Here are the statement and proof Hartshorne II 8.15. I got the following questions concerning the proof:
Concerning non-singularity of abstract variety, it is required to prove the iff relation for every point of $X$, why the consideration only at the closed points can be generalized?
I cannot see how (8.14A) can be applied explicitly.
Below are the statements of (8.14A) and (Ex. 5.7):
They are likely to be trivial questions for most of you, but it will help a lot for me to understand the context. Thank you very much in advance.
If $x$ is a non-closed point, then there is a closed point $y$ in its closure, and then $\mathcal{O}_{X,x}$ is just the localization of $\mathcal{O}_{X,y}$ at the prime ideal corresponding to $x$ (proof: we can restrict to an affine open set containing $x$, and then this is just the fact that if $A$ is a ring and $P\subseteq Q$ are prime ideals, then $(A_Q)_P\cong A_P$). So, if we know that $\mathcal{O}_{X,y}$ is regular for all closed points $y$, that implies $\mathcal{O}_{X,x}$ is regular for all points $x$.