Theorems · Theorem · algebraic geometry

AlgebraicGeometry.Proj.isLocallyFraction_comapStructureSheafFun

∀ {A : Type u_1} {B : Type u_2} {σ : Type u_4} {τ : Type u_5} [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))
  (U : TopologicalSpace.Opens (ProjectiveSpectrum 𝒜)) (V : TopologicalSpace.Opens (ProjectiveSpectrum ℬ))
  (hUV : V.carrier ⊆ ⇑(AlgebraicGeometry.ProjectiveSpectrum.comap f hf) ⁻¹' U.carrier)
  (s : (x : ↥U) → HomogeneousLocalization.AtPrime 𝒜 (↑x).asHomogeneousIdeal.toSubmodule),
  (AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.isLocallyFraction 𝒜).pred s →
    (AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.isLocallyFraction ℬ).pred
      (AlgebraicGeometry.Proj.comapStructureSheafFun f hf U V hUV s)

Defined in

Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Functor

Cited by

0 results in Mathlib

Foundations

Depth 114 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingSetLikeAddSubgroupClassCommRingSetLikeAddSubgroupClassGradedRingGradedRing