Theorems · Definition · algebraic geometry

AlgebraicGeometry.projIsoSpecTopComponent

{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 ∈ 𝒜 m →
                    0 < m →
                      (↑((AlgebraicGeometry.Proj.toLocallyRingedSpace 𝒜).restrict ⋯).toPresheafedSpace ≅
                        ↑(AlgebraicGeometry.Spec.locallyRingedSpaceObj
                              (CommRingCat.of (HomogeneousLocalization.Away 𝒜 f))).toPresheafedSpace)

The homeomorphism Proj|D(f) ≅ Spec A⁰_f defined by - φ : Proj|D(f) ⟶ Spec A⁰_f by sending x to A⁰_f ∩ span {g / 1 | g ∈ x} - ψ : Spec A⁰_f ⟶ Proj|D(f) by sending q to {a | aᵢᵐ/fⁱ ∈ q}.

Defined in

Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme

Cited by

1 results in Mathlib

Foundations

Depth 144 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingSetLikeAddSubgroupClassGradedRing