Theorems · Theorem · algebraic geometry

AlgebraicGeometry.Scheme.nonempty_of_isLimit

∀ {I : Type u} [inst : CategoryTheory.Category.{u, u} I] (D : CategoryTheory.Functor I AlgebraicGeometry.Scheme)
  (c : CategoryTheory.Limits.Cone D) (hc : CategoryTheory.Limits.IsLimit c) [CategoryTheory.IsCofilteredOrEmpty I]
  [∀ {i j : I} (f : i ⟶ j), AlgebraicGeometry.IsAffineHom (D.map f)] [∀ (i : I), Nonempty ↥(D.obj i)]
  [∀ (i : I), CompactSpace ↥(D.obj i)], Nonempty ↥c.pt

Suppose we have a cofiltered diagram of nonempty quasi-compact schemes, whose transition maps are affine. Then the limit is also nonempty.

Defined in

Mathlib.AlgebraicGeometry.AffineTransitionLimit

Cited by

1 results in Mathlib

Foundations

Depth 176 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.IsCofilteredOrEmptyAlgebraicGeometry.IsAffineHomNonemptyCompactSpace