| 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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|
