| import Mathlib.RingTheory.Flat.Basic |
| import Mathlib.RingTheory.Length |
| import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic |
| import Mathlib.RingTheory.TensorProduct.Quotient |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace PiExponentJets.W25 |
|
|
| open TensorProduct |
|
|
| variable {A B : Type*} [CommRing A] [CommRing B] [Algebra A B] |
| [IsLocalRing A] [IsLocalRing B] |
|
|
| theorem simple_baseChange_of_map_maximalIdeal |
| (hmax : (IsLocalRing.maximalIdeal A).map (algebraMap A B) = |
| IsLocalRing.maximalIdeal B) |
| {M : Type*} [AddCommGroup M] [Module A M] [IsSimpleModule A M] : |
| IsSimpleModule B (B ⊗[A] M) := by |
| obtain ⟨J, hJ, ⟨e⟩⟩ := |
| (isSimpleModule_iff_quot_maximal (R := A) (M := M)).mp inferInstance |
| have hJmax : J = IsLocalRing.maximalIdeal A := IsLocalRing.eq_maximalIdeal hJ |
| subst J |
| have : IsSimpleModule B (B ⧸ IsLocalRing.maximalIdeal B) := |
| isSimpleModule_iff_isCoatom.mpr |
| (Ideal.isMaximal_def.mp (inferInstance : (IsLocalRing.maximalIdeal B).IsMaximal)) |
| let eB : (B ⊗[A] M) ≃ₗ[B] (B ⧸ IsLocalRing.maximalIdeal B) := |
| (e.baseChange A B M (A ⧸ IsLocalRing.maximalIdeal A)).trans |
| ((Algebra.TensorProduct.quotIdealMapEquivTensorQuot B |
| (IsLocalRing.maximalIdeal A)).toLinearEquiv.symm.trans |
| (Submodule.quotEquivOfEq _ _ hmax)) |
| exact IsSimpleModule.congr eB |
|
|
| theorem length_baseChange_of_flat_local [Module.Flat A B] |
| (hmax : (IsLocalRing.maximalIdeal A).map (algebraMap A B) = |
| IsLocalRing.maximalIdeal B) |
| {M : Type*} [AddCommGroup M] [Module A M] (hM : IsFiniteLength A M) : |
| Module.length B (B ⊗[A] M) = Module.length A M := by |
| induction hM with |
| | of_subsingleton => |
| simp only [Module.length_eq_zero] |
| | @of_simple_quotient M _ _ N hsimple hN ih => |
| have : IsSimpleModule B (B ⊗[A] (M ⧸ N)) := |
| simple_baseChange_of_map_maximalIdeal hmax |
| have hexact : Function.Exact N.subtype N.mkQ := by |
| rw [LinearMap.exact_iff, Submodule.range_subtype, Submodule.ker_mkQ] |
| have hinj : Function.Injective (N.subtype.baseChange B) := |
| Module.Flat.lTensor_preserves_injective_linearMap N.subtype |
| (Submodule.subtype_injective N) |
| have hsurj : Function.Surjective (N.mkQ.baseChange B) := |
| LinearMap.lTensor_surjective B (Submodule.mkQ_surjective N) |
| have hbaseexact : Function.Exact (N.subtype.baseChange B) (N.mkQ.baseChange B) := |
| Module.Flat.lTensor_exact B hexact |
| have hA := Module.length_eq_add_of_exact N.subtype N.mkQ |
| (Submodule.subtype_injective N) (Submodule.mkQ_surjective N) hexact |
| have hB := Module.length_eq_add_of_exact (N.subtype.baseChange B) |
| (N.mkQ.baseChange B) hinj hsurj hbaseexact |
| calc |
| Module.length B (B ⊗[A] M) = |
| Module.length B (B ⊗[A] N) + Module.length B (B ⊗[A] (M ⧸ N)) := hB |
| _ = Module.length A N + Module.length A (M ⧸ N) := by |
| rw [ih, Module.length_eq_one B (B ⊗[A] (M ⧸ N)), |
| Module.length_eq_one A (M ⧸ N)] |
| _ = Module.length A M := hA.symm |
|
|
| theorem finiteLength_baseChange_of_flat_local [Module.Flat A B] |
| (hmax : (IsLocalRing.maximalIdeal A).map (algebraMap A B) = |
| IsLocalRing.maximalIdeal B) |
| {M : Type*} [AddCommGroup M] [Module A M] (hM : IsFiniteLength A M) : |
| IsFiniteLength B (B ⊗[A] M) := by |
| apply Module.length_ne_top_iff.mp |
| rw [length_baseChange_of_flat_local hmax hM] |
| exact Module.length_ne_top_iff.mpr hM |
|
|
| theorem length_quotient_map_of_flat_local [Module.Flat A B] |
| (hmax : (IsLocalRing.maximalIdeal A).map (algebraMap A B) = |
| IsLocalRing.maximalIdeal B) |
| (I : Ideal A) (hI : IsFiniteLength A (A ⧸ I)) : |
| Module.length B (B ⧸ I.map (algebraMap A B)) = Module.length A (A ⧸ I) := by |
| calc |
| Module.length B (B ⧸ I.map (algebraMap A B)) = |
| Module.length B (B ⊗[A] (A ⧸ I)) := |
| (Algebra.TensorProduct.quotIdealMapEquivTensorQuot B I).toLinearEquiv.length_eq |
| _ = Module.length A (A ⧸ I) := length_baseChange_of_flat_local hmax hI |
|
|
| end PiExponentJets.W25 |
|
|
| end |
|
|
| end OAI |
|
|
