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feat(RingTheory/Algebraic): tower law for Module.finrank over domains #41614

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5 changes: 4 additions & 1 deletion Mathlib/LinearAlgebra/Dimension/Free.lean
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Expand Up @@ -61,7 +61,10 @@ theorem rank_mul_rank (A : Type v) [AddCommMonoid A]
convert! lift_rank_mul_lift_rank F K A <;> rw [lift_id]

/-- Tower law: if `A` is a `K`-module and `K` is an extension of `F` then
$\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$. -/
$\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$.
See `Module.finrank_mul_finrank'` for a variant over a tower of domains that assumes the rings are
module-finite rather than the modules being free. -/
theorem Module.finrank_mul_finrank : finrank F K * finrank K A = finrank F A := by
simp_rw [finrank]
rw [← toNat_lift.{w} (Module.rank F K), ← toNat_lift.{v} (Module.rank K A), ← toNat_mul,
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22 changes: 19 additions & 3 deletions Mathlib/RingTheory/Algebraic/Integral.lean
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Expand Up @@ -5,12 +5,11 @@ Authors: Johan Commelin
-/
module

public import Mathlib.Algebra.Ring.Hom.InjSurj
public import Mathlib.LinearAlgebra.Dimension.Localization
public import Mathlib.RingTheory.Algebraic.Basic
public import Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic
public import Mathlib.RingTheory.Localization.BaseChange

import Mathlib.RingTheory.Polynomial.Subring
public import Mathlib.RingTheory.Polynomial.Subring

/-!
# Algebraic elements and integral elements
Expand Down Expand Up @@ -561,6 +560,23 @@ theorem rank_fractionRing [IsDomain S] :
Module.rank (FractionRing R) (FractionRing S) = Module.rank R S :=
rank_of_isFractionRing ..

attribute [local instance] FractionRing.liftAlgebra in
/-- Tower law for `Module.finrank` in a tower of domains `R → S → B` that is module-finite at each
step. This is a variant of `Module.finrank_mul_finrank` that assumes the rings are domains (and
module-finite) instead of the modules being free. -/
theorem _root_.Module.finrank_mul_finrank' (T : Type*) [CommRing T] [IsDomain T] [Algebra S T]
[Algebra R T] [IsScalarTower R S T]
[FaithfulSMul S T] [Module.Finite R S] [Module.Finite S T] :

@tb65536 tb65536 Jul 11, 2026

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You can replace the finiteness assumptions with [Algebra.IsAlgebraic S T] (since algebraic is transitive).

@tb65536 tb65536 Jul 11, 2026

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Actually, I haven't golfed this yet, but you can remove a lot of the assumptions from finrank_of_isFractionRing!

theorem _root_.Module.rank_eq_zero_of_not_faithfulSMul
    {R S : Type*} [Semiring R] [AddCommMonoid S] [Module R S]
    (h : ¬ FaithfulSMul R S) : Module.rank R S = 0 := by
  contrapose! h
  rw [← Cardinal.one_le_iff_ne_zero, Module.one_le_rank_iff] at h
  obtain ⟨f, hf⟩ := h
  refine ⟨fun {x y} h ↦ hf ?_⟩
  have := f.map_smul x 1
  have := f.map_smul y 1
  simp_all

theorem _root_.Module.finrank_eq_zero_of_not_faithfulSMul
    {R S : Type*} [Semiring R] [AddCommMonoid S] [Module R S]
    (h : ¬ FaithfulSMul R S) : Module.finrank R S = 0 :=
  Module.finrank_eq_zero_of_rank_eq_zero (Module.rank_eq_zero_of_not_faithfulSMul h)

theorem _root_.Transcendental.linearIndependent_pow
    {R S : Type*} [CommRing R] [CommRing S] [Algebra R S] {x : S}
    (h : Transcendental R x) : LinearIndependent R (x ^ ·) := by
  sorry

theorem _root_.Module.finrank_eq_zero_of_not_isAlgebraic
    {R S : Type*} [CommRing R] [Nontrivial R] [CommRing S] [Algebra R S]
    (h : ¬ Algebra.IsAlgebraic R S) : Module.finrank R S = 0 := by
  rw [Algebra.isAlgebraic_def, not_forall] at h
  obtain ⟨x, hx : Transcendental R x⟩ := h
  exact Cardinal.toNat_apply_of_aleph0_le hx.linearIndependent_pow.aleph0_le_rank

theorem _root_.Algebra.IsAlgebraic.finrank_of_isFractionRing' (R R' S S' : Type*)
    [CommRing R] [CommRing R'] [CommRing S] [CommRing S']
    [Algebra R S] [Algebra R R'] [Algebra S S'] [Algebra R' S'] [Algebra R S']
    [IsScalarTower R R' S'] [IsScalarTower R S S']
    [IsFractionRing R R'] [IsFractionRing S S']
    [NoZeroDivisors S] : Module.finrank R' S' = Module.finrank R S := by
  rcases iff_iff_and_or_not_and_not.mp (IsFractionRing.nontrivial_iff_nontrivial R R')
    with ⟨_, _⟩ | ⟨_, _⟩; swap
  · rw [not_nontrivial_iff_subsingleton] at *
    simp
  by_cases hfs : FaithfulSMul R S; swap
  · rw [Module.finrank_eq_zero_of_not_faithfulSMul hfs]
    replace hfs : ¬ FaithfulSMul R' S' := by
      contrapose! hfs
      rw [faithfulSMul_iff_algebraMap_injective] at *
      have : Function.Injective (algebraMap S S') := FaithfulSMul.algebraMap_injective S S'
      apply Function.Injective.of_comp (f := algebraMap S S')
      rw [← RingHom.coe_comp, ← IsScalarTower.algebraMap_eq,
        IsScalarTower.algebraMap_eq R R' S']
      apply hfs.comp
      exact FaithfulSMul.algebraMap_injective R R'
    rw [Module.finrank_eq_zero_of_not_faithfulSMul hfs]
  · have : NoZeroDivisors R := NoZeroDivisors.of_faithfulSMul R S
    have : NoZeroDivisors R' := IsLocalization.noZeroDivisors R⁰
    by_cases halg : Algebra.IsAlgebraic R S
    · apply finrank_of_isFractionRing
    · rw [Module.finrank_eq_zero_of_not_isAlgebraic halg]
      replace halg : ¬ Algebra.IsAlgebraic R' S' := by
        contrapose! halg
        have : Algebra.IsAlgebraic R R' := IsLocalization.isAlgebraic R' R⁰
        have : Algebra.IsAlgebraic R S' := Algebra.IsAlgebraic.trans R R' S'
        apply Algebra.IsAlgebraic.tower_bot_of_injective (R := R) (S := S) (A := S')
        exact FaithfulSMul.algebraMap_injective S S'
      rw [Module.finrank_eq_zero_of_not_isAlgebraic halg]

@tb65536 tb65536 Jul 11, 2026

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(although actually there might just be a more direct proof with localizations)

Module.finrank R S * Module.finrank S T = Module.finrank R T := by
have : FaithfulSMul R T := .trans R S T
have : IsDomain R := (FaithfulSMul.algebraMap_injective R T).isDomain
have : IsDomain S := (FaithfulSMul.algebraMap_injective S T).isDomain
have : Module.Finite R T := Module.Finite.trans S T
rw [← finrank_of_isFractionRing R (FractionRing R) S (FractionRing S),
← finrank_of_isFractionRing S (FractionRing S) T (FractionRing T),
← finrank_of_isFractionRing R (FractionRing R) T (FractionRing T),
Module.finrank_mul_finrank (FractionRing R) (FractionRing S) (FractionRing T)]

end Algebra.IsAlgebraic

section Polynomial
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