Theorems · Theorem · algebraic geometry
AlgebraicGeometry.isLocallyNoetherian_iff_of_affine_openCover
∀ {X : AlgebraicGeometry.Scheme} (𝒰 : X.OpenCover) [∀ (i : 𝒰.I₀), AlgebraicGeometry.IsAffine (𝒰.X i)],
AlgebraicGeometry.IsLocallyNoetherian X ↔ ∀ (i : 𝒰.I₀), IsNoetherianRing ↑((𝒰.X i).presheaf.obj (Opposite.op ⊤))A version of isLocallyNoetherian_iff_of_iSup_eq_top using Scheme.OpenCover.
- Defined in
- Mathlib.AlgebraicGeometry.Noetherian
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 160 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- AlgebraicGeometry.IsAffine
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites36
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
- Set.Elemproof · cited by 7,166
- TopCat.carrierstatement · cited by 3,184
- AlgebraicGeometry.Schemestatement and proof · cited by 2,540
- iSupproof · cited by 2,415
- 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
Cited by2
Results whose statement or proof uses this declaration.
- AlgebraicGeometry.isLocallyNoetherian_iff_openCoverproof · cited by 2
- AlgebraicGeometry.isNoetherian_iff_of_finite_affine_openCoverproof · cited by 0