Theorems · Definition · algebraic geometry

AlgebraicGeometry.Proj.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)) →
                                    V.carrier ⊆ ⇑(AlgebraicGeometry.ProjectiveSpectrum.comap f hf) ⁻¹' U.carrier →
                                      ((x : ↥U) →
                                          HomogeneousLocalization.AtPrime 𝒜 (↑x).asHomogeneousIdeal.toSubmodule) →
                                        (y : ↥V) → HomogeneousLocalization.AtPrime ℬ (↑y).asHomogeneousIdeal.toSubmodule

The underlying function of Proj ℬ ⟶ Proj 𝒜 on the level of structure sheaves.

Defined in

Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Functor

Cited by

3 results in Mathlib

Foundations

Depth 113 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingSetLikeAddSubgroupClassCommRingSetLikeAddSubgroupClassGradedRingGradedRing