Theorems · Theorem · commutative algebra

Module.bijective_of_surjective_of_rankAtStalk_eq

∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} {N : Type u_3} [inst_1 : AddCommGroup M] [inst_2 : Module R M]
  [Module.Finite R M] [Module.Flat R M] [inst_5 : AddCommGroup N] [inst_6 : Module R N] [Module.Finite R N]
  [Module.Flat R N] {φ : M →ₗ[R] N},
  Function.Surjective ⇑φ →
    (∀ (m : Ideal R) [inst_9 : m.IsMaximal],
        Module.rankAtStalk M { asIdeal := m, isPrime := ⋯ } = Module.rankAtStalk N { asIdeal := m, isPrime := ⋯ }) →
      Function.Bijective ⇑φ

Defined in: Mathlib.RingTheory.Flat.LocallyFree
Cited by: 1 results in Mathlib
Foundations: Depth 119 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing AddCommGroup Module Module.Finite Module.Flat AddCommGroup Module Module.Finite Module.Flat

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Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
Modulestatement and proof · cited by 20,661
RingHom.idstatement and proof · cited by 18,349
CommRingstatement and proof · cited by 17,173
AddCommGroupstatement and proof · cited by 12,871
LinearMapstatement and proof · cited by 10,215
Idealstatement and proof · cited by 4,748
Module.Finitestatement and proof · cited by 1,032
Function.Bijectivestatement · cited by 863
Ideal.IsPrimestatement · cited by 827
Eq.leproof · cited by 605
Ideal.primeComplproof · cited by 462

Cited by1

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Module.Free.away_of_finite_of_flat_of_rankAtStalk_constantproof · cited by 0