Unramified morphism of schemes is locally of finite presentation

Question

Let $f:X\to Y$ be a morphism of schemes and $x\in X$.

Is it true that $f$ unramified in $x\Rightarrow f$ is locally of finite presentation? If yes, I don't see how to prove it?

Thank you for your help

Answer 1

An unramified morphism is defined to be locally of finite type, which is equivalent to locally of finite presentation in the case that the target of the morphism is locally noetherian.

Here's a reference for the last fact: Stacks 01TX. The proof is standard: a ring map being of finite presentation is equivalent to finite type plus the kernel being finitely generated. As a ring finite type over a noetherian ring is again noetherian and thus has all ideals finitely generated, the result follows after translating from rings to schemes.