1

We say that a morphism of schemes $f : Y \to X$ is finite if $X$ may be covered by affine open sets $\text{Spec}(B)$ such that each $f^{-1}(\text{Spec}(B))$ is affine, say of the form $\text{Spec}(A)$, where $A$ is finitely-generated as a $B$-module.

It seems like the natural analogue would then be to say that a morphism of schemes $f : Y \to X$ is of finite type if $X$ may be covered by affine open sets $\text{Spec}(B)$ such that each $f^{-1}(\text{Spec}(B))$ is affine, say of the form $\text{Spec}(A)$, where $A$ is finitely-generated as a $B$-algebra.

However, the usual definition is that $f : Y \to X$ is locally of finite type if $X$ may be covered by affine open sets $\text{Spec}(B)$ such that each $f^{-1}(\text{Spec}(B))$ is covered by affine open sets $\text{Spec}(A)$ where each $A$ is finitely-generated as a $B$-algebra. Then we say that $f : Y \to X$ is of finite type if each $\text{Spec}(B)$ in the previous definition can be covered by finitely many $\text{Spec}(A)$'s with $A$ finitely-generated as a $B$-algebra.

I therefore have two questions.

Question 1: Why can't we simply impose the condition that each $f^{-1}(\text{Spec}(B))$ is an affine open set in the definition of finite type?

Question 2: Why is it worthwhile to distinguish between morphisms that are "locally of finite type" and "of finite type"? In particular, what is an example of a morphism that is locally of finite type but not of finite type?

user514014
  • 442

1 Answers1

2

For question one, consider the map from $\mathbb{P}^1$ to a point. This is locally of finite type (and of finite type) but there's no way to cover the base with affines such that the pre-images are affine.

For question two, for most applications (to classical algebraic geometry, number theory, etc.) we indeed do not much care about the difference. (EDIT: I had a nonsense counterexample earlier... here is a good one: example of locally finite type not finite type)

As a word of caution, despite the seeming similarity between the two definitions, they're really quite different. "Of finite type" includes any morphism of polynomial rings over a fixed base ring, so basically anything you care about in classical algebraic geometry and more-or-less anything you care about in non-classical algebraic geometry as well.

hunter
  • 29,847