Non-singularity of an algebraic variety can be characterised in intrinsic terms by the fact that all local rings are regular local rings.


By a theorem of Serre, any localization of a regular local ring at a prime ideal is again a regular local ring.


If ones proves that the local ring at any non-closed point is a localization of a local ring at a closed point, by the previous theorem it suffices to check non-singularity at closed points.


I am confused as to how to prove the former statement.



Local rings of non-closed points
Since problem is local assume scheme is $Spec(R)$.Nonclosed point is prime ideal $P$ and can find maximal ideal $M \supset P$. Then $R\_P=(R\_M)\_{PR\_M}$ and use Serre theorem and hypothese that 
$R\_M$ is regular.


