Time for my latest dumb algebraic geometry question as I try to self learn through Hartshorne. I am currently trying to do the exercises in chapter 1 without looking at solutions. One method I have used several times now seems suspect to me and I wanted to run through the general process. Say I have an arbitrary variety $X$. In our case it is general enough to say it's quasi-projective. I want to consider the local ring at a point $P \in X$, say $\mathcal{O}_{P}$, defined as being equivalence classes of pairs $\left <\phi_{U}, U \right>$ in the usual way.
Is it valid for me to take an affine neighbourhood $V$ of $P$ and so all equivalence classes in $\mathcal{O}_{P}$ will have a representation of the form $\left< \phi _{U \cap V}, U \cap V \right>$ and so now my local ring is determined completely by the affine neighbourhood, and so it simply the coordinate ring of $V$ localized at $\mathfrak{m}_{P}$. Is this a valid argument? I am skeptical since then it would seem that every local ring on any arbitrary variety is nothing more than the localization of an affine algebra at a maximal ideal. But I thought this would only work for affine varieties, since that is where you get the nice correspondence with affine algebras. It just seems that if what I have done is correct then local rings don't seem to carry much information, if that makes sense?
I apologize if this is a poorly framed or vague question, it's just that my intuition seems to be doubting me.
Any feedback would be appreciated.
Thanks