Theorems · Theorem · algebraic geometry

AlgebraicGeometry.Proj.localRingHom_comp_stalkIso_apply

∀ {A B σ τ : Type u} [inst : CommRing A] [inst_1 : SetLike σ A] [inst_2 : AddSubgroupClass σ A] [inst_3 : CommRing B]
  [inst_4 : SetLike τ B] [inst_5 : AddSubgroupClass τ B] {𝒜 : ℕ → σ} {ℬ : ℕ → τ} [inst_6 : GradedRing 𝒜]
  [inst_7 : GradedRing ℬ] (f : 𝒜 →+*ᵍ ℬ)
  (hf : HomogeneousIdeal.irrelevant ℬ ≤ HomogeneousIdeal.map f (HomogeneousIdeal.irrelevant 𝒜))
  (p : ProjectiveSpectrum ℬ)
  (x : ↑((AlgebraicGeometry.Proj 𝒜).presheaf.stalk ((AlgebraicGeometry.ProjectiveSpectrum.comap f hf) p))),
  (CategoryTheory.ConcreteCategory.hom (AlgebraicGeometry.Proj.stalkIso ℬ p).inv)
      ((HomogeneousLocalization.localRingHom f
          ((AlgebraicGeometry.ProjectiveSpectrum.comap f hf) p).asHomogeneousIdeal.toIdeal p.asHomogeneousIdeal.toIdeal
          ⋯)
        ((CategoryTheory.ConcreteCategory.hom
            (AlgebraicGeometry.Proj.stalkIso 𝒜 ((AlgebraicGeometry.ProjectiveSpectrum.comap f hf) p)).hom)
          x)) =
    (CategoryTheory.ConcreteCategory.hom
        (AlgebraicGeometry.PresheafedSpace.Hom.stalkMap (AlgebraicGeometry.Proj.sheafedSpaceMap f hf).hom p))
      x

Defined in

Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Functor

Cited by

0 results in Mathlib

Foundations

Depth 153 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingSetLikeAddSubgroupClassCommRingSetLikeAddSubgroupClassGradedRingGradedRing