Theorems · Definition · algebraic geometry

AlgebraicGeometry.morphismRestrictRestrict

{X Y : AlgebraicGeometry.Scheme} →
  (f : X ⟶ Y) →
    (U : Y.Opens) →
      (V : (↑U).Opens) →
        CategoryTheory.Arrow.mk (f ∣_ U ∣_ V) ≅
          CategoryTheory.Arrow.mk (f ∣_ (AlgebraicGeometry.Scheme.Hom.opensFunctor U.ι).obj V)

Restricting a morphism twice is isomorphic to one restriction.

Defined in: Mathlib.AlgebraicGeometry.Restrict
Cited by: 2 results in Mathlib
Foundations: Depth 149 from the axioms · uses propext, Classical.choice, Quot.sound

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Cites27

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
CategoryTheory.Functor.objstatement and proof · cited by 19,642
CategoryTheory.Isostatement · cited by 3,963
TopCat.carrierstatement · cited by 3,184
AlgebraicGeometry.Schemestatement and proof · cited by 2,540
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
AlgebraicGeometry.LocallyRingedSpace.toSheafedSpacestatement · cited by 1,892
AlgebraicGeometry.Scheme.toLocallyRingedSpacestatement · cited by 1,734
AlgebraicGeometry.Scheme.Opensstatement and proof · cited by 1,149

Cited by3

Results whose statement or proof uses this declaration.

AlgebraicGeometry.isClosedImmersion_diagonal_restrict_diagonalCoverDiagonalRangeproof · cited by 1
AlgebraicGeometry.morphismRestrictRestrictBasicOpenproof · cited by 1
AlgebraicGeometry.Scheme.Hom.exists_isIso_morphismRestrict_toNormalizationproof · cited by 1