Theorems · Definition · algebraic geometry

ProjectiveSpectrum.vanishingIdeal

{A : Type u_1} →
  {σ : Type u_2} →
    [inst : CommRing A] →
      [inst_1 : SetLike σ A] →
        [inst_2 : AddSubmonoidClass σ A] →
          {𝒜 : ℕ → σ} → [inst_3 : GradedRing 𝒜] → Set (ProjectiveSpectrum 𝒜) → HomogeneousIdeal 𝒜

The vanishing ideal of a set t of points of the projective spectrum of a commutative ring R is the intersection of all the relevant homogeneous prime ideals in the set t. An element f of A can be thought of as a dependent function on the projective spectrum of 𝒜. At a point x (a homogeneous prime ideal) the function (i.e., element) f takes values in the quotient ring A modulo the prime ideal x. In this manner, vanishingIdeal t is exactly the ideal of A consisting of all "functions" that vanish on all of t.

Defined in

Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology

Cited by

20 results in Mathlib

Foundations

Depth 95 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingSetLikeAddSubmonoidClassGradedRing