Spectrum of a localized ring

Question

I must find the Spec of the localized ring k[x,y] at the ideal (x,y).

I know that the spec of localized ring k[x] at (x) is {(0),(x)}. Is there any similar attribute?

Answer 1

If $A$ is a commutative ring and $\mathfrak{p}$ is a prime ideal of $A$, then prime ideals of $A_\mathfrak{p}$ correspond to the prime ideals of $A$ which are contained in $\mathfrak{p}$. If $k$ is algebraically closed, the prime ideals of $k[x,y]$ are $(0)$, $(f)$ for some irreducible polynomial $f$ and the maximal ideals $(x-a,y-b)$ for some $a,b \in k$. Of course $(0)$ is contained in any prime ideal. We have $(f) \subseteq (x,y)$ iff $f(0,0)=0$, i.e. $f$ has no constant term, and $(x-a,y-b) \subseteq (x,y)$ iff $a=b=0$. This describes the prime ideals in $k[x,y]_{(x,y)}$. I doubt that it can be simplified. Geometrically, the spectrum consists of all curves in $\mathbb{A}^2$ which go through the origin, as well as the origin itsself, and the generic point.