Theorems · Theorem · commutative algebra

Module.FaithfullyFlat.zero_iff_lTensor_zero

∀ (R : Type u) (M : Type v) [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
  [h : Module.FaithfullyFlat R M] {N : Type u_1} [inst_3 : AddCommGroup N] [inst_4 : Module R N] {N' : Type u_2}
  [inst_5 : AddCommGroup N'] [inst_6 : Module R N'] (f : N →ₗ[R] N'), f = 0 ↔ LinearMap.lTensor M f = 0

If M is a faithfully flat module, then 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

5 results in Mathlib

Foundations

Depth 107 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleModule.FaithfullyFlatAddCommGroupModuleAddCommGroupModule