isReduced_of_isIntegral — Mathlib

English

If X is integral, then the ring of global sections Γ(X, ⊤) is an integral domain.

Русский

Если X интегральна, то Γ(X, ⊤) является доменной шпаргалкой.

LaTeX

$$$\text{IsIntegral}(X) \Rightarrow \text{Domain}(\Gamma(X, \top))$$$

Lean4

instance (priority := 900) isReduced_of_isIntegral [IsIntegral X] : IsReduced X :=
  by
  constructor
  intro U
  rcases U.1.eq_empty_or_nonempty with h | h
  · have : U = ⊥ := SetLike.ext' h
    haveI : Subsingleton Γ(X, U) := CommRingCat.subsingleton_of_isTerminal (X.sheaf.isTerminalOfEqEmpty this)
    infer_instance
  · haveI : Nonempty U := by simpa
    infer_instance

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