Theorems · Theorem · algebraic geometry

AlgebraicGeometry.ProjectiveSpectrum.Proj.toOpen_toSpec_val_c_app

∀ {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)
  (U :
    (TopologicalSpace.Opens
        ↑↑(AlgebraicGeometry.Spec.locallyRingedSpaceObj
                (CommRingCat.of (HomogeneousLocalization.Away 𝒜 f))).toPresheafedSpace)ᵒᵖ),
  CategoryTheory.CategoryStruct.comp
      (AlgebraicGeometry.Scheme.ΓSpecIso (CommRingCat.of (HomogeneousLocalization.Away 𝒜 f))).inv
      (CategoryTheory.CategoryStruct.comp
        ((AlgebraicGeometry.Spec.locallyRingedSpaceObj (CommRingCat.of (HomogeneousLocalization.Away 𝒜 f))).presheaf.map
          (CategoryTheory.homOfLE ⋯).op)
        ((AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec 𝒜 f).c.app U)) =
    CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToΓ 𝒜 f)
      (((AlgebraicGeometry.Proj.toLocallyRingedSpace 𝒜).restrict ⋯).presheaf.map (CategoryTheory.homOfLE ⋯).op)

Defined in

Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme

Cited by

1 results in Mathlib

Foundations

Depth 136 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingSetLikeAddSubgroupClassGradedRing