isEmbedding_comap_of_surjective — Mathlib

English

If f is a bijective ring homomorphism, then comap f is a homeomorphism between prime spectra.

Русский

Если f—биективный гомоморфизм колец, то comap f является гомоморфизмом по топологиям спектров.

LaTeX

$$IsHomeomorph (comap f).$$

Lean4

/-- The embedding has closed range if the domain (and therefore the codomain) is a ring,
  see `PrimeSpectrum.isClosedEmbedding_comap_of_surjective`.
  On the other hand, `comap (Nat.castRingHom (ZMod 2))` does not have closed range. -/
theorem isEmbedding_comap_of_surjective (hf : Surjective f) : IsEmbedding (comap f) :=
  (isEmbedding_iff _).2 ⟨comap_isInducing_of_surjective _ _ hf, comap_injective_of_surjective f hf⟩

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