“Locally of finite type” is a local property

Previously we showed that morphisms locally of finite type are preserved under base change. We can use this to show that

(*) Given a morphism of schemes p: X \to Y, the preimage of any affine \text{Spec }A \subset Y can be covered by affines such that the corresponding ring maps are of finite type.

Alternatively, if we define a morphism locally of finite type to be one that satisfies (*), then what we are saying is that such a property can be checked on a cover; we can replace “any affine” with “an affine in a cover of affines”.

Let’s try to prove (*). First, we base change to \text{Spec }A. Since the morphism p^{-1}(\text{Spec }A) \to \text{Spec }A is also locally of finite type, we can cover \text{Spec }A by affines \text{Spec }A_i such that their preimages can be covered by the spectra of finitely-generated A_i-algebras B_{ik}. However, we don’t know if these are finitely-generated A-algebras! To fix this, we base change to even smaller affines. Cover \text{Spec }A_i by basic open sets \text{Spec }A[f^{-1}]. This gives us a cover of each \text{Spec }B_{ik} by basic open sets of the form \text{Spec }(A[f^{-1}] \otimes_{A_i} B_{ik}). Since A_i \to B_{ik} is of finite type, A[f^{-1}] \to A[f^{-1}] \otimes_{A_i} B_{ik} is of finite type. Since A \to A[f^{-1}] is clearly of finite type, A \to A[f^{-1}] \otimes_{A_i} B_{ik} is of finite type, giving us the desired cover of p^{-1}(\text{Spec }A). The following diagram may be illustrative (every square is a pullback)