Theorems · Theorem · algebraic geometry

AlgebraicGeometry.Scheme.exists_le_and_germ_injective

∀ (X : AlgebraicGeometry.Scheme) (x : ↥X) [X.IsGermInjectiveAt x] (V : X.Opens),
  x ∈ V →
    ∃ U,
      ∃ (hx : x ∈ U),
        AlgebraicGeometry.IsAffineOpen U ∧
          U ≤ V ∧ Function.Injective ⇑(CategoryTheory.ConcreteCategory.hom (X.presheaf.germ U x hx))

Defined in: Mathlib.AlgebraicGeometry.SpreadingOut
Cited by: 3 results in Mathlib
Foundations: Depth 157 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: AlgebraicGeometry.Scheme.IsGermInjectiveAt

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Cites25

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

DFunLike.coestatement and proof · cited by 62,936
CategoryTheory.Functor.objstatement and proof · cited by 19,642
RingHomstatement · cited by 10,189
Oppositestatement · cited by 8,081
CategoryTheory.ConcreteCategory.homstatement and proof · cited by 4,022
TopCat.carrierstatement and proof · 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 and proof · cited by 2,020
AlgebraicGeometry.SheafedSpace.toPresheafedSpacestatement and proof · cited by 1,988
AlgebraicGeometry.LocallyRingedSpace.toSheafedSpacestatement and proof · cited by 1,892

Cited by3

Results whose statement or proof uses this declaration.

AlgebraicGeometry.exists_lift_of_germInjectiveproof · cited by 1
AlgebraicGeometry.spread_out_unique_of_isGermInjectiveproof · cited by 1
AlgebraicGeometry.isGermInjectiveAt_iff_of_isOpenImmersionproof · cited by 1