Theorems · Theorem · algebraic geometry

PrimeSpectrum.exists_image_comap_of_finite_of_free

∀ {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)
  [Module.Finite R (A ⧸ Ideal.span s)] [Module.Free R (A ⧸ Ideal.span s)],
  ∃ t,
    PrimeSpectrum.comap (algebraMap R A) '' (PrimeSpectrum.zeroLocus s \ PrimeSpectrum.zeroLocus {f}) =
      (PrimeSpectrum.zeroLocus ↑t)ᶜ

Let A be an R-algebra. If A ⧸ I is finite free over R, then the image of Z(I) ∩ D(f) ⊆ Spec S in Spec R is compact open.

Defined in

Mathlib.RingTheory.Spectrum.Prime.Polynomial

Cited by

1 results in Mathlib

Foundations

Depth 129 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraModule.FiniteModule.Free