File size: 2,517 Bytes
b797a5b | 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 | 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
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