Theorems · Definition · algebraic geometry

ProjectiveSpectrum.zeroLocus

{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 A → Set (ProjectiveSpectrum 𝒜)

The zero locus of a set s of elements of a commutative ring A is the set of all relevant homogeneous prime ideals of the ring that contain the set s. 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, zeroLocus s is exactly the subset of ProjectiveSpectrum 𝒜 where all "functions" in s vanish simultaneously.

Defined in

Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology

Cited by

42 results in Mathlib

Foundations

Depth 98 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingSetLikeAddSubmonoidClassGradedRing