Theorems · Definition · algebraic geometry

AlgebraicGeometry.Proj.openCoverOfMapIrrelevantEqTop

{σ : Type u_1} →
  {A : Type u} →
    [inst : CommRing A] →
      [inst_1 : SetLike σ A] →
        [inst_2 : AddSubgroupClass σ A] →
          (𝒜 : ℕ → σ) →
            [inst_3 : GradedRing 𝒜] →
              {X : AlgebraicGeometry.Scheme} →
                (f : A →+* ↑(X.presheaf.obj (Opposite.op ⊤))) →
                  Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤ → X.OpenCover

Given a graded ring A and a map f : A →+* Γ(X, ⊤) such that the image of the irrelevant ideal under f generates the whole ring, the set of D(f(r)) for homogeneous r of positive degree forms an open cover on X.

Defined in

Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Cited by

3 results in Mathlib

Foundations

Depth 142 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingSetLikeAddSubgroupClassGradedRing