Theorems · Theorem · algebraic geometry

AlgebraicGeometry.Proj.awayToSection_comp_appLE_assoc

∀ {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)) {i : ℕ} {s : A},
  s ∈ 𝒜 i →
    ∀ {Z : CommRingCat}
      (h : (AlgebraicGeometry.Proj ℬ).presheaf.obj (Opposite.op (AlgebraicGeometry.Proj.basicOpen ℬ (f s))) ⟶ Z),
      CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Proj.awayToSection 𝒜 s)
          (CategoryTheory.CategoryStruct.comp
            (AlgebraicGeometry.Scheme.Hom.appLE (AlgebraicGeometry.Proj.map f hf) (AlgebraicGeometry.Proj.basicOpen 𝒜 s)
              (AlgebraicGeometry.Proj.basicOpen ℬ (f s)) ⋯)
            h) =
        CategoryTheory.CategoryStruct.comp (CommRingCat.ofHom (HomogeneousLocalization.Away.map f s))
          (CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Proj.awayToSection ℬ (f s)) h)

Defined in

Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Functor

Cited by

0 results in Mathlib

Foundations

Depth 158 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingSetLikeAddSubgroupClassCommRingSetLikeAddSubgroupClassGradedRingGradedRing