Lemma of the day

Let h : X –> Y, g : Y –> B be morphisms of algebraic spaces with composition f : X –> B. Let b ∈ |B| and let Spec(k) → B be a morphism in the equivalence class of b. Assume

  1. X → B is a proper morphism,
  2. Y → B is separated and locally of finite type,
  3. one of the following is true:
    1. the image of |X_k| → |Y_k| is finite,
    2. the image of |f|^{−1}({b}) in |Y| is finite and B is decent.

Then there is an open subspace B′ ⊂ B containing b such that X_{B′} → Y_{B′} factors through a closed subspace Z ⊂ Y_{B′} finite over B′. See Lemma Tag 0AEJ.

Slogan: Collapsing a fibre of a proper family forces nearby ones to collapse too.