noetherianSpace_of_isAffine — Mathlib

English

An affine open inside a Noetherian scheme has Noetherian underlying space when Γ(X,U) is Noetherian.

Русский

Афинное открытое внутри Ноetherian схемы имеет Ноetherian основанное пространство, когда Γ(X,U) — Ноetherian.

LaTeX

$$$[\text{IsAffineOpen}(U)] \land [\text{IsNoetherianRing}(\Gamma(X,U))] \Rightarrow \text{NoetherianSpace}(U).$$$

Lean4

theorem noetherianSpace_of_isAffine [IsAffine X] [IsNoetherianRing Γ(X, ⊤)] : NoetherianSpace X :=
  (noetherianSpace_iff_of_homeomorph X.isoSpec.inv.homeomorph).mp inferInstance

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