Theorems · Theorem · algebraic geometry

AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.preimage_basicOpen

∀ {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] (f : A) (z : HomogeneousLocalization.NumDenSameDeg 𝒜 (Submonoid.powers f)),
  AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.toFun f ⁻¹'
      ↑(PrimeSpectrum.basicOpen (HomogeneousLocalization.mk z)) =
    Subtype.val ⁻¹' ↑(ProjectiveSpectrum.basicOpen 𝒜 ↑z.num)

Defined in: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme
Cited by: 2 results in Mathlib
Foundations: Depth 135 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing SetLike AddSubgroupClass GradedRing

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Cites36

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Setstatement · cited by 53,352
CategoryTheory.Functor.objstatement · cited by 19,642
CommRingstatement and proof · cited by 17,173
SetLike.coestatement · cited by 8,199
Set.preimagestatement · cited by 4,946
TopCat.carrierstatement and proof · cited by 3,184
CommRingCatstatement · cited by 2,333
Set.extproof · cited by 2,266
TopologicalSpace.Opensstatement · cited by 2,040
AlgebraicGeometry.PresheafedSpace.carrierstatement and proof · cited by 2,020
AlgebraicGeometry.SheafedSpace.toPresheafedSpacestatement and proof · cited by 1,988
AlgebraicGeometry.LocallyRingedSpace.toSheafedSpacestatement and proof · cited by 1,892

Cited by2

Results whose statement or proof uses this declaration.

AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_preimage_basicOpenproof · cited by 1
AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec_preimage_basicOpenproof · cited by 0