Theorems · Theorem · commutative algebra

LinearMap.lTensor_injective_of_exact_of_flat

∀ {R : Type u_1} [inst : CommRing R] {M : Type u_3} [inst_1 : AddCommGroup M] [inst_2 : Module R M] {N : Type u_4}
  {P : Type u_5} [inst_3 : AddCommGroup N] [inst_4 : AddCommGroup P] [inst_5 : Module R N] [inst_6 : Module R P]
  [Module.Flat R P] (f : N →ₗ[R] P),
  Function.Surjective ⇑f →
    ∀ (g : M →ₗ[R] N),
      Function.Injective ⇑g →
        Function.Exact ⇑g ⇑f →
          ∀ (A : Type u_6) [inst_8 : AddCommGroup A] [inst_9 : Module R A], Function.Injective ⇑(LinearMap.lTensor A g)

Given a short exact sequence 0 → M → N → P → 0 with P flat, then any A ⊗ M → A ⊗ N is injective.

Defined in

Mathlib.RingTheory.Flat.Equalizer

Cited by

0 results in Mathlib

Foundations

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

Assumes

CommRingAddCommGroupModuleAddCommGroupAddCommGroupModuleModuleModule.FlatAddCommGroupModule