Theorems · Theorem · algebraic geometry

PrimeSpectrum.mem_image_comap_zeroLocus_sdiff

∀ {R : Type u_2} {A : Type u_1} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A] (f : A) (s : Set A)
  (x : PrimeSpectrum R),
  x ∈ PrimeSpectrum.comap (algebraMap R A) '' (PrimeSpectrum.zeroLocus s \ PrimeSpectrum.zeroLocus {f}) ↔
    ¬IsNilpotent ((algebraMap A (TensorProduct R (A ⧸ Ideal.span s) x.asIdeal.ResidueField)) f)

Let A be an R-algebra. 𝔭 : Spec R is in the image of Z(I) ∩ D(f) ⊆ Spec S if and only if f is not nilpotent on κ(𝔭) ⊗ A ⧸ I.

Defined in

Mathlib.RingTheory.Spectrum.Prime.Polynomial

Cited by

2 results in Mathlib

Foundations

Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebra