Theorems · Theorem · algebraic geometry
AlgebraicGeometry.iSup_basicOpen_of_span_eq_top
∀ {X : AlgebraicGeometry.Scheme} (U : X.Opens) (s : Set ↑(X.presheaf.obj (Opposite.op U))),
Ideal.span s = ⊤ → ⨆ i ∈ s, X.basicOpen i = UGiven a spanning set of Γ(X, U), the corresponding basic open sets cover U.
See IsAffineOpen.basicOpen_union_eq_self_iff for the inverse direction for affine open sets.
- Defined in
- Mathlib.AlgebraicGeometry.AffineScheme
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 153 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites53
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Setstatement and proof · cited by 53,352
- CategoryTheory.Functor.objstatement and proof · cited by 19,642
- Top.topstatement and proof · cited by 9,680
- CategoryTheory.Functor.mapproof · cited by 8,698
- SetLike.coeproof · cited by 8,199
- Oppositestatement · cited by 8,081
- Set.Elemproof · cited by 7,166
- Set.imageproof · cited by 5,609
- Idealstatement and proof · cited by 4,748
- CategoryTheory.ConcreteCategory.homproof · cited by 4,022
- TopCat.carrierstatement and proof · cited by 3,184
Cited by2
Results whose statement or proof uses this declaration.
- AlgebraicGeometry.HasRingHomProperty.iff_exists_appLE_locallyproof · cited by 3
- AlgebraicGeometry.isAffine_of_isAffineOpen_basicOpenproof · cited by 2