Theorems · Theorem · commutative algebra

Module.FaithfullyFlat.iff_zero_iff_lTensor_zero

∀ (R : Type u) (M : Type v) [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M],
  Module.FaithfullyFlat R M ↔
    Module.Flat R M ∧
      ∀ {N : Type (max u v)} [inst_3 : AddCommGroup N] [inst_4 : Module R N] {N' : Type (max u v)}
        [inst_5 : AddCommGroup N'] [inst_6 : Module R N'] (f : N →ₗ[R] N'), LinearMap.lTensor M f = 0 ↔ f = 0

An R-module M is faithfully flat iff it is flat and for all linear maps f, the map id ⊗ f = 0, if and only if f = 0.

Defined in

Mathlib.RingTheory.Flat.FaithfullyFlat.Basic

Cited by

1 results in Mathlib

Foundations

Depth 108 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModule