TITLE: example of locally finite type not finite type
QUESTION [5 upvotes]: Let $A$ be a ring then, a homomorphism $A\rightarrow A[x_1,\cdots,x_n]$ induces a finite type morphism between spectrums.
I want to find the map which is a locally finite type but not finite type....

REPLY [9 votes]: A morphism is called of finite type if it is quasi-compact and locally of finite type. If $f_i : X_i \to Y$ are morphisms of finite type for $i \in I$, then clearly the induced morphism $f : \coprod_{i \in I} X_i \to Y$ is locally of finite type, but it is usually not of finite type when $I$ is infinite. So, for example, if $A$ is any commutative ring $ \neq 0$, then the canonical morphism $\coprod_{n \geq 0} \mathrm{Spec}(A) \to \mathrm{Spec}(A)$ locally of finite type, but it is not of finite type.