Theorems · Theorem · commutative algebra

HomogeneousLocalization.Away.isLocalization_mul

∀ {A : Type u_2} {σ : Type u_3} [inst : CommRing A] [inst_1 : SetLike σ A] [inst_2 : AddSubgroupClass σ A] {𝒜 : ℕ → σ}
  [inst_3 : GradedRing 𝒜] {e d : ℕ} {f : A} (hf : f ∈ 𝒜 d) {g : A} (hg : g ∈ 𝒜 e) {x : A} (hx : x = f * g),
  d ≠ 0 → IsLocalization.Away (HomogeneousLocalization.Away.isLocalizationElem hf hg) (HomogeneousLocalization.Away 𝒜 x)

Let t := g ^ d / f ^ e, then A_{(fg)} = A_{(f)}[1/t].

Defined in

Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Cited by

2 results in Mathlib

Foundations

Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingSetLikeAddSubgroupClassGradedRing