1) n).hom r) | |
| lemma chain_linear_comp_zero (C : ChainComplex (ModuleCat.{u} R) ℕ) (n : ℕ) : | |
| (C.d (n + 1) n).hom.comp (C.d (n + 2) (n + 1)).hom = 0 := by | |
| simpa only [ModuleCat.hom_comp, ModuleCat.hom_zero] using | |
| congrArg (fun f => f.hom) (C.d_comp_d (n + 2) (n + 1) n) | |
| theorem scalarConeMap_comp (C : ChainComplex (ModuleCat.{u} R) ℕ) (r : R) (n : ℕ) : | |
| scalarConeMap C r (n + 1) ≫ scalarConeMap C r n = 0 := by | |
| cases n with | |
| | zero => | |
| apply ModuleCat.hom_ext | |
| exact coneBottom_comp _ _ r (chain_linear_comp_zero C 0) | |
| | succ n => | |
| apply ModuleCat.hom_ext | |
| exact coneDifferential_comp _ _ _ r | |
| (chain_linear_comp_zero C (n + 1)) (chain_linear_comp_zero C n) | |
| def scalarConeComplex (C : ChainComplex (ModuleCat.{u} R) ℕ) (r : R) : | |
| ChainComplex (ModuleCat.{u} R) ℕ := | |
| ChainComplex.of (scalarConeObject C) (scalarConeMap C r) (scalarConeMap_comp C r) | |
| theorem scalarCone_interior_exact (C : ChainComplex (ModuleCat.{u} R) ℕ) (r : R) | |
| (n : ℕ) | |
| (hnext : LinearMap.range (C.d (n + 3) (n + 2)).hom = | |
| LinearMap.ker (C.d (n + 2) (n + 1)).hom) | |
| (hprev : LinearMap.range (C.d (n + 2) (n + 1)).hom = | |
| LinearMap.ker (C.d (n + 1) n).hom) : | |
| LinearMap.range (scalarConeMap C r (n + 2)).hom = | |
| LinearMap.ker (scalarConeMap C r (n + 1)).hom := | |
| coneDifferential_exact _ _ _ r hnext hprev | |
| end PiExponentSiegelAux.W30 | |
| end OAI | |