Theorems · Theorem · commutative algebra

injective_of_isLocalization_of_span_eq_top

∀ {R : Type u_1} {S : Type u_2} [inst : CommSemiring R] [inst_1 : CommSemiring S] {s : Set R},
  Ideal.span s = ⊤ →
    ∀ (Rᵣ : ↑s → Type u_3) [inst_2 : (r : ↑s) → CommSemiring (Rᵣ r)] [inst_3 : (r : ↑s) → Algebra R (Rᵣ r)]
      (Sᵣ : ↑s → Type u_4) [inst_4 : (r : ↑s) → CommSemiring (Sᵣ r)] [inst_5 : (r : ↑s) → Algebra S (Sᵣ r)]
      (f : R →+* S) [inst_6 : ∀ (r : ↑s), IsLocalization.Away (↑r) (Rᵣ r)]
      [inst_7 : ∀ (r : ↑s), IsLocalization.Away (f ↑r) (Sᵣ r)],
      (∀ (r : ↑s), Function.Injective ⇑(IsLocalization.Away.map (Rᵣ r) (Sᵣ r) f ↑r)) → Function.Injective ⇑f

Defined in

Mathlib.RingTheory.LocalProperties.Exactness

Cited by

1 results in Mathlib

Foundations

Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringCommSemiringCommSemiringAlgebraCommSemiringAlgebraIsLocalization.AwayIsLocalization.Away