Theorems · Theorem · algebraic geometry
AlgebraicGeometry.isLocallyNoetherian_of_affine_cover
∀ {X : AlgebraicGeometry.Scheme} {ι : Sort u_1} {S : ι → ↑X.affineOpens},
⨆ i, ↑(S i) = ⊤ →
(∀ (i : ι), IsNoetherianRing ↑(X.presheaf.obj (Opposite.op ↑(S i)))) → AlgebraicGeometry.IsLocallyNoetherian XIf a scheme X has a cover by affine opens whose sections are Noetherian rings,
then X is locally Noetherian.
- Defined in
- Mathlib.AlgebraicGeometry.Noetherian
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 159 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites38
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 and proof · cited by 19,642
- Finsetproof · cited by 13,712
- Top.topstatement and proof · cited by 9,680
- SetLike.coeproof · cited by 8,199
- 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
Cited by2
Results whose statement or proof uses this declaration.
- AlgebraicGeometry.isLocallyNoetherian_iff_of_affine_openCoverproof · cited by 2
- AlgebraicGeometry.isLocallyNoetherian_iff_of_iSup_eq_topproof · cited by 1