Theorems · Theorem · commutative algebra

Module.Free.away_of_finite_of_flat_of_rankAtStalk_constant

∀ {R : Type u_1} [inst : CommRing R] (M : Type u_2) [inst_1 : AddCommGroup M] [inst_2 : Module R M] [Module.Finite R M]
  [Module.Flat R M] (p : Ideal R) [inst_5 : p.IsPrime],
  (∀ (m : Ideal R) [inst_6 : m.IsMaximal],
      Module.rankAtStalk M { asIdeal := m, isPrime := ⋯ } = Module.rankAtStalk M { asIdeal := p, isPrime := ⋯ }) →
    ∃ a ∉ p, Module.Free (Localization.Away a) (LocalizedModule.Away a M)

Let M be a finite flat R-module, p be a prime ideal of R. If rankAtStalk M is constant, then there exists a ∉ p such that M is free after localization away from a.

Defined in

Mathlib.RingTheory.Flat.LocallyFree

Cited by

0 results in Mathlib

Foundations

Depth 125 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleModule.FiniteModule.FlatIdeal.IsPrime