Theorems · Theorem · algebraic geometry
AlgebraicGeometry.Spec.map_preimage
∀ {R S : CommRingCat} (f : AlgebraicGeometry.Spec S ⟶ AlgebraicGeometry.Spec R),
AlgebraicGeometry.Spec.map (AlgebraicGeometry.Spec.preimage f) = f- Cited by
- 11 results in Mathlib
- Foundations
- Depth 142 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Quiver.Homstatement and proof · cited by 32,603
- AlgebraicGeometry.Schemestatement · cited by 2,540
- CommRingCatstatement and proof · cited by 2,333
- AlgebraicGeometry.Specstatement and proof · cited by 626
- AlgebraicGeometry.Spec.mapstatement · cited by 332
- CategoryTheory.Functor.map_preimageproof · cited by 64
- AlgebraicGeometry.Scheme.Specproof · cited by 57
- AlgebraicGeometry.Spec.preimagestatement · cited by 14
Cited by12
Results whose statement or proof uses this declaration.
- AlgebraicGeometry.Spec.map_surjectiveproof · cited by 25
- AlgebraicGeometry.Spec.homEquivproof · cited by 5
- AlgebraicGeometry.Spec.preimage_compproof · cited by 2
- AlgebraicGeometry.IsIntegralHom.iff_universallyClosed_and_isAffineHomproof · cited by 2
- AlgebraicGeometry.SpecMap_residueFieldIsoBase_invproof · cited by 2
- AlgebraicGeometry.Spec.preimage_mapproof · cited by 1
- AlgebraicGeometry.eq_of_SpecMap_comp_eq_of_isAffineOpenproof · cited by 1
- AlgebraicGeometry.pointsPi_surjective_of_isAffineproof · cited by 1
- AlgebraicGeometry.SurjectiveOnStalks.isEmbedding_pullbackproof · cited by 0
- AlgebraicGeometry.Spec.preimage_idproof · cited by 0
- AlgebraicGeometry.Proj.valuativeCriterion_existenceproof · cited by 0