# Confused about localization for affine varieties and affine schemes?

I have some basic familiarity and intuition with algebraic varieties and am wanting to start on schemes. I am looking at section 2.2 of Hartshorne, in particular the definition of the sheaf of rings of $\text{spec}A$. For a ring $A$, this consists of functions, \begin{equation} \label{regulartionfunction} s: U \longrightarrow \bigsqcup_{\mathfrak{p} \in U}A_{p} \end{equation} satisfying certain local conditions. I am trying to connect this back to the case of affine varieties where the maps are from an open set $U$ into the ground field, say $k$. Suppose $A$ is some polynomial ring, for simplicity start with $\mathbb{C}[x]$. The points of an affine variety correspond to the maximal ideals $\mathfrak{m}$, which look like $\left< x-a \right>$. My reasoning (which I think has some kind of problem), was as follows:

The prime ideals of $\mathbb{C}[x]_{\mathfrak{m}}$ are in one-one correspondence with the prime ideals of $\mathbb{C}[x]$ contained in $\mathfrak{m}$. Since the prime ideals of $\mathbb{C}[x]$ are precisely the maximal ones, then $\mathbb{C}[x]_{\mathfrak{m}}$ has precisely one maximal ideal, which is $(0)$, hence making it a field. If that is correct, then bringing this in line with above definition for schemes in the case that $A = \mathbb{C}[x]$ would result in $s$ being a map from $U$ onto a field.

However, this reasoning seems to break down for polynomial rings in more than one variable. Also, I am not even convinced my above reasoning is correct, since I only seemed to use the fact that $\mathbb{C}[x]$ was a PID, and am not sure all localizations of PIDs are fields.

So if anyone could shed some light on how the map $s$ reduces to the regular functions we are familiar with for affine varieties, and also sure up my reasoning about localizations of polynomial rings by maximal ideals I would be very appreciative.

Thanks

Luke

• Apologies, the undefined references are supposed to refer to the only equation environment I've defined. – Luke Nov 26 '16 at 5:55
• – vidyarthi Nov 26 '16 at 7:05

Your description of the ideal structure of $\mathbb{C}[x]_\mathfrak{m}$ is incorrect. $\mathbb{C}[x]_\mathfrak{m}$ is a local ring which is not a field- it contains the ideal $\mathfrak{m}$ which is different from the zero ideal and not the whole ring. (Exercise: verify this with $\mathbb{C}[x]_{(x)}$, where every element not divisible by $x$ is a unit- show that $1\notin(x)$ and $x\notin(0)$.)