Theorems · Theorem · commutative algebra

TensorProduct.map_injective_of_flat_flat

∀ {R : Type u_1} [inst : CommSemiring R] {M : Type u_2} {N : Type u_3} [inst_1 : AddCommMonoid M]
  [inst_2 : AddCommMonoid N] [inst_3 : Module R M] [inst_4 : Module R N] {P : Type u_4} {Q : Type u_5}
  [inst_5 : AddCommMonoid P] [inst_6 : Module R P] [inst_7 : AddCommMonoid Q] [inst_8 : Module R Q] (f : P →ₗ[R] M)
  (g : Q →ₗ[R] N) [Module.Flat R M] [Module.Flat R Q],
  Function.Injective ⇑f → Function.Injective ⇑g → Function.Injective ⇑(TensorProduct.map f g)

Tensor product of injective maps are injective under some flatness conditions. Also see TensorProduct.map_injective_of_flat_flat' and TensorProduct.map_injective_of_flat_flat_of_isDomain for different flatness conditions.

Defined in

Mathlib.RingTheory.Flat.Basic

Cited by

4 results in Mathlib

Foundations

Depth 103 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringAddCommMonoidAddCommMonoidModuleModuleAddCommMonoidModuleAddCommMonoidModuleModule.FlatModule.Flat