Theorems · Theorem · commutative algebra

injective_of_isLocalized_span

∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
  [inst_3 : AddCommMonoid N] [inst_4 : Module R N] (s : Set R),
  Ideal.span s = ⊤ →
    ∀ (Mₚ : ↑s → Type u_5) [inst_5 : (r : ↑s) → AddCommMonoid (Mₚ r)] [inst_6 : (r : ↑s) → Module R (Mₚ r)]
      (f : (r : ↑s) → M →ₗ[R] Mₚ r) [inst_7 : ∀ (r : ↑s), IsLocalizedModule.Away (↑r) (f r)] (Nₚ : ↑s → Type u_6)
      [inst_8 : (r : ↑s) → AddCommMonoid (Nₚ r)] [inst_9 : (r : ↑s) → Module R (Nₚ r)] (g : (r : ↑s) → N →ₗ[R] Nₚ r)
      [inst_10 : ∀ (r : ↑s), IsLocalizedModule.Away (↑r) (g r)] (F : M →ₗ[R] N),
      (∀ (r : ↑s), Function.Injective ⇑((IsLocalizedModule.map (Submonoid.powers ↑r) (f r) (g r)) F)) →
        Function.Injective ⇑F

Defined in

Mathlib.RingTheory.LocalProperties.Exactness

Cited by

4 results in Mathlib

Foundations

Depth 77 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringAddCommMonoidModuleAddCommMonoidModuleAddCommMonoidModuleIsLocalizedModule.AwayAddCommMonoidModuleIsLocalizedModule.Away