Theorems · Theorem · commutative algebra

Module.IsLocalRing.linearCombination_bijective_of_flat

∀ {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.Finite R M] [Module.Flat R M] {ι : Type u} (v : ι → M),
  Function.Bijective
      ⇑(Finsupp.linearCombination (IsLocalRing.ResidueField R)
          (⇑((TensorProduct.mk R (IsLocalRing.ResidueField R) M) 1) ∘ v)) →
    Function.Bijective ⇑(Finsupp.linearCombination R v)

Defined in: Mathlib.RingTheory.LocalRing.Module
Cited by: 1 results in Mathlib
Foundations: Depth 113 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing AddCommGroup Module IsLocalRing 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 · cited by 18,349
CommRingstatement and proof · cited by 17,173
AddCommGroupstatement and proof · cited by 12,871
LinearMapstatement · cited by 10,215
Top.topproof · cited by 9,680
Submoduleproof · cited by 7,192
Finsuppstatement · cited by 5,255
TensorProductstatement · cited by 2,545
TensorProduct.tmulproof · cited by 1,182
Module.Finitestatement and proof · cited by 1,032

Cited by1

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Module.nonempty_basis_of_flat_of_finrank_eqproof · cited by 1