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e3e0a0d | 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 Mathlib
import OAI.RingTheory.Multiplicity.HomologyLengthDevissage
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits HomologicalComplex
namespace Lech
universe u
variable {R : Type u} [CommRing R]
lemma additive_length_pi (T : ModuleCat.{u} R ⥤ ModuleCat.{u} R) [T.Additive]
{ι : Type} [Fintype ι] (M : ι → ModuleCat.{u} R) :
Module.length R (T.obj (ModuleCat.of R (∀ i,M i))) =
∑ i,Module.length R (T.obj (M i)) := by
let e := T.mapIso (ModuleCat.biproductIsoPi M).symm ≪≫
T.mapBiproduct M ≪≫ ModuleCat.biproductIsoPi (fun i : ι => T.obj (M i))
exact e.toLinearEquiv.length_eq.trans (Module.length_pi_of_fintype _ _)
lemma additive_length_finite_free [Nontrivial R] (T : ModuleCat.{u} R ⥤ ModuleCat.{u} R) [T.Additive]
(M : ModuleCat.{u} R) [Module.Free R M] [Module.Finite R M] :
Module.length R (T.obj M) = Module.finrank R M •
Module.length R (T.obj (ModuleCat.of R R)) := by
classical
let b := Module.Free.chooseBasis R M
let := Module.Free.ChooseBasisIndex.fintype R M
let e := (b.reindex (Fintype.equivFin _)).equivFun.toModuleIso
rw [(T.mapIso e).toLinearEquiv.length_eq]
rw [additive_length_pi T (fun _ : Fin (Fintype.card (Module.Free.ChooseBasisIndex R M)) => ModuleCat.of R R)]
simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin]
congr 1
exact (Module.finrank_eq_card_basis b).symm
end Lech
end
end OAI
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