Let $k$ be a field. Then $k[[x,y]]$ is a complete local noetherian regular domain of dimension $2$. What are the prime ideals?


I've browsed through the paper "Prime ideals in power series rings" (Jimmy T. Arnold), but it does not give a satisfactory answer. Perhaps there is none. Of course you might think it is more natural to consider only certain prime ideals (for example open/closed ones w.r.t. the adic topology), but I'm interested in the whole spectrum.


A first approximation is the subring $k[[x]] \otimes\_k k[[y]]$. If we know its spectrum, perhaps we can compute the fibers of $\text{Spec } k[[x,y]] \to \text{Spec } k[[x]] \otimes\_k k[[y]]$. Now the spectrum of the tensor product consists of $(x),(y),(x,y)$ and $\text{Spec } k((x)) \otimes\_k k((y))$. The latter one is still very complicated, I think. For example we have the kernel of $k((x)) \otimes\_k k((y)) \to k((x))$. Also, for every $p \in k[[x]]$, we have the prime ideal $(y - p)$.



What are the prime ideals of k[[x,y]]?
The ring $k[[x,y]]$ is a local UFD of dimension 2; so its prime ideals are the zero ideal, the maximal ideal, and all the ideals generated by an irreducible element. They are all closed (all ideals in a noetherian local ring are closed). Classifying them is an extremely complicated business, already when $k = \mathbb C$.


The ring $k[[x]] \otimes\_k k[[y]]$ is truly nasty, it is not even noetherian, and I doubt it would help.


