TITLE: on the generic points of a scheme
QUESTION [8 upvotes]: This question may be a little bit metaphysical:are there any important properties about the generic points on a scheme?Or rather,why do we introduce the concept of generic point?I am not very clear the importance of generic points in the study of schemes.So,I would appreciate it if someone would like to say something on generic points.

REPLY [4 votes]: A quick remark: 
Classical varieties correspond to algebras of finite type over a (algebraically closed) field. Their points correspond to maximal ideals. But there are lots of reasons why one needs to consider more general rings: the concept of specialisation requires fields of transcendance degree $>0$, arithmetic geometry requires algebras over rings of integers, deformation theory requires artinian rings like $k[t]/(t^n)$ etc... Unfortunatly under a general morphism of rings $\phi:A\to B$ the inverse image of a maximal ideal is NOT a maximal ideal. So you do not have a map $\phi^{-1}: Spm(B) \to Spm(A)$ between maximal spectra as you would for algebras of finite type over a field (thanks to the Nullstellensatz). But you do have a map $\phi^{-1}: Spec(B) \to Spec(A)$ between prime spectra. For a classical complex variety this means than you need to add exactly one point for every non maximal prime ideal, i.e. every irreductible closed subset other than the usual (closed) points. And for this small effort you get a theory that is much much more flexible.