Mathlib Map

Theorems · Definition · algebraic geometry

AlgebraicGeometry.pointsPi

{ι : Type u} →
  (R : ι → CommRingCat) →
    (X : AlgebraicGeometry.Scheme) →
      (AlgebraicGeometry.Spec (CommRingCat.of ((i : ι) → ↑(R i))) ⟶ X) → (i : ι) → AlgebraicGeometry.Spec (R i) ⟶ X

The canonical map X(Π Rᵢ) ⟶ Π X(Rᵢ). This is injective if X is quasi-separated, surjective if X is affine, or if X is compact and each Rᵢ is local.

Defined in
Mathlib.AlgebraicGeometry.PointsPi
Cited by
3 results in Mathlib
Foundations
Depth 130 from the axioms · uses propext, Classical.choice, Quot.sound
Mathlib docsSourceLoogleLeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

AlgebraicGeometry.pointsPi_surjective_of_isAffine · cited by 1AlgebraicGeometry.pointsP…AlgebraicGeometry.pointsPi_injective · cited by 0AlgebraicGeometry.pointsP…AlgebraicGeometry.pointsPi_surjective · cited by 0AlgebraicGeometry.pointsP…Quiver.Hom · cited by 32603Quiver.HomCategoryTheory.CategoryStruct.comp · cited by 17999CategoryStruct.compAlgebraicGeometry.Scheme · cited by 2540AlgebraicGeometry.SchemeCommRingCat · cited by 2333CommRingCatCommRingCat.carrier · cited by 1096CommRingCat.carrierAlgebraicGeometry.Spec · cited by 626AlgebraicGeometry.SpecAlgebraicGeometry.Spec.map · cited by 332Spec.mapCommRingCat.ofHom · cited by 259CommRingCat.ofHomPi.evalRingHom · cited by 44Pi.evalRingHomAlgebraicGeometry.pointsPiCITED BYCITES

Cites9

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by3

Results whose statement or proof uses this declaration.