Theorems · Theorem · algebraic geometry

AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.of_stalk_iso

∀ {X Y : AlgebraicGeometry.LocallyRingedSpace} (f : X ⟶ Y),
  Topology.IsOpenEmbedding ⇑(CategoryTheory.ConcreteCategory.hom f.base) →
    ∀
      [stalk_iso :
        ∀ (x : ↑↑X.toPresheafedSpace), CategoryTheory.IsIso (AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap f x)],
      AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion f

Suppose X Y : SheafedSpace C, where C is a concrete category, whose forgetful functor reflects isomorphisms, preserves limits and filtered colimits. Then a morphism X ⟶ Y that is a topological open embedding is an open immersion iff every stalk map is an iso.

Defined in

Mathlib.Geometry.RingedSpace.OpenImmersion

Cited by

1 results in Mathlib

Foundations

Depth 95 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.IsIso