Theorems · Theorem · algebraic geometry

AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff_of_mem_mul

∀ {A : Type u_1} {σ : Type u_2} [inst : CommRing A] [inst_1 : SetLike σ A] [inst_2 : AddSubgroupClass σ A] {𝒜 : ℕ → σ}
  [inst_3 : GradedRing 𝒜] {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m),
  0 < m →
    ∀
      (q :
        ↑↑(AlgebraicGeometry.Spec.locallyRingedSpaceObj
                (CommRingCat.of (HomogeneousLocalization.Away 𝒜 f))).toPresheafedSpace)
      (a : A) {n : ℕ} (hn : a ∈ 𝒜 (n * m)),
      a ∈ AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier f_deg q ↔
        HomogeneousLocalization.mk { deg := m * n, num := ⟨a, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ } ∈ q.asIdeal

Defined in

Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme

Cited by

1 results in Mathlib

Foundations

Depth 122 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingSetLikeAddSubgroupClassGradedRing