Theorems · Theorem · algebraic geometry

AlgebraicGeometry.isField_of_universallyClosed

∀ {X : AlgebraicGeometry.Scheme} (K : Type u) [inst : Field K] (f : X ⟶ AlgebraicGeometry.Spec (CommRingCat.of K))
  [AlgebraicGeometry.IsIntegral X] [AlgebraicGeometry.UniversallyClosed f], IsField ↑(X.presheaf.obj (Opposite.op ⊤))

If X is an integral scheme that is universally closed over Spec K, then Γ(X, ⊤) is a field.

Defined in

Mathlib.AlgebraicGeometry.Morphisms.Proper

Cited by

1 results in Mathlib

Foundations

Depth 227 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldAlgebraicGeometry.IsIntegralAlgebraicGeometry.UniversallyClosed