Theorems · Theorem · algebraic geometry
AlgebraicGeometry.isLocallyNoetherian_iff_of_iSup_eq_top
∀ {X : AlgebraicGeometry.Scheme} {ι : Sort u_1} {S : ι → ↑X.affineOpens},
⨆ i, ↑(S i) = ⊤ →
(AlgebraicGeometry.IsLocallyNoetherian X ↔ ∀ (i : ι), IsNoetherianRing ↑(X.presheaf.obj (Opposite.op ↑(S i))))A scheme is locally Noetherian if and only if it is covered by affine opens whose sections are Noetherian rings. See [Har77], Proposition II.3.2.
- Defined in
- Mathlib.AlgebraicGeometry.Noetherian
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 160 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites22
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- CategoryTheory.Functor.objstatement · cited by 19,642
- Top.topstatement and proof · cited by 9,680
- Oppositestatement · cited by 8,081
- Set.Elemstatement and proof · cited by 7,166
- TopCat.carrierstatement · cited by 3,184
- AlgebraicGeometry.Schemestatement and proof · cited by 2,540
- iSupstatement and proof · 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 · cited by 1,988
Cited by1
Results whose statement or proof uses this declaration.
- AlgebraicGeometry.isNoetherian_iff_of_finite_iSup_eq_topproof · cited by 0