How to determine the local ring

Question

In general, how does one determine a local ring. And in particular, how would one do it for $O_{A}(A $ \ $ \{(0)\})$, where A is 1-dim affine space in $\mathbb{C}$?

Answer 1

In general, when $X$ is an affine variety and you are interested in $\mathcal O_X(X\setminus Z)$ with $Z=Z(f)$ a hypersurface, then $\mathcal O_X(X\setminus Z)=\mathcal O_X(X)_f$ is the localization of $\mathcal O_X(X)$ at $f$. In your case, $Z=\{ 0\} = Z(x)$, so $\mathcal O_A(A\setminus \{0\})=\mathcal O_A(A)_x = k[x]_x = k[x,x^{-1}].$

However, note that this is not a local ring! It is a localization at $x$, but a local ring is a ring with a unique maximal ideal and $k[x,x^{-1}]$ has more than one maximal ideal.

Note that this is not easily done when $Z$ is not a hypersurface. For example, $\mathcal O_{A^2}(A^2\setminus\{0\})$ is the same as $\mathcal O_{A^2}(A^2)$, namely the polynomial ring in two variables.