isOpenMap_comap_C — Mathlib

English

Noetherian Artinian case implies discrete topology on Spec via the Krull-dimension criterion.

Русский

В случае Artinian Noetherian спектр имеет дискретную топологию по критерию размерности Крулля.

LaTeX

$$$\\text{DiscreteTopology}(\\mathrm{PrimeSpectrum} R)$$$

Lean4

theorem isOpenMap_comap_C : IsOpenMap (comap (R := R) C) :=
  by
  intro U hU
  obtain ⟨S, hS, rfl⟩ := isTopologicalBasis_basic_opens.open_eq_sUnion hU
  rw [Set.image_sUnion]
  apply isOpen_sUnion
  rintro _ ⟨t, ht, rfl⟩
  obtain ⟨r, rfl⟩ := hS ht
  simp only [image_comap_C_basicOpen]
  exact (isClosed_zeroLocus _).isOpen_compl

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