Theorems · Theorem · algebraic geometry
AlgebraicGeometry.isNoetherian_iff_of_finite_affine_openCover
∀ {X : AlgebraicGeometry.Scheme} {𝒰 : X.OpenCover} [Finite 𝒰.I₀] [∀ (i : 𝒰.I₀), AlgebraicGeometry.IsAffine (𝒰.X i)],
AlgebraicGeometry.IsNoetherian X ↔ ∀ (i : 𝒰.I₀), IsNoetherianRing ↑((𝒰.X i).presheaf.obj (Opposite.op ⊤))A version of isNoetherian_iff_of_finite_iSup_eq_top using Scheme.OpenCover.
- Defined in
- Mathlib.AlgebraicGeometry.Noetherian
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 161 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites25
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CategoryTheory.Functor.objstatement and proof · cited by 19,642
- Top.topstatement and proof · cited by 9,680
- Oppositestatement · cited by 8,081
- TopCat.carrierstatement · cited by 3,184
- Finitestatement and proof · cited by 3,029
- AlgebraicGeometry.Schemestatement and proof · cited by 2,540
- CommRingCatstatement · cited by 2,333
- TopologicalSpace.Opensstatement · cited by 2,040
- AlgebraicGeometry.PresheafedSpace.carrierstatement · cited by 2,020
- AlgebraicGeometry.SheafedSpace.toPresheafedSpacestatement and proof · cited by 1,988
- AlgebraicGeometry.LocallyRingedSpace.toSheafedSpacestatement and proof · cited by 1,892
- AlgebraicGeometry.Scheme.toLocallyRingedSpacestatement and proof · cited by 1,734
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