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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 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | 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
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