Theorems · Definition · algebraic geometry

AlgebraicGeometry.openToLocalization

{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] →
              (U : TopologicalSpace.Opens ↑(ProjectiveSpectrum.top 𝒜)) →
                (x : ↑(ProjectiveSpectrum.top 𝒜)) →
                  x ∈ U →
                    ((AlgebraicGeometry.ProjectiveSpectrum.Proj.structureSheaf 𝒜).obj.obj (Opposite.op U) ⟶
                      CommRingCat.of (HomogeneousLocalization.AtPrime 𝒜 x.asHomogeneousIdeal.toIdeal))

The ring homomorphism that takes a section of the structure sheaf of Proj on the open set U, implemented as a subtype of dependent functions to localizations at homogeneous prime ideals, and evaluates the section on the point corresponding to a given homogeneous prime ideal.

Defined in

Mathlib.AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf

Cited by

1 results in Mathlib

Foundations

Depth 122 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingSetLikeAddSubgroupClassGradedRing