Theorems · Theorem · commutative algebra

Module.exists_basis_of_basis_baseChange

∀ {R : Type u_1} {M : Type u_2} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
  [inst_3 : IsLocalRing R] [Module.FinitePresentation R M] {ι : Type u_5} (v : ι → M),
  LinearIndependent (IsLocalRing.ResidueField R) (⇑((TensorProduct.mk R (IsLocalRing.ResidueField R) M) 1) ∘ v) →
    Submodule.span (IsLocalRing.ResidueField R)
          (Set.range (⇑((TensorProduct.mk R (IsLocalRing.ResidueField R) M) 1) ∘ v)) =
        ⊤ →
      Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype (IsLocalRing.maximalIdeal R))) →
        ∃ b, ∀ (i : ι), b i = v i

If M is of finite presentation over a local ring (R, 𝔪, k) such that 𝔪 ⊗ M → M is injective, then every family of elements that is a k-basis of k ⊗ M is an R-basis of M.

Defined in

Mathlib.RingTheory.LocalRing.Module

Cited by

1 results in Mathlib

Foundations

Depth 120 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleIsLocalRingModule.FinitePresentation