Theorems · Theorem · commutative algebra

IsAdicComplete.eq_liftRingHom

∀ {R : Type u_1} {S : Type u_2} [inst : NonAssocSemiring R] [inst_1 : CommRing S] (I : Ideal S)
  [inst_2 : IsAdicComplete I S] (f : (n : ℕ) → R →+* S ⧸ I ^ n)
  (hf : ∀ {m n : ℕ} (hle : m ≤ n), (Ideal.Quotient.factorPow I hle).comp (f n) = f m) (F : R →+* S),
  (∀ (n : ℕ), (Ideal.Quotient.mk (I ^ n)).comp F = f n) → F = IsAdicComplete.liftRingHom I f ⋯

Uniqueness of the lift. Given a compatible family of linear maps f n : R →ₗ[R] S ⧸ (I ^ n). If F : R →+* S makes the following diagram commute `` R | \ F| \ f n | \ v v S --> S ⧸ (I ^ n) ` Then it is the map IsAdicComplete.lift`.

Defined in

Mathlib.RingTheory.AdicCompletion.RingHom

Cited by

0 results in Mathlib

Foundations

Depth 111 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NonAssocSemiringCommRingIsAdicComplete