In algebraic geometry's books, there are many propositions like "let $f : X \to Y$ be a morphism of locally finite presentation, then such-and-such...", but every time I face on such propositions, the question comes up to my mind.
I'm a beginner in studying algebraic geometry, but I feel that almost all schemes in practice are noetherian, like that almost all rings are noetherian. And these propositions becomes so clear if schemes are noetherian. So I want to restrict the situations to noetherian schemes, if it will not have any problems.
So the question is: are there non-noetherian schemes in practice?
p.s. I want to study arithmetic geometry, so if there is, please give me any examples of this field.